{"id":3451,"date":"2018-12-11T13:51:58","date_gmt":"2018-12-11T18:51:58","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-applications-with-rational-equations\/"},"modified":"2018-12-11T13:51:58","modified_gmt":"2018-12-11T18:51:58","slug":"solve-applications-with-rational-equations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-applications-with-rational-equations\/","title":{"raw":"Solve Applications with Rational Equations","rendered":"Solve Applications with Rational Equations"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Solve proportions<\/li><li>Solve similar figure applications<\/li><li>Solve uniform motion applications<\/li><li>Solve work applications<\/li><li>Solve direct variation problems<\/li><li>Solve inverse variation problems<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167835419617\" class=\"be-prepared\"><p id=\"fs-id1167832053944\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167834526371\" type=\"1\"><li>Solve: \\(2\\left(n-1\\right)-4=-10.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836399284\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>An express train and a charter bus leave Chicago to travel to Champaign. The express train can make the trip in two hours and the bus takes five hours for the trip. The speed of the express train is 42 miles per hour faster than the speed of the bus. Find the speed of the bus.<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/36adea73-2201-46d3-b9b6-d13ef7df78b2#fs-id1167836309166\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Solve \\(\\frac{1}{3}x+\\frac{1}{4}x=\\frac{5}{6}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167833239741\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834162026\"><h3 data-type=\"title\">Solve Proportions<\/h3><p>When two rational expressions are equal, the equation relating them is called a <span data-type=\"term\">proportion<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167835165941\"><div data-type=\"title\">Proportion<\/div><p id=\"fs-id1167826873771\">A <strong data-effect=\"bold\">proportion<\/strong> is an equation of the form \\(\\frac{a}{b}=\\frac{c}{d},\\) where \\(b\\ne 0,d\\ne 0.\\)<\/p><p id=\"fs-id1167830908992\">The proportion is read \u201c<em data-effect=\"italics\">a<\/em> is to <em data-effect=\"italics\">b<\/em> as <em data-effect=\"italics\">c<\/em> is to <em data-effect=\"italics\">d.<\/em>\u201d<\/p><\/div><p id=\"fs-id1167834137555\">The equation \\(\\frac{1}{2}=\\frac{4}{8}\\) is a proportion because the two fractions are equal. The proportion \\(\\frac{1}{2}=\\frac{4}{8}\\) is read \u201c1 is to 2 as 4 is to 8.\u201d<\/p><p>Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved rational equations. We\u2019ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167831112213\"><div data-type=\"problem\" id=\"fs-id1167835234960\"><p id=\"fs-id1167835339103\">Solve: \\(\\frac{n}{n+14}=\\frac{5}{7}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835330048\"><table class=\"unnumbered unstyled can-break\" summary=\"Solve n divided by the quantity n plus 14 is equal to 5 divided by 7. Notice that the least common denominator is the product of 7 and the quantity n plus 14. Multiply each side of the equation by the least common denominator and remove the common factors on each side. The result is 7 n is equal to 5 times the quantity n plus 14. Simplify the equation on the right side. The result is 7 n is equal to 5 n plus 70. By further simplifying, the equation becomes 2 n is equal to 70. When the equation is solved, the result is n is equal to 35. Check the solution by substituting into the original equation. The result is 35 divided by the sum of 35 and 14 is equal to 5 divided by 7. Determine whether the equation is true by simplifying. Is 35 divided by 49 is equal to 5 divided by 7 a true equation? Show the common factors on the right side. Is the quantity 5 times 7 divided by the quantity 7 times 7 is equal to 5 divided by 7 true? 5 divided by 7 is equal to 5 divided by 7 is a true equation.\" data-label=\"\"><tbody><tr><td><\/td><td><\/td><td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-align=\"left\">Multiply both sides by LCD.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835335325\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-align=\"left\">Remove common factors on each side.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831910788\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-align=\"left\">Solve for <em data-effect=\"italics\">n<\/em>.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835533903\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835188971\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"3\" data-align=\"left\">Check.<\/td><\/tr><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835237211\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831116538\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835345556\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr><td data-align=\"left\">Show common factors.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834422480\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834138284\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167835241089\">Solve the proportion: \\(\\frac{y}{y+55}=\\frac{3}{8}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831880440\"><p id=\"fs-id1167835345046\">\\(y=33\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826996879\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167835367206\">Solve the proportion: \\(\\frac{z}{z-84}=-\\frac{1}{5}.\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167834430708\">\\(z=14\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835309313\">Notice in the last example that when we cleared the fractions by multiplying by the LCD, the result is the same as if we had cross-multiplied.<\/p><span data-type=\"media\" data-alt=\"The equation n divided by the quantity n plus 14 is equal to 5 divided by 7 can be solved by multiplying each side by the least common denominator, 7 times the quantity n plus 14. Multiplying by the least common denominator is a way to clear the fractions. The result is 7 n is equal to 5 times the quantity n plus 14. The equation n divided by the quantity n plus 14 is equal to 5 divided by 7 can also be solved using cross multiplication. Multiply n and 7. Multiply the quantity n plus 14 and 5. The result is also 7 n is equal to 5 times the quantity n plus 14. Cross multiplication also clears fractions.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_018_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equation n divided by the quantity n plus 14 is equal to 5 divided by 7 can be solved by multiplying each side by the least common denominator, 7 times the quantity n plus 14. Multiplying by the least common denominator is a way to clear the fractions. The result is 7 n is equal to 5 times the quantity n plus 14. The equation n divided by the quantity n plus 14 is equal to 5 divided by 7 can also be solved using cross multiplication. Multiply n and 7. Multiply the quantity n plus 14 and 5. The result is also 7 n is equal to 5 times the quantity n plus 14. Cross multiplication also clears fractions.\"><\/span><p id=\"fs-id1167835309022\">For any proportion, \\(\\frac{a}{b}=\\frac{c}{d},\\) we get the same result when we clear the fractions by multiplying by the LCD as when we cross-multiply.<\/p><span data-type=\"media\" data-alt=\"Multiply each side of a proportion a divided by b is equal to c divided by d by the least common denominator, b d, to clear the fractions. The result is a d is equal to b c. Cross multiply to clear the fractions in the proportion a divided by b is equal to c divided by d. The cross products are a times d and b times c. The result is also a d is equal to b c.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_019_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiply each side of a proportion a divided by b is equal to c divided by d by the least common denominator, b d, to clear the fractions. The result is a d is equal to b c. Cross multiply to clear the fractions in the proportion a divided by b is equal to c divided by d. The cross products are a times d and b times c. The result is also a d is equal to b c.\"><\/span><p id=\"fs-id1167835335405\">To solve applications with proportions, we will follow our usual strategy for solving applications But when we set up the proportion, we must make sure to have the units correct\u2014the units in the numerators must match each other and the units in the denominators must also match each other.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835304047\"><div data-type=\"problem\" id=\"fs-id1167835305367\"><p id=\"fs-id1167835343502\">When pediatricians prescribe acetaminophen to children, they prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of the child\u2019s weight. If Zoe weighs 80 pounds, how many milliliters of acetaminophen will her doctor prescribe?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834300223\"><table id=\"fs-id1167835377332\" class=\"unnumbered unstyled\" summary=\"Identify what must be found and a variable to represent it. The number of milliliters of acetaminophen the doctor will prescribe must be found. Let a represent the number of milliliters of acetaminophen. Write a sentence that gives the information to find a. If 5 milliliters are prescribed for every 25 pounds, find how much will be prescribed for 80 pounds. Translate the sentence into a proportion. Be careful of the units, making sure the proportion shows the number of milliliters divided by the number of pounds for the rational expressions on each side. The proportion to find how much acetaminophen is prescribed for 80 pounds is 5 divided by 25 is equal to a divided by 80. Multiply each side of the proportion by the least common denominator, which is 400. The result is 400 times the quantity 5 divided by 25 is equal to 400 times the quantity a divided by 80. Remove the common factors on each side. The common factor on the left side is 25. The common factor on the right side is 8. The equation that result is 16 times 5 is equal to 5 a. Do not simplify on the left side of the equation. Think about the next step. Solve for a by dividing each side by 5. The result is 16 is equal to a. Check that the answer is reasonable. Notice that 80 is about 3 times 25. The amount of acetaminophen should be about 3 times 5. So, 16 milliliters makes sense. To be sure, substitute a is equal to 16 in the original proportion. Is 5 divided by 25 equal to 16 divided by 80? The proportion simplifies to 1 divided by 5 is equal to 1 divided by 5, which is true. The solution, 16 is equal to a, checks. Write a complete sentence. The pediatrician would prescribe 16 milliliters of acetaminophen to Zoe.\" data-label=\"\"><tbody><tr><td data-align=\"left\">Identify what we are asked to find,<div data-type=\"newline\"><br><\/div>and choose a variable to represent it.<\/td><td data-align=\"left\">How many ml of acetaminophen will the<div data-type=\"newline\"><br><\/div>doctor prescribe?<\/td><\/tr><tr><td><\/td><td data-align=\"left\">Let <em data-effect=\"italics\">a<\/em> = ml of acetaminophen.<\/td><\/tr><tr><td data-align=\"left\">Write a sentence that gives the<div data-type=\"newline\"><br><\/div>information to find it.<\/td><td data-align=\"left\">If 5 ml is prescribed for every<div data-type=\"newline\"><br><\/div>25 pounds, how much will be<div data-type=\"newline\"><br><\/div>prescribed for 80 pounds?<\/td><\/tr><tr><td data-align=\"left\">Translate into a proportion\u2014be<div data-type=\"newline\"><br><\/div>careful of the units.<\/td><td data-align=\"left\"><\/td><\/tr><tr><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834403463\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Multiply both sides by the LCD, 400.<\/td><td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Remove common factors on each side.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831880394\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify, but don\u2019t multiply on the left. Notice<div data-type=\"newline\"><br><\/div>what the next step will be.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835226082\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Solve for <em data-effect=\"italics\">a<\/em>.<\/td><td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Check.<div data-type=\"newline\"><br><\/div>Is the answer reasonable?<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><td><\/td><\/tr><tr><td data-align=\"left\">Write a complete sentence.<\/td><td data-align=\"left\">The pediatrician would prescribe 16 ml of<div data-type=\"newline\"><br><\/div>acetaminophen to Zoe.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835478978\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835233288\"><div data-type=\"problem\" id=\"fs-id1167835371012\"><p id=\"fs-id1167830959928\">Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child\u2019s weight. How many milliliters of acetaminophen will the doctor prescribe for Emilia, who weighs 60 pounds?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835346003\"><p>The pediatrician will prescribe 12 ml of acetaminophen to Emilia.<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835336774\"><div data-type=\"problem\" id=\"fs-id1167832031142\"><p id=\"fs-id1167834330002\">For every 1 kilogram (kg) of a child\u2019s weight, pediatricians prescribe 15 milligrams (mg) of a fever reducer. If Isabella weighs 12 kg, how many milligrams of the fever reducer will the pediatrician prescribe?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835335689\"><p id=\"fs-id1167834191169\">The pediatrician will prescribe 180 mg of fever reducer to Isabella.<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\"><h3 data-type=\"title\">Solve similar figure applications<\/h3><p id=\"fs-id1167835237568\">When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with <span data-type=\"term\">similar figures<\/span>. If two figures have exactly the same shape, but different sizes, they are said to be similar. One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides have the same ratio.<\/p><div data-type=\"note\" id=\"fs-id1167834439091\"><div data-type=\"title\">Similar Figures<\/div><p id=\"fs-id1167835368059\">Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides have the same ratio.<\/p><\/div><p>For example, the two triangles in <a href=\"#CNX_IntAlg_Figure_07_05_003\" class=\"autogenerated-content\">(Figure)<\/a> are similar. Each side of \\(\\text{\u0394}ABC\\) is four times the length of the corresponding side of \\(\\text{\u0394}XYZ.\\)<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_07_05_003\"><span data-type=\"media\" id=\"fs-id1167826869931\" data-alt=\"The first figure is triangle A B C with side A B 12 units long, side B C 16 units long, and side A C 20 units long. The second figure is triangle X Y Z with side X Y 3 units long, side Y X 4 units long, and side X Z is 5 units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. 16 divided by 4 is equal to 20 divided 5 is equal to 12 divided by 3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is triangle A B C with side A B 12 units long, side B C 16 units long, and side A C 20 units long. The second figure is triangle X Y Z with side X Y 3 units long, side Y X 4 units long, and side X Z is 5 units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. 16 divided by 4 is equal to 20 divided 5 is equal to 12 divided by 3.\"><\/span><\/div><p>This is summed up in the Property of Similar Triangles.<\/p><div data-type=\"note\"><div data-type=\"title\">Property of Similar Triangles<\/div><p id=\"fs-id1167831928919\">If \\(\\text{\u0394}ABC\\) is similar to \\(\\text{\u0394}XYZ,\\) then their corresponding angle measure are equal and their corresponding sides have the same ratio.<\/p><span data-type=\"media\" data-alt=\"The first figure is triangle A B C with side A B c units long, side B C a units long, and side A C b units long. The second figure is triangle X Y Z with side X Y x units long, side Y Z x units long, and side X Z y units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. a divided by x is equal to b divided by y is equal to c divided by z.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is triangle A B C with side A B c units long, side B C a units long, and side A C b units long. The second figure is triangle X Y Z with side X Y x units long, side Y Z x units long, and side X Z y units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. a divided by x is equal to b divided by y is equal to c divided by z.\"><\/span><\/div><p id=\"fs-id1167834335084\">To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.<\/p><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\"><p>On a map, San Francisco, Las Vegas, and Los Angeles form a triangle. The distance between the cities is measured in inches. The figure on the left below represents the triangle formed by the cities on the map. If the actual distance from Los Angeles to Las Vegas is 270 miles, find the distance from Los Angeles to San Francisco.<\/p><span data-type=\"media\" id=\"fs-id1167832057115\" data-alt=\"The first figure is a triangle labeled \u201cDistances on map.\u201d The triangle is formed by San Francisco, Las Vegas, and Los Angeles. The distance between San Francisco and Las Vegas is 2.1 inches. The distance between Las Vegas and Los Angeles is 1 inch. The distance between Los Angeles and San Francisco is 1.3 inches. The second figure is a triangle labeled \u201cActual Distances.\u201d The triangle is formed by San Francisco, Las Vegas, and Los Angeles. The distance between Las Vegas and Los Angeles is 270 miles. The distance between Los Angeles and San Francisco is labeled x.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is a triangle labeled \u201cDistances on map.\u201d The triangle is formed by San Francisco, Las Vegas, and Los Angeles. The distance between San Francisco and Las Vegas is 2.1 inches. The distance between Las Vegas and Los Angeles is 1 inch. The distance between Los Angeles and San Francisco is 1.3 inches. The second figure is a triangle labeled \u201cActual Distances.\u201d The triangle is formed by San Francisco, Las Vegas, and Los Angeles. The distance between Las Vegas and Los Angeles is 270 miles. The distance between Los Angeles and San Francisco is labeled x.\"><\/span><\/div><div data-type=\"solution\" id=\"fs-id1167834397757\"><p>Since the triangles are similar, the corresponding sides are proportional.<\/p><table class=\"unnumbered unstyled\" summary=\"Read the problem. Draw the figures and label it with the given information. Recall that the figures are shown above. Identify what we are looking for. It is the actual distance from Los Angeles to San Francisco. Name the variable to represent the distance. Let x be equal to the distance from Los Angeles to San Francisco. Translate the problem into an equation. Since the triangles are similar, the corresponding sides are proportional. Make the numerators miles and the numerator inches. The equation is x miles divided by 1.3 inches is equal to 270 miles divided by 1 inch. Solve the equation, 1.3 times the quantity x divided by 1.3 is equal to 1.3 times the quantity 270 divided by 1. The solution is x is equal to 351. Check the solution. On the map, the distance from Los Angeles to San Francisco is more than the distance from Los Angeles to Las Vegas. Since 351 is more than 270, the answer makes sense. Now check x is equal to 351 in the original proportion. Use a calculator. Is 351 miles divided by 1.3 inches equal to 270 miles divided by 1 inch? When simplified, the equation becomes 270 miles divided by 1 inch is equal to 270 miles divided by 1 inch, which is true. Answer the question. The distance from Los Angeles to San Francisco is 351 miles.\" data-label=\"\"><tbody><tr><td data-align=\"left\"><strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figures and label<div data-type=\"newline\"><br><\/div>it with the given information.<\/td><td data-align=\"left\">The figures are shown above.<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/td><td data-align=\"left\">the actual distance from Los Angeles<div data-type=\"newline\"><br><\/div>to San Francisco<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Name<\/strong> the variables.<\/td><td data-align=\"left\">Let <em data-effect=\"italics\">x<\/em> = distance from Los Angeles<div data-type=\"newline\"><br><\/div>to San Francisco.<\/td><\/tr><tr><td data-align=\"left\"><strong data-effect=\"bold\">Translate<\/strong> into an equation.<div data-type=\"newline\"><br><\/div>Since the triangles are similar, the<div data-type=\"newline\"><br><\/div>corresponding sides are proportional. We\u2019ll<div data-type=\"newline\"><br><\/div>make the numerators \u201cmiles\u201d and<div data-type=\"newline\"><br><\/div>the denominators \u201cinches\u201d.<\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835186937\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\"><strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td><td data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835347899\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\"><strong data-effect=\"bold\">Check.<\/strong><div data-type=\"newline\"><br><\/div>On the map, the distance from Los Angeles<div data-type=\"newline\"><br><\/div>to San Francisco is more than<div data-type=\"newline\"><br><\/div>the distance from Los Angeles to<div data-type=\"newline\"><br><\/div>Las Vegas. Since 351 is more than 270<div data-type=\"newline\"><br><\/div>the answer makes sense.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835307534\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Answer<\/strong> the question.<\/td><td data-align=\"left\">The distance from Los Angeles to<div data-type=\"newline\"><br><\/div>San Francisco is 351 miles.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><p id=\"fs-id1167835362231\">On the map, Seattle, Portland, and Boise form a triangle. The distance between the cities is measured in inches. The figure on the left below represents the triangle formed by the cities on the map. The actual distance from Seattle to Boise is 400 miles.<\/p><span data-type=\"media\" id=\"fs-id1167835288353\" data-alt=\"The figure is a triangle formed by Portland, Seattle, and Boise. The distance between Portland and Seattle is 1.5 inches. The distance between Seattle and Boise is 4 inches. The distance between Boise and Portland is 3.5 inches.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a triangle formed by Portland, Seattle, and Boise. The distance between Portland and Seattle is 1.5 inches. The distance between Seattle and Boise is 4 inches. The distance between Boise and Portland is 3.5 inches.\"><\/span><div data-type=\"note\" id=\"fs-id1167835361550\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835283341\"><div data-type=\"problem\"><p id=\"fs-id1167834161709\">Find the actual distance from Seattle to Portland.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835370464\"><p id=\"fs-id1167835305350\">The distance is 150 miles.<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834137694\"><div data-type=\"problem\" id=\"fs-id1167835226378\"><p id=\"fs-id1167834190063\">Find the actual distance from Portland to Boise.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835410965\"><p id=\"fs-id1167835365483\">The distance is 350 miles.<\/p><\/div><\/div><\/div><p id=\"fs-id1167835347011\">We can use similar figures to find heights that we cannot directly measure.<\/p><div data-type=\"example\" id=\"fs-id1167835369388\" class=\"textbox textbox--examples\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167831106741\"><p id=\"fs-id1167835381415\">Tyler is 6 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a tree was 24 feet long. Find the height of the tree.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832053612\"><table id=\"fs-id1167827987927\" class=\"unnumbered unstyled\" summary=\"Read the problem and draw a figure. The figure is a triangle formed by the height of a tree labeled h and the shadow of the tree, which is 24 feet long. Within the triangle is a smaller triangle formed by the height of Tyler, which is 6 feet and Tyler\u2019s shadow, which is 8 feet. We are looking for h, the height of the tree. We will use similar triangles to write an equation. The equation is h divided by 24 is equal to 6 divided by 8. The small triangle is similar to the large triangle. Solve the proportion by multiplying each side by the least common denominator, 24. 24 times the quantity 6 divided by 8 is equal to 24 times the quantity h divided by 24. Simplify the equation. The result is 18 is equal to h. Check the answer. Tyler\u2019s height is less than his shadow\u2019s, so it makes sense that the tree\u2019s height is less than the length of its shadow. Check that h is equal to 18 in the original equation. Is 6 divided by 8 equal to 18 divided by 24? When simplified, the equation becomes 3 divided by 4 is equal to 3 divided by 4, which checks.\" data-label=\"\"><tbody><tr><td data-align=\"left\">Read the problem and draw a figure.<\/td><td><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">We are looking for <em data-effect=\"italics\">h<\/em>, the height of the tree.<\/td><td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">We will use similar triangles to write an equation.<\/td><td><\/td><\/tr><tr><td data-align=\"left\">The small triangle is similar to the large triangle.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834300681\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Solve the proportion.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834422775\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830960983\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td><\/td><\/tr><tr><td data-align=\"left\">Check.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835338242\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835348414\"><div data-type=\"problem\" id=\"fs-id1167835417710\"><p id=\"fs-id1167835181752\">A telephone pole casts a shadow that is 50 feet long. Nearby, an 8 foot tall traffic sign casts a shadow that is 10 feet long. How tall is the telephone pole?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832051470\"><p>The telephone pole is 40 feet tall.<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835280790\"><div data-type=\"problem\"><p id=\"fs-id1167826993906\">A pine tree casts a shadow of 80 feet next to a 30 foot tall building which casts a 40 feet shadow. How tall is the pine tree?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831881158\"><p id=\"fs-id1167830964488\">The pine tree is 60 feet tall.<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832042392\"><h3 data-type=\"title\">Solve Uniform Motion Applications<\/h3><p id=\"fs-id1167835339943\">We have solved uniform motion problems using the formula \\(D=rt\\) in previous chapters. We used a table like the one below to organize the information and lead us to the equation.<\/p><span data-type=\"media\" id=\"fs-id1167834194145\" data-alt=\"This chart has two columns and three rows. The first row is a header row and the second column is labeled \u201cRate times Time is equal to Distance.\u201d There is nothing in the rest of the chart.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This chart has two columns and three rows. The first row is a header row and the second column is labeled \u201cRate times Time is equal to Distance.\u201d There is nothing in the rest of the chart.\"><\/span><p id=\"fs-id1167835335379\">The formula \\(D=rt\\) assumes we know <em data-effect=\"italics\">r<\/em> and <em data-effect=\"italics\">t<\/em> and use them to find <em data-effect=\"italics\">D<\/em>. If we know <em data-effect=\"italics\">D<\/em> and <em data-effect=\"italics\">r<\/em> and need to find <em data-effect=\"italics\">t<\/em>, we would solve the equation for <em data-effect=\"italics\">t<\/em> and get the formula \\(t=\\frac{D}{r}.\\)<\/p><p id=\"fs-id1167834095294\">We have also explained how flying with or against the wind affects the speed of a plane. We will revisit that idea in the next example.<\/p><div data-type=\"example\" id=\"fs-id1167835274969\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835337384\"><div data-type=\"problem\" id=\"fs-id1167834587428\"><p>An airplane can fly 200 miles into a 30 mph headwind in the same amount of time it takes to fly 300 miles with a 30 mph tailwind. What is the speed of the airplane?<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167830865629\">This is a uniform motion situation. A diagram will help us visualize the situation.<\/p><table id=\"fs-id1167831920753\" class=\"unnumbered unstyled can-break\" summary=\"A figure shows the uniform motion of an airplane. An arrow labeled \u201c300 miles with the wind; r plus 30\u201d runs parallel to the wind which is labeled \u201cWind; 30 miles per hour.\u201d A second arrow also runs parallel to the wind, is labeled \u201c200 miles against the wind; r minus 30,\u201d and points opposite the first arrow. We fill in the chart to organize the information. The chart has two columns and three rows. The first row is a header row and the second column is labeled \u201cRate times Time is equal to Distance.\u201d The second column is subdivided into three columns for \u201cRate,\u201d \u201cTime,\u201d and \u201cDistance.\u201d The first column is a header column with the first row labeled \u201cHeadwind\u201d and the second row labeled \u201cTailwind.\u201d We are looking for the speed of the airplane. Let r be equal to the speed of the airplane. When the plane flies with the wind, the wind increases its speed and so the rate is r plus 30. When the plane flies against the wind, the wind decreases its speed and the rate is r minus 30. Write the headwind speed, r minus 30, and the tailwind speed r plus 30 in the \u201cRate\u201d column of the chart. Write the distances 200 and 300 in the \u201cDistance\u201d column of the chart. Since D is equal to r times t, we solve for t and get t is equal to D divided by r. We divided the distance by the rate in each row, and place the expressions 200 divided by the quantity r minus 30 for the headwind and 300 divided by the quantity r plus 30 in the \u201cTime\u201d column. We know the times are equal and so we write our equation, 200 divided by the quantity r minus 30 is equal to 300 divided by the quantity r plus 30. We multiply both sides of the equation by the least common denominator, the quantity r plus 30 times the quantity r minus 30. The result is 200 times the quantity r plus 30 is equal to 300 times the quantity r minus 30. Simplify. The equation becomes 200 r plus 6,000 is equal to 300 r minus 9,000, which becomes 15,000 is equal to 100 r. The result is 150 is equal to r. Check. Is 150 miles per hour a reasonable speed for an airplane? Yes. If the plane is traveling 150 miles per hour and the wind is 30 miles per hour, the tailwind is 150 plus 30, which is equal to 180 miles per hour and 300 divided by 180 is equal to five-thirds hours. Also, the headwind is 150 minus 30, which is equal to 120 miles per hour and 200 divided by 120 is equal to Five-thirds hours. The times are equal, so the plane was traveling 150 miles per hour.\" data-label=\"\"><tbody><tr><td colspan=\"2\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835331004\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-align=\"left\">We fill in the chart to organize the information.<\/td><\/tr><tr><td data-align=\"left\">We are looking for the speed of the airplane.<\/td><td data-align=\"left\">Let <em data-effect=\"italics\">r<\/em> = the speed of the airplane.<\/td><\/tr><tr><td colspan=\"2\" data-align=\"left\">When the plane flies with the wind,<div data-type=\"newline\"><br><\/div>the wind increases its speed and so the rate is <em data-effect=\"italics\">r<\/em> + 30.<\/td><\/tr><tr><td colspan=\"2\" data-align=\"left\">When the plane flies against the wind,<div data-type=\"newline\"><br><\/div>the wind decreases its speed and the rate is <em data-effect=\"italics\">r<\/em> \u2212 30.<\/td><\/tr><tr><td data-align=\"left\">Write in the rates.<div data-type=\"newline\"><br><\/div>Write in the distances.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>Since \\(D=r\u00b7t,\\) we solve for \\(t\\) and get \\(t=\\frac{D}{r}.\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>We divide the distance by the rate in each row, and place the expression in the time column.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831884957\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">We know the times are equal and so we write<div data-type=\"newline\"><br><\/div>our equation.<\/td><td data-align=\"left\">\\(\\phantom{\\rule{7.1em}{0ex}}\\frac{200}{r-30}=\\frac{300}{r+30}\\)<\/td><\/tr><tr><td data-align=\"left\">We multiply both sides by the LCD.<\/td><td data-align=\"left\">\\(\\left(r+30\\right)\\left(r-30\\right)\\left(\\frac{200}{r-30}\\right)=\\left(r+30\\right)\\left(r-30\\right)\\left(\\frac{300}{r+30}\\right)\\)<\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\">\\(\\phantom{\\rule{4.45em}{0ex}}\\left(r+30\\right)\\left(200\\right)=\\left(r-30\\right)300\\)<\/td><\/tr><tr><td><\/td><td data-align=\"left\">\\(\\phantom{\\rule{4.8em}{0ex}}200r+6000=300r-9000\\)<\/td><\/tr><tr><td data-align=\"left\">Solve.<\/td><td data-align=\"left\">\\(\\phantom{\\rule{7.3em}{0ex}}15000=100r\\)<\/td><\/tr><tr><td data-align=\"left\"><strong data-effect=\"bold\">Check.<\/strong><div data-type=\"newline\"><br><\/div>Is 150 mph a reasonable speed for an airplane? Yes.<div data-type=\"newline\"><br><\/div>If the plane is traveling 150 mph and the wind is 30 mph,<div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}\\text{Tailwind}\\hfill &amp; &amp; 150+30=180\\phantom{\\rule{0.2em}{0ex}}\\text{mph}\\hfill &amp; \\frac{300}{180}=\\frac{5}{3}\\phantom{\\rule{0.2em}{0ex}}\\text{hours}\\hfill \\\\ \\text{Headwind}\\hfill &amp; &amp; 150-30=120\\phantom{\\rule{0.2em}{0ex}}\\text{mph}\\hfill &amp; \\frac{200}{120}=\\frac{5}{3}\\phantom{\\rule{0.2em}{0ex}}\\text{hours}\\hfill \\end{array}\\)<\/td><td><\/td><\/tr><tr><td data-align=\"left\">The times are equal, so it checks.<\/td><td data-align=\"left\">The plane was traveling 150 mph.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826927156\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831880816\"><div data-type=\"problem\"><p id=\"fs-id1167831919776\">Link can ride his bike 20 miles into a 3 mph headwind in the same amount of time he can ride 30 miles with a 3 mph tailwind. What is Link\u2019s biking speed?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830699550\"><p id=\"fs-id1167835530027\">Link\u2019s biking speed is 15 mph.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835420100\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167826813961\"><div data-type=\"problem\" id=\"fs-id1167835365574\"><p id=\"fs-id1167831923144\">Danica can sail her boat 5 miles into a 7 mph headwind in the same amount of time she can sail 12 miles with a 7 mph tailwind. What is the speed of Danica\u2019s boat without a wind?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835188060\"><p id=\"fs-id1167835346588\">The speed of Danica\u2019s boat is 17 mph.<\/p><\/div><\/div><\/div><p id=\"fs-id1167831823931\">In the next example, we will know the total time resulting from travelling different distances at different speeds.<\/p><div data-type=\"example\" id=\"fs-id1167834593545\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167830757319\"><div data-type=\"problem\" id=\"fs-id1167834228747\"><p id=\"fs-id1167834146995\">Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835333772\"><p id=\"fs-id1167834367171\">This is a uniform motion situation. A diagram will help us visualize the situation.<\/p><table id=\"fs-id1167835348572\" class=\"unnumbered unstyled can-break\" summary=\"A figure shows the uniform motion of an airplane. An arrow labeled \u201c300 miles with the wind; r plus 30\u201d runs parallel to the wind which is labeled \u201cWind; 30 miles per hour.\u201d A second arrow also runs parallel to the wind, is labeled \u201c200 miles against the wind; r minus 30,\u201d and points opposite the first arrow. We fill in the chart to organize the information. The chart has two columns and three rows. The first row is a header row and the second column is labeled \u201cRate times Time is equal to Distance.\u201d The second column is subdivided into three columns for \u201cRate,\u201d \u201cTime,\u201d and \u201cDistance.\u201d The first column is a header column with the first row labeled \u201cHeadwind\u201d and the second row labeled \u201cTailwind.\u201d We are looking for the speed of the airplane. Let r be equal to the speed of the airplane. When the plane flies with the wind, the wind increases its speed and so the rate is r plus 30. When the plane flies against the wind, the wind decreases its speed and the rate is r minus 30. Write the headwind speed, r minus 30, and the tailwind speed r plus 30 in the \u201cRate\u201d column of the chart. Write the distances 200 and 300 in the \u201cDistance\u201d column of the chart. Since D is equal to r times t, we solve for t and get t is equal to D divided by r. We divided the distance by the rate in each row, and place the expressions 200 divided by the quantity r minus 30 for the headwind and 300 divided by the quantity r plus 30 in the \u201cTime\u201d column. We know the times are equal and so we write our equation, 200 divided by the quantity r minus 30 is equal to 300 divided by the quantity r plus 30. We multiply both sides of the equation by the least common denominator, the quantity r plus 30 times the quantity r minus 30. The result is 200 times the quantity r plus 30 is equal to 300 times the quantity r minus 30. Simplify. The equation becomes 200 r plus 6,000 is equal to 300 r minus 9,000, which becomes 15,000 is equal to 100 r. The result is 150 is equal to r. Check. Is 150 miles per hour a reasonable speed for an airplane? Yes. If the plane is traveling 150 miles per hour and the wind is 30 miles per hour, the tailwind is 150 plus 30, which is equal to 180 miles per hour and 300 divided by 180 is equal to five-thirds hours. Also, the headwind is 150 minus 30, which is equal to 120 miles per hour and 200 divided by 120 is equal to Five-thirds hours. The times are equal, so the plane was traveling 150 miles per hour.\" data-label=\"\"><tbody><tr><td colspan=\"2\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835269037\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-align=\"left\">We fill in the chart to organize the information.<\/td><\/tr><tr><td data-align=\"left\">We are looking for Jazmine\u2019s running speed.<\/td><td data-align=\"left\">Let <em data-effect=\"italics\">r<\/em> = Jazmine\u2019s running speed.<\/td><\/tr><tr><td data-align=\"left\">Her biking speed is 4 miles faster than her<div data-type=\"newline\"><br><\/div>running speed.<\/td><td data-align=\"left\"><em data-effect=\"italics\">r<\/em> + 4 = her biking speed<\/td><\/tr><tr><td data-align=\"left\">The distances are given, enter them into the chart.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>Since \\(D=r\u00b7t,\\) we solve for \\(t\\) and get \\(t=\\frac{D}{r}.\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>We divide the distance by the rate in each row, and place the expression in the time column.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835234049\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Write a word sentence.<\/td><td data-align=\"left\">Her time plus the time biking is 3 hours.<\/td><\/tr><tr><td data-align=\"left\">Translate the sentence to get the equation.<\/td><td data-align=\"center\">\\(\\frac{8}{r}+\\frac{24}{r+4}\\phantom{\\rule{0.5em}{0ex}}=\\phantom{\\rule{0.5em}{0ex}}3\\phantom{\\rule{1.8em}{0ex}}\\)<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\begin{array}{ccc}\\hfill r\\left(r+4\\right)\\left(\\frac{8}{r}+\\frac{24}{r+4}\\right)&amp; =\\hfill &amp; 3\u00b7r\\left(r+4\\right)\\hfill \\\\ \\hfill 8\\left(r+4\\right)+24r&amp; =\\hfill &amp; 3r\\left(r+4\\right)\\hfill \\\\ \\hfill 8r+32+24r&amp; =\\hfill &amp; 3{r}^{2}+12r\\hfill \\\\ \\hfill 32+32r&amp; =\\hfill &amp; 3{r}^{2}+12r\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; 3{r}^{2}-20r-32\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; \\left(3r+4\\right)\\left(r-8\\right)\\hfill \\end{array}\\)<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left(3r+4\\right)=0\\phantom{\\rule{1.2em}{0ex}}\\left(r-8\\right)=0\\)<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(\\overline{)r=-\\frac{4}{3}}\\phantom{\\rule{1.2em}{0ex}}r=8\\)<\/td><\/tr><tr><td data-align=\"left\">Check.<div data-type=\"newline\"><br><\/div>A negative speed does not make sense in this problem,<div data-type=\"newline\"><br><\/div>so \\(r=8\\) is the solution.<div data-type=\"newline\"><br><\/div>Is 8 mph a reasonable running speed? Yes.<div data-type=\"newline\"><br><\/div>If Jazmine\u2019s running rate is 4, then her biking rate,<div data-type=\"newline\"><br><\/div>\\(r+4,\\) which is \\(8+4=12.\\)<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{cccc}\\text{Run}\\phantom{\\rule{0.2em}{0ex}}8\\phantom{\\rule{0.2em}{0ex}}\\text{mph}\\hfill &amp; \\frac{8\\phantom{\\rule{0.2em}{0ex}}\\text{miles}}{8\\phantom{\\rule{0.2em}{0ex}}\\text{mph}}\\hfill &amp; =\\hfill &amp; 1\\phantom{\\rule{0.2em}{0ex}}\\text{hour}\\hfill \\\\ \\text{Bike}\\phantom{\\rule{0.2em}{0ex}}12\\phantom{\\rule{0.2em}{0ex}}\\text{mph}\\hfill &amp; \\frac{24\\phantom{\\rule{0.2em}{0ex}}\\text{miles}}{12\\phantom{\\rule{0.2em}{0ex}}\\text{mph}}\\hfill &amp; =\\hfill &amp; 2\\phantom{\\rule{0.2em}{0ex}}\\text{hours}\\hfill \\end{array}\\)<\/td><td><\/td><\/tr><tr><td data-align=\"left\">\\(\\phantom{\\rule{8.7em}{0ex}}\\)Total 3 hours.<\/td><td data-align=\"left\">Jazmine\u2019s running speed is 8 mph.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835254604\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835377806\"><div data-type=\"problem\" id=\"fs-id1167835300559\"><p id=\"fs-id1167834377240\">Dennis went cross-country skiing for 6 hours on Saturday. He skied 20 mile uphill and then 20 miles back downhill, returning to his starting point. His uphill speed was 5 mph slower than his downhill speed. What was Dennis\u2019 speed going uphill and his speed going downhill?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835225817\"><p id=\"fs-id1167826937400\">Dennis\u2019s uphill speed was 10 mph and his downhill speed was 5 mph.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835420116\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835332653\"><div data-type=\"problem\" id=\"fs-id1167834135092\"><p id=\"fs-id1167834064113\">Joon drove 4 hours to his home, driving 208 miles on the interstate and 40 miles on country roads. If he drove 15 mph faster on the interstate than on the country roads, what was his rate on the country roads?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834462842\"><p id=\"fs-id1167834533387\">Joon\u2019s rate on the country roads is 50 mph.<\/p><\/div><\/div><\/div><p id=\"fs-id1167835339905\">Once again, we will use the uniform motion formula solved for the variable \\(t.\\)<\/p><div data-type=\"example\" id=\"fs-id1167831823993\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834219719\"><div data-type=\"problem\"><p id=\"fs-id1167832057221\">Hamilton rode his bike downhill 12 miles on the river trail from his house to the ocean and then rode uphill to return home. His uphill speed was 8 miles per hour slower than his downhill speed. It took him 2 hours longer to get home than it took him to get to the ocean. Find Hamilton\u2019s downhill speed.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835306833\"><p id=\"fs-id1167835302060\">This is a uniform motion situation. A diagram will help us visualize the situation.<\/p><table id=\"fs-id1167834184998\" class=\"unnumbered unstyled can-break\" summary=\"A figure shows the uniform motion of a bike. An arrow is labeled \u201c12 miles.\u201d A second arrow in the opposite direction is labeled \u201c8 miles per hour slower; 2 hours longer.\u201d We fill in the chart to organize the information. The first row is a header row and the second column is labeled \u201cRate times Time is equal to Distance.\u201d The second column is subdivided into three columns for \u201cRate,\u201d \u201cTime,\u201d and \u201cDistance.\u201d The first column is a header column with the first row labeled \u201cDownhill\u201d and the second row labeled \u201cUphill.\u201d We are looking for Hamilton\u2019s downhill speed. Let h be equal to Hamilton\u2019s downhill speed. His uphill speed is 8 miles per hour slower. Let h minus 8 be equal to Hamilton\u2019s uphill speed. Write the rates, h and the quantity h minus 8, in the \u201cRate\u201d column. The distance is the same in both directions, 12 miles. Since D is equal to r times t, we solve for t and get t is equal to D divided by r. We divide the distance by the rate in each row and place the expressions in the \u201ctime\u201d column. The downhill time is 12 divided by h. The uphill time is 12 divided by the quantity h minus 8. Write a word sentence about the time. He took 2 hours longer uphill than downhill. The uphill time is 2 more than the downhill time. Translate the sentence to get the equation. The equation is 12 divided by the quantity h minus 8 is equal to the sum of 12 divided by h and 2. Solve the equation by multiplying each side by the least common denominator h times the quantity h minus 8. The result is h times the quantity h minus 8 times 12 divided by the quantity h minus 8 is equal to h times the quantity h minus 8 times the sum of 12 divided by h and 2. When completely simplified, the result is 0 is 0 is equal to 2 h squared minus 16 h minus 96. Notice that the factor 2 can be removed on the right side. The equation becomes 0 is equal to 2 times the quantity h squared minus 8 h minus 48. Factoring the right side, the equation becomes 0 is equal to 2 times the quantity h minus 12 times the quantity h plus 4, which means h minus 12 is equal to 0 or h plus 4 is equal to 0. h is equal to 12 can be a solution, but h is equal to negative 4 cannot. Check. Is 12 miles per hour a s reasonable speed for biking downhill. Yes. The downhill speed is 12 miles per hour, so the time is 12 miles divided by 12 miles per hour is equal to 1 hour. The uphill speed is 12 minus 8 is equal to 4 miles per hour, so the time is 12 miles divided by 4 miles per hour is equal to 3 hours. The uphill time is 2 hours more than the downhill time. Hamilton\u2019s downhill speed is 12 miles per hour.\" data-label=\"\"><tbody><tr><td colspan=\"2\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831071365\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_012a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td colspan=\"2\" data-align=\"left\">We fill in the chart to organize the information.<\/td><\/tr><tr><td data-align=\"left\">We are looking for Hamilton\u2019s downhill speed.<\/td><td data-align=\"left\">Let <em data-effect=\"italics\">h<\/em> = Hamilton\u2019s downhill speed.<\/td><\/tr><tr><td data-align=\"left\">His uphill speed is 8 miles per hour slower.<div data-type=\"newline\"><br><\/div>Enter the rates into the chart.<\/td><td data-align=\"left\"><em data-effect=\"italics\">h<\/em> \u2212 8 = Hamilton\u2019s uphill speed<\/td><\/tr><tr><td data-align=\"left\">The distance is the same in both directions.<div data-type=\"newline\"><br><\/div>12 miles.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>Since \\(D=r\u00b7t,\\) we solve for \\(t\\) and get \\(t=\\frac{D}{r}.\\)<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>We divide the distance by the rate in each row, and place the expression in the time column.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835376704\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Write a word sentence about the line.<\/td><td data-align=\"left\">He took 2 hours longer uphill than downhill.<div data-type=\"newline\"><br><\/div>The uphill time is 2 more than the downhill time.<\/td><\/tr><tr><td data-align=\"left\">Translate the sentence to get the equation.<\/td><td data-align=\"left\">\\(\\phantom{\\rule{4.2em}{0ex}}\\frac{12}{h-8}\\phantom{\\rule{0.2em}{0ex}}=\\phantom{\\rule{0.2em}{0ex}}\\frac{12}{h}+2\\)<\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve.<\/td><td data-align=\"left\">\\(\\begin{array}{c}\\begin{array}{ccc}\\hfill h\\left(h-8\\right)\\left(\\frac{12}{h-8}\\right)&amp; =\\hfill &amp; h\\left(h-8\\right)\\left(\\frac{12}{h}+2\\right)\\hfill \\\\ \\hfill 12h&amp; =\\hfill &amp; 12\\left(h-8\\right)+2h\\left(h-8\\right)\\hfill \\\\ \\hfill 12h&amp; =\\hfill &amp; 12h-96+2{h}^{2}-16h\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; 2{h}^{2}-16h-96\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; 2\\left({h}^{2}-8h-48\\right)\\hfill \\\\ \\hfill 0&amp; =\\hfill &amp; 2\\left(h-12\\right)\\left(h+4\\right)\\hfill \\end{array}\\hfill \\\\ \\\\ \\\\ \\phantom{\\rule{3.97em}{0ex}}h-12\\phantom{\\rule{0.2em}{0ex}}=\\phantom{\\rule{0.2em}{0ex}}0\\phantom{\\rule{1em}{0ex}}h+4\\phantom{\\rule{0.2em}{0ex}}=\\phantom{\\rule{0.2em}{0ex}}0\\hfill \\\\ \\phantom{\\rule{6.2em}{0ex}}h\\phantom{\\rule{0.2em}{0ex}}=\\phantom{\\rule{0.2em}{0ex}}12\\phantom{\\rule{2em}{0ex}}\\overline{)h\\phantom{\\rule{0.2em}{0ex}}=\\phantom{\\rule{0.2em}{0ex}}\\text{\u2212}4}\\hfill \\end{array}\\)<\/td><\/tr><tr><td data-align=\"left\">Check.<div data-type=\"newline\"><br><\/div>Is 12 mph a reasonable speed for biking downhill? Yes.<div data-type=\"newline\"><br><\/div>\\(\\begin{array}{ccccc}\\hfill \\text{Downhill}&amp; \\text{12 mph}\\hfill &amp; \\frac{\\text{12 miles}}{\\text{12 mph}}\\hfill &amp; =\\hfill &amp; \\text{1 hour}\\hfill \\\\ \\hfill \\text{Uphill}&amp; 12-8=\\text{4 mph}\\hfill &amp; \\frac{\\text{12 miles}}{\\text{4 mph}}\\hfill &amp; =\\hfill &amp; \\text{3 hours.}\\hfill \\end{array}\\)<\/td><\/tr><tr><td data-align=\"left\">The uphill time is 2 hours more that the downhill time.<\/td><td><\/td><\/tr><tr><td><\/td><td data-align=\"left\">Hamilton\u2019s downhill speed is 12 mph.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167830699110\"><div data-type=\"problem\" id=\"fs-id1167826874280\"><p id=\"fs-id1167832066052\">Kayla rode her bike 75 miles home from college one weekend and then rode the bus back to college. It took her 2 hours less to ride back to college on the bus than it took her to ride home on her bike, and the average speed of the bus was 10 miles per hour faster than Kayla\u2019s biking speed. Find Kayla\u2019s biking speed.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831923485\"><p id=\"fs-id1167835381684\">Kayla\u2019s biking speed was 15 mph.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826880259\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835334970\"><div data-type=\"problem\" id=\"fs-id1167834084978\"><p id=\"fs-id1167830914996\">Victoria jogs 12 miles to the park along a flat trail and then returns by jogging on an 20 mile hilly trail. She jogs 1 mile per hour slower on the hilly trail than on the flat trail, and her return trip takes her two hours longer. Find her rate of jogging on the flat trail.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826782738\"><p id=\"fs-id1167835479367\">Victoria jogged 6 mph on the flat trail.<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835497506\"><h3 data-type=\"title\">Solve Work Applications<\/h3><p id=\"fs-id1167835305013\">The weekly gossip magazine has a big story about the Princess\u2019 baby and the editor wants the magazine to be printed as soon as possible. She has asked the printer to run an extra printing press to get the printing done more quickly. Press #1 takes 6 hours to do the job and Press #2 takes 12 hours to do the job. How long will it take the printer to get the magazine printed with both presses running together?<\/p><p id=\"fs-id1167832060221\">This is a typical \u2018work\u2019 application. There are three quantities involved here\u2014the time it would take each of the two presses to do the job alone and the time it would take for them to do the job together.<\/p><p id=\"fs-id1165927750470\">If Press #1 can complete the job in 6 hours, in one hour it would complete \\(\\frac{1}{6}\\) of the job.<\/p><p id=\"fs-id1165928009731\">If Press #2 can complete the job in 12 hours, in one hour it would complete \\(\\frac{1}{12}\\) of the job.<\/p><p id=\"fs-id1167834138137\">We will let <em data-effect=\"italics\">t<\/em> be the number of hours it would take the presses to print the magazines with both presses running together. So in 1 hour working together they have completed \\(\\frac{1}{t}\\) of the job.<\/p><p id=\"fs-id1167832134011\">We can model this with the word equation and then translate to a rational equation. To find the time it would take the presses to complete the job if they worked together, we solve for \\(t.\\)<\/p><p id=\"fs-id1167834376982\">A chart will help us organize the information. We are looking for how many hours it would take to complete the job with both presses running together.<\/p><table id=\"fs-id1167835569736\" class=\"unnumbered unstyled can-break\" summary=\"A figure shows the uniform motion of a bike. An arrow is labeled \u201c12 miles.\u201d A second arrow in the opposite direction is labeled \u201c8 miles per hour slower; 2 hours longer.\u201d We fill in the chart to organize the information. The first row is a header row and the second column is labeled \u201cRate times Time is equal to Distance.\u201d The second column is subdivided into three columns for \u201cRate,\u201d \u201cTime,\u201d and \u201cDistance.\u201d The first column is a header column with the first row labeled \u201cDownhill\u201d and the second row labeled \u201cUphill.\u201d We are looking for Hamilton\u2019s downhill speed. Let h be equal to Hamilton\u2019s downhill speed. His uphill speed is 8 miles per hour slower. Let h minus 8 be equal to Hamilton\u2019s uphill speed. Write the rates, h and the quantity h minus 8, in the \u201cRate\u201d column. The distance is the same in both directions, 12 miles. Since D is equal to r times t, we solve for t and get t is equal to D divided by r. We divide the distance by the rate in each row and place the expressions in the \u201ctime\u201d column. The downhill time is 12 divided by h. The uphill time is 12 divided by the quantity h minus 8. Write a word sentence about the time. He took 2 hours longer uphill than downhill. The uphill time is 2 more than the downhill time. Translate the sentence to get the equation. The equation is 12 divided by the quantity h minus 8 is equal to the sum of 12 divided by h and 2. Solve the equation by multiplying each side by the least common denominator h times the quantity h minus 8. The result is h times the quantity h minus 8 times 12 divided by the quantity h minus 8 is equal to h times the quantity h minus 8 times the sum of 12 divided by h and 2. When completely simplified, the result is 0 is 0 is equal to 2 h squared minus 16 h minus 96. Notice that the factor 2 can be removed on the right side. The equation becomes 0 is equal to 2 times the quantity h squared minus 8 h minus 48. Factoring the right side, the equation becomes 0 is equal to 2 times the quantity h minus 12 times the quantity h plus 4, which means h minus 12 is equal to 0 or h plus 4 is equal to 0. h is equal to 12 can be a solution, but h is equal to negative 4 cannot. Check. Is 12 miles per hour a s reasonable speed for biking downhill. Yes. The downhill speed is 12 miles per hour, so the time is 12 miles divided by 12 miles per hour is equal to 1 hour. The uphill speed is 12 minus 8 is equal to 4 miles per hour, so the time is 12 miles divided by 4 miles per hour is equal to 3 hours. The uphill time is 2 hours more than the downhill time. Hamilton\u2019s downhill speed is 12 miles per hour.\" data-label=\"\"><tbody><tr><td data-align=\"left\">Let <em data-effect=\"italics\">t<\/em> = the number of hours needed to<div data-type=\"newline\"><br><\/div>complete the job together.<\/td><td><\/td><\/tr><tr><td data-align=\"left\" data-valign=\"top\">Enter the hours per job for Press #1,<div data-type=\"newline\"><br><\/div>Press #2, and when they work together.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>If a job on Press #1 takes 6 hours, then in 1 hour \\(\\frac{1}{6}\\) of the job is completed.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>Similarly find the part of the job completed\/hours for Press #2 and when thet both together.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>Write a word sentence.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832134116\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\">The part completed by Press #1 plus the part completed by Press #2 equals the amount completed together.<\/td><\/tr><tr><td data-align=\"left\">Translate into an equation.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834308133\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Solve.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826798768\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Mutiply by the LCD, 12<em data-effect=\"italics\">t<\/em><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832056805\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_014d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835418136\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_014e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"center\">When both presses are running it<div data-type=\"newline\"><br><\/div>takes 4 hours to do the job.<\/td><\/tr><\/tbody><\/table><p>Keep in mind, it should take less time for two presses to complete a job working together than for either press to do it alone.<\/p><div data-type=\"example\" id=\"fs-id1167835325647\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835287848\"><div data-type=\"problem\" id=\"fs-id1167826996733\"><p>Suppose Pete can paint a room in 10 hours. If he works at a steady pace, in 1 hour he would paint \\(\\frac{1}{10}\\) of the room. If Alicia would take 8 hours to paint the same room, then in 1 hour she would paint \\(\\frac{1}{8}\\) of the room. How long would it take Pete and Alicia to paint the room if they worked together (and didn\u2019t interfere with each other\u2019s progress)?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834432456\"><p id=\"fs-id1167835530396\">This is a \u2018work\u2019 application. A chart will help us organize the information. We are looking for the numbers of hours it will take them to paint the room together.<\/p><p id=\"fs-id1167834282673\">In one hour Pete did \\(\\frac{1}{10}\\) of the job. Alicia did \\(\\frac{1}{8}\\) of the job. And together they did \\(\\frac{1}{t}\\) of the job.<\/p><table id=\"fs-id1167835370917\" class=\"unnumbered unstyled can-break\" summary=\"Let t be equal to the number of hours needed to paint the room together. The chart has three columns and four rows. The first row is a header row and it labels the first column \u201cNumber of hours to complete the job.\u201d and the second column \u201cPart of job completed or hour\u201d. The first column is a header column and it labels the first row \u201cPete\u201d, the second row \u201cAlicia\u201d, and the third row \u201cTogether\u201d. Enter the hours per job for Pete, Alicia, and when they work together. They are 10, 8, and t. In 1 hour of working together, they have completed 1 divided by t of the job. Similarly find the part of the job completed by Pete, and then by Alicia. They are one-tenth of the job and one-eighth of the job. Write a word sentence. The work completed by Pete plus the work completed by Alicia equals the total work completed. When the sentence is translated into an equation, the result is one-tenth plus one-eighth is equal to 1 divided by t. To solve the equation, multiply by the least common denominator, 40 t. The result is 40 t times the sum of one-tenth and one-eighth is equal to 40 t times the quantity 1 divided by t. Distribute 40 t. When simplified, the equation becomes 4 t plus 5 t is equal to 40. The solution is t is equal to 40 divided by 9. We\u2019ll write it as a mixed number so that we can convert it to hours and minutes. The solution is t is equal to 4 and four-ninths hours. Remember 1 hour is equal to 60 minutes. Multiply, and then round to the nearest minute to convert the fraction. The result is t is equal to 4 hours plus four-ninths times 60 minutes. So, t is equal to 4 hours plus 27 minutes. It would take Pete and Alicia about 4 hours and 27 minutes to paint the room.\" data-label=\"\"><tbody><tr><td data-align=\"left\">Let <em data-effect=\"italics\">t<\/em> be the number of hours needed<div data-type=\"newline\"><br><\/div>to paint the room together.<\/td><td><\/td><\/tr><tr><td data-align=\"left\">Enter the hours per job for Pete, Alicia, and when they work together.<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div>In 1 hour working together, they have completed \\(\\frac{1}{t}\\) of the job.<div data-type=\"newline\"><br><\/div>Similarly, find the part of the job<div data-type=\"newline\"><br><\/div>completed\/hour by Pete and then by Alicia.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835390377\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Write a word sentence.<\/td><td data-align=\"left\">The work completed by Pete plus the work<div data-type=\"newline\"><br><\/div>completed by Alicia equals the total<div data-type=\"newline\"><br><\/div>work completed.<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{5em}{0ex}}\\)Work completed by:<div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167835319199\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <span data-type=\"media\" id=\"fs-id1167835300385\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span> <\/td><\/tr><tr><td data-align=\"left\">Multiply by the LCD, 40<em data-effect=\"italics\">t<\/em>.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835284770\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Distribute.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835301933\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify and solve.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835417811\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">We\u2019ll write as a mixed number<div data-type=\"newline\"><br><\/div>so that we can convert it to hours<div data-type=\"newline\"><br><\/div>and minutes.<\/td><td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Remember, 1 hour = 60 minutes.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830704125\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Multiply, and then round to the<div data-type=\"newline\"><br><\/div>nearest minute.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834196502\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"center\">It would take Pete and Alica about<div data-type=\"newline\"><br><\/div>4 hours and 27 minutes to paint the room.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832065583\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831107028\"><div data-type=\"problem\" id=\"fs-id1167831107030\"><p id=\"fs-id1167835511112\">One gardener can mow a golf course in 4 hours, while another gardener can mow the same golf course in 6 hours. How long would it take if the two gardeners worked together to mow the golf course?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835530131\"><p id=\"fs-id1167834131823\">When the two gardeners work together it takes 2 hours and 24 minutes.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831908325\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831959454\"><div data-type=\"problem\" id=\"fs-id1167831959457\"><p id=\"fs-id1167831040305\">Daria can weed the garden in 7 hours, while her mother can do it in 3. How long will it take the two of them working together?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834219754\"><p id=\"fs-id1167834527691\">When Daria and her mother work together it takes 2 hours and 6 minutes.<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1167834196208\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835186400\"><div data-type=\"problem\" id=\"fs-id1167835186402\"><p id=\"fs-id1167831955876\">Ra\u2019shon can clean the house in 7 hours. When his sister helps him it takes 3 hours. How long does it take his sister when she cleans the house alone?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831955879\"><p id=\"fs-id1167835326432\">This is a work problem. A chart will help us organize the information.<\/p><p id=\"fs-id1167835254281\">We are looking for how many hours it would take Ra\u2019shon\u2019s sister to complete the job by herself.<\/p><table id=\"fs-id1167826978190\" class=\"unnumbered unstyled can-break\" summary=\"Let s be equal to the number of hours Ra\u2019shon\u2019s sister takes to clean the house alone. The chart has three columns and four rows. The first row is a header row and it labels the first column \u201cNumber of hours to clean the house\u201d and the second column \u201cPart of job completed or hour\u201d. The first column is a header column and it labels the first row \u201cRa\u2019shon\u201d, the second row \u201cHis sister\u201d, and the third row \u201cTogether\u201d. Enter the hours per job for Ra\u2019shon, his sister, and when they work together. They are 7, s, and 3. If Ras\u2019shon takes 7 hours, then in 1 hour, one-seventh of the job is completed. If Ra\u2019shon\u2019s sister takes s hours, then in 1 hour, 1 divided by s of the job is completed. Write a word sentence. The part completed by Ras\u2019shon plus the part completed by his sister equals the amount completed together. Translate to an equation. One-seventh is equal to 1 divided by s is equal to one-third. Solve the equation. Multiply by the least common denominator, 21 s. The result is 21 s times sum of one-seventh and one divided by s is equal to 21 s times one-third. When simplified, the equation becomes 3 s plus 21 is equal to 7 s. When solved for s, the result is s is equal to 5 and one-fourth hours. There are 60 minutes in 1 hour, so s is equal to 5 hours plus one-fourth times 60 minutes. That means s is equal to 5 hours plus 15 minutes. It would take Ra\u2019Shon\u2019s sister 5 hours and 15 minutes to clean the house alone.\" data-label=\"\"><tbody><tr><td data-align=\"left\">Let <em data-effect=\"italics\">s<\/em> be the number of hours Ra\u2019shon\u2019s<div data-type=\"newline\"><br><\/div>sister takes to clean the house alone.<\/td><td><\/td><\/tr><tr><td data-align=\"left\">Enter the hours per job for Ra\u2019shon, his<div data-type=\"newline\"><br><\/div>sister, and when they work together.<div data-type=\"newline\"><br><\/div>If Ra\u2019shon takes 7 hours, then in 1 hour \\(\\frac{1}{7}\\)<div data-type=\"newline\"><br><\/div>of the job is completed.<div data-type=\"newline\"><br><\/div>If Ra\u2019shon\u2019s sister takes <em data-effect=\"italics\">s<\/em> hours, then in<div data-type=\"newline\"><br><\/div>1 hour \\(\\frac{1}{s}\\) of the job is completed.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835198865\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Write a word sentence.<\/td><td data-align=\"left\">The part completed by Ra\u2019shon plus the part<div data-type=\"newline\"><br><\/div>by his sister equals the amount completed together.<\/td><\/tr><tr><td data-align=\"left\">Translate to an equation.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835280078\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Solve.<\/td><td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Multiply by the LCD, 21<em data-effect=\"italics\">s<\/em>.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835368800\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835509964\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Write as a mixed number to<div data-type=\"newline\"><br><\/div>convert it to hours and minutes.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834191154\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">There are 60 minutes in 1 hour.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835309911\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\">It would take Ra\u2019shon\u2019s sister 5 hours and<div data-type=\"newline\"><br><\/div>15 minutes to clean the house alone.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831025386\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835343869\"><div data-type=\"problem\" id=\"fs-id1167835343871\"><p>Alice can paint a room in 6 hours. If Kristina helps her it takes them 4 hours to paint the room. How long would it take Kristina to paint the room by herself?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835354966\"><p id=\"fs-id1167835340158\">Kristina can paint the room in 12 hours.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835422085\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834257815\"><div data-type=\"problem\" id=\"fs-id1167834196340\"><p id=\"fs-id1167834196342\">Tracy can lay a slab of concrete in 3 hours, with Jordan\u2019s help they can do it in 2 hours. If Jordan works alone, how long will it take?<\/p><\/div><div data-type=\"solution\"><p>It will take Jordan 6 hours.<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835351315\"><h3 data-type=\"title\">Solve Direct Variation Problems<\/h3><p id=\"fs-id1167831106792\">When two quantities are related by a proportion, we say they are <em data-effect=\"italics\">proportional<\/em> to each other. Another way to express this relation is to talk about the <em data-effect=\"italics\">variation<\/em> of the two quantities. We will discuss direct variation and inverse variation in this section.<\/p><p id=\"fs-id1167826880152\">Lindsay gets paid ?15 per hour at her job. If we let <em data-effect=\"italics\">s<\/em> be her salary and <em data-effect=\"italics\">h<\/em> be the number of hours she has worked, we could model this situation with the equation<\/p><div data-type=\"equation\" id=\"fs-id1167830836766\" class=\"unnumbered\" data-label=\"\">\\(s=15h\\)<\/div><p id=\"fs-id1167835344749\">Lindsay\u2019s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay\u2019s salary <em data-effect=\"italics\">varies directly<\/em> with the number of hours she works. Two variables vary directly if one is the product of a constant and the other.<\/p><div data-type=\"note\" id=\"fs-id1167835374810\"><div data-type=\"title\">Direct Variation<\/div><p id=\"fs-id1167835332296\">For any two variables <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>, <em data-effect=\"italics\">y<\/em> varies directly with <em data-effect=\"italics\">x<\/em> if<\/p><div data-type=\"equation\" id=\"fs-id1167835304679\" class=\"unnumbered\" data-label=\"\">\\(y=kx,\\phantom{\\rule{0.2em}{0ex}}\\text{where}\\phantom{\\rule{0.2em}{0ex}}k\\ne 0\\)<\/div><p id=\"fs-id1167832058585\">The constant <em data-effect=\"italics\">k<\/em> is called the constant of variation.<\/p><\/div><p id=\"fs-id1167835229188\">In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y.<\/em> Then we can use that equation to find values of <em data-effect=\"italics\">y<\/em> for other values of <em data-effect=\"italics\">x<\/em>.<\/p><p id=\"fs-id1167834346195\">We\u2019ll list the steps here.<\/p><div data-type=\"note\" id=\"fs-id1167834346198\" class=\"howto\"><div data-type=\"title\">Solve direct variation problems.<\/div><ol id=\"fs-id1167835417621\" type=\"1\" class=\"stepwise\"><li>Write the formula for direct variation.<\/li><li>Substitute the given values for the variables.<\/li><li>Solve for the constant of variation.<\/li><li>Write the equation that relates <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> using the constant of variation.<\/li><\/ol><\/div><p>Now we\u2019ll solve an application of direct variation.<\/p><div data-type=\"example\" id=\"fs-id1167832150888\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167832150890\"><div data-type=\"problem\" id=\"fs-id1167835330703\"><p id=\"fs-id1167835330705\">When Raoul runs on the treadmill at the gym, the number of calories, <em data-effect=\"italics\">c<\/em>, he burns varies directly with the number of minutes, <em data-effect=\"italics\">m<\/em>, he uses the treadmill. He burned 315 calories when he used the treadmill for 18 minutes.<\/p><p id=\"fs-id1167834186254\"><span class=\"token\">\u24d0<\/span> Write the equation that relates <em data-effect=\"italics\">c<\/em> and <em data-effect=\"italics\">m<\/em>. <span class=\"token\">\u24d1<\/span> How many calories would he burn if he ran on the treadmill for 25 minutes?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835354913\"><p id=\"fs-id1167835354915\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167834279347\" class=\"unnumbered unstyled can-break\" summary=\"The number of calories, c, varies directly with the number of minutes, m, on a treadmill, and c is equal to 315 and m is equal to 18. Write the formula for direct variation, y is equal to k times x. We will use c in place of y and m in place of x. So, the equation is c is equal to k times m instead. Substitute the given values for the variables. The result is 315 is equal to k times 18. Solve for the constant of variation by dividing each side of the equation by 18. The result is 17.5 is equal to k. Write the equation that relates c and m. The equation is c is equal to k times m. Substitute the constant of variation. The result is c is equal to 17.5 times m.\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\">The number of calories, <em data-effect=\"italics\">c<\/em>, varies directly with<div data-type=\"newline\"><br><\/div>the number of minutes, <em data-effect=\"italics\">m<\/em>, on the treadmill,<div data-type=\"newline\"><br><\/div>and \\(c=315\\) when \\(m=18\\).<\/td><\/tr><tr><td data-align=\"left\">Write the formula for direct variation.<\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830700596\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">We will use <em data-effect=\"italics\">c<\/em> in place of <em data-effect=\"italics\">y<\/em> and <em data-effect=\"italics\">m<\/em> in place of <em data-effect=\"italics\">x<\/em>.<\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835339267\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Substitute the given values for the variables.<\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831148800\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Solve for the constant of variation.<\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831103299\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167831879987\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Write the equation that relates <em data-effect=\"italics\">c<\/em> and <em data-effect=\"italics\">m<\/em>.<\/td><td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834053655\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Substitute in the constant of variation.<\/td><td data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167834432629\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167832081890\" class=\"unnumbered unstyled can-break\" summary=\"Find c when m is equal to 25. Write the equation that relates c to m. the equation is c is equal to 17.5 m. Substitute the given value for m. The result is c is equal to 17.5 times 25. Simplify the equation. The result is c is equal to 4375. Raoul would burn 437.5 calories if he used the treadmill for 25 minutes.\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\">\\(\\phantom{\\rule{0.05em}{0ex}}\\)Find <em data-effect=\"italics\">c<\/em> when <em data-effect=\"italics\">m<\/em> = 25.<\/td><\/tr><tr><td data-align=\"left\">Write the equation that relates <em data-effect=\"italics\">c<\/em> and <em data-effect=\"italics\">m<\/em>.\\(\\phantom{\\rule{1.4em}{0ex}}\\)<\/td><td data-align=\"center\">\\(\\phantom{\\rule{0.1em}{0ex}}\\)<span data-type=\"media\" id=\"fs-id1167835380271\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_016a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Substitute the given value for <em data-effect=\"italics\">m<\/em>.<\/td><td data-align=\"center\">\\(\\phantom{\\rule{0.1em}{0ex}}\\)<span data-type=\"media\" id=\"fs-id1167834433091\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"center\">\\(\\phantom{\\rule{0.1em}{0ex}}\\)<span data-type=\"media\" id=\"fs-id1167835236743\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\">\\(\\phantom{\\rule{0.1em}{0ex}}\\)Raoul would burn 437.5 calories if<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{0.1em}{0ex}}\\)he used the treadmill for 25 minutes.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835533838\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832054682\"><div data-type=\"problem\" id=\"fs-id1167832054685\"><p id=\"fs-id1167832054687\">The number of calories, <em data-effect=\"italics\">c<\/em>, burned varies directly with the amount of time, <em data-effect=\"italics\">t<\/em>, spent exercising. Arnold burned 312 calories in 65 minutes exercising.<\/p><p id=\"fs-id1167835369286\"><span class=\"token\">\u24d0<\/span> Write the equation that relates <em data-effect=\"italics\">c<\/em> and <em data-effect=\"italics\">t<\/em>. <span class=\"token\">\u24d1<\/span> How many calories would he burn if he exercises for 90 minutes?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835333023\"><p id=\"fs-id1167834133512\"><span class=\"token\">\u24d0<\/span>\\(c=4.8t\\)<span class=\"token\">\u24d1<\/span> He would burn 432 calories.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835362159\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834532325\"><div data-type=\"problem\" id=\"fs-id1167834532327\"><p id=\"fs-id1167834532329\">The distance a moving body travels, <em data-effect=\"italics\">d<\/em>, varies directly with time, <em data-effect=\"italics\">t<\/em>, it moves. A train travels 100 miles in 2 hours<\/p><p id=\"fs-id1167834247063\"><span class=\"token\">\u24d0<\/span> Write the equation that relates <em data-effect=\"italics\">d<\/em> and <em data-effect=\"italics\">t<\/em>. <span class=\"token\">\u24d1<\/span> How many miles would it travel in 5 hours?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835306694\"><p id=\"fs-id1167831116771\"><span class=\"token\">\u24d0<\/span>\\(d=50t\\)<span class=\"token\">\u24d1<\/span> It would travel 250 miles.<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835355996\"><h3 data-type=\"title\">Solve Inverse Variation Problems<\/h3><p>Many applications involve two variable that <em data-effect=\"italics\">vary inversely<\/em>. As one variable increases, the other decreases. The equation that relates them is \\(y=\\frac{k}{x}.\\)<\/p><div data-type=\"note\" id=\"fs-id1167835363556\"><div data-type=\"title\">Inverse Variation<\/div><p id=\"fs-id1167835363561\">For any two variables <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>, <em data-effect=\"italics\">y<\/em> varies inversely with <em data-effect=\"italics\">x<\/em> if<\/p><div data-type=\"equation\" id=\"fs-id1167835514630\" class=\"unnumbered\" data-label=\"\">\\(y=\\frac{k}{x},\\phantom{\\rule{0.2em}{0ex}}\\text{where}\\phantom{\\rule{0.2em}{0ex}}k\\ne 0\\)<\/div><p>The constant <em data-effect=\"italics\">k<\/em> is called the constant of variation.<\/p><\/div><p id=\"fs-id1167835417942\">The word \u2018inverse\u2019 in inverse variation refers to the multiplicative inverse. The multiplicative inverse of <em data-effect=\"italics\">x<\/em> is \\(\\frac{1}{x}.\\)<\/p><p id=\"fs-id1167835320238\">We solve inverse variation problems in the same way we solved direct variation problems. Only the general form of the equation has changed. We will copy the procedure box here and just change \u2018direct\u2019 to \u2018inverse\u2019.<\/p><div data-type=\"note\" id=\"fs-id1167835489429\" class=\"howto\"><div data-type=\"title\">Solve inverse variation problems.<\/div><ol id=\"fs-id1167831880447\" type=\"1\" class=\"stepwise\"><li>Write the formula for inverse variation.<\/li><li>Substitute the given values for the variables.<\/li><li>Solve for the constant of variation.<\/li><li>Write the equation that relates <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> using the constant of variation.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1167835512316\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835512318\"><div data-type=\"problem\" id=\"fs-id1167835512320\"><p id=\"fs-id1167834252390\">The frequency of a guitar string varies inversely with its length. A 26 in.-long string has a frequency of 440 vibrations per second.<\/p><p id=\"fs-id1167834252394\"><span class=\"token\">\u24d0<\/span> Write the equation of variation. <span class=\"token\">\u24d1<\/span> How many vibrations per second will there be if the string\u2019s length is reduced to 20 inches by putting a finger on a fret?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834189087\"><p id=\"fs-id1167834189089\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167831847028\" class=\"unnumbered unstyled can-break\" summary=\"The frequency varies inversely with the length. Name the variables. Let f be equal to frequency and L be equal to length. Write the formula for inverse variation. The formula is y is equal to k divided by x. We will use f in place of y and L in place of x. The equation is f is equal to the quantity k divided by L instead. Substitute the given values for the variables. f is equal to 440 when L is equal to 26. The result is 440 is equal k divided by 26. Solve for the constant of variation. Multiplying each side by the least common denominator 26, the result is 26 times 440 is equal to 26 times the quantity k divided by 26. Solving for k, the result is 11,440 is equal to k. Write the equation that relates f and L. The equation is f is equal to the quantity k divided by L. Substitute the constant of variation. The result is f is equal to the quantity 11,440 divided by L\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\">The frequency varies<div data-type=\"newline\"><br><\/div>inversely with the length.<\/td><\/tr><tr><td data-align=\"left\">Name the variables.<\/td><td data-align=\"left\">Let <em data-effect=\"italics\">f<\/em> = frequency.<div data-type=\"newline\"><br><\/div>\\(\\phantom{\\rule{1em}{0ex}}\\)<em data-effect=\"italics\">L<\/em> = length<\/td><\/tr><tr><td data-align=\"left\">Write the formula for inverse variation.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834556260\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">We will use <em data-effect=\"italics\">f<\/em> in place of <em data-effect=\"italics\">y<\/em> and <em data-effect=\"italics\">L<\/em> in place of <em data-effect=\"italics\">x<\/em>.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826994506\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Substitute the given values for the variables.\\(\\phantom{\\rule{4em}{0ex}}\\)<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834534801\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167835334003\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Solve for the constant of variation<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835510166\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167834539288\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Write the equation that relates <em data-effect=\"italics\">f<\/em> and <em data-effect=\"italics\">L<\/em>.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834179765\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr><td data-align=\"left\">Substitute the constant of variation<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834156945\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167834156588\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\phantom{\\rule{8.2em}{0ex}}\\text{Find}\\phantom{\\rule{0.2em}{0ex}}f\\phantom{\\rule{0.2em}{0ex}}\\text{when}\\phantom{\\rule{0.2em}{0ex}}L=20.\\hfill \\\\ \\text{Write the equation that relates}\\phantom{\\rule{0.2em}{0ex}}f\\text{and}\\phantom{\\rule{0.2em}{0ex}}L.\\hfill &amp; &amp; &amp; \\phantom{\\rule{8.2em}{0ex}}f=\\frac{11,440}{L}\\hfill \\\\ \\text{Substitute the given value for}\\phantom{\\rule{0.2em}{0ex}}L.\\hfill &amp; &amp; &amp; \\phantom{\\rule{8.2em}{0ex}}f=\\frac{11,440}{20}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{8.2em}{0ex}}f=572\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{8.2em}{0ex}}\\begin{array}{c}\\text{A}\\phantom{\\rule{0.2em}{0ex}}20\\text{\u2033}\\text{-guitar string has frequency 572}\\hfill \\\\ \\text{vibrations per second.}\\hfill \\end{array}\\hfill \\end{array}\\)<\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834382551\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834382555\"><div data-type=\"problem\" id=\"fs-id1167831922745\"><p id=\"fs-id1167831922747\">The number of hours it takes for ice to melt varies inversely with the air temperature. Suppose a block of ice melts in 2 hours when the temperature is 65 degrees Celsius.<\/p><p id=\"fs-id1167835229122\"><span class=\"token\">\u24d0<\/span> Write the equation of variation. <span class=\"token\">\u24d1<\/span> How many hours would it take for the same block of ice to melt if the temperature was 78 degrees?<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167835254300\"><span class=\"token\">\u24d0<\/span>\\(h=\\frac{130}{t}\\)<span class=\"token\">\u24d1<\/span>\\(1\\frac{2}{3}\\) hours<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835243949\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835243953\"><div data-type=\"problem\" id=\"fs-id1167835288095\"><p id=\"fs-id1167835288097\">Xander\u2019s new business found that the daily demand for its product was inversely proportional to the price, \\(p.\\) When the price is ?5, the demand is 700 units.<\/p><p id=\"fs-id1167835326113\"><span class=\"token\">\u24d0<\/span> Write the equation of variation. <span class=\"token\">\u24d1<\/span> What is the demand if the price is raised to ?7?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834185666\"><p id=\"fs-id1167834185668\"><span class=\"token\">\u24d0<\/span>\\(x=\\frac{3500}{p}\\)<span class=\"token\">\u24d1<\/span> 500 units<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832068313\" class=\"media-2\"><p id=\"fs-id1167832068317\">Access this online resource for additional instruction and practice with applications of rational expressions<\/p><ul id=\"fs-id1171790665329\" data-display=\"block\"><li id=\"fs-id1167834340056\"><a href=\"https:\/\/openstax.org\/l\/37AppRatExp\">Applications of Rational Expressions<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167834517390\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167831910687\" data-bullet-style=\"bullet\"><li>A proportion is an equation of the form \\(\\frac{a}{b}=\\frac{c}{d},\\) where \\(b\\ne 0,d\\ne 0.\\) The proportion is read \u201c<em data-effect=\"italics\">a<\/em> is to <em data-effect=\"italics\">b<\/em> as <em data-effect=\"italics\">c<\/em> is to <em data-effect=\"italics\">d.<\/em>\u201d<\/li><li><strong data-effect=\"bold\">Property of Similar Triangles<\/strong><div data-type=\"newline\"><br><\/div> If \\(\\text{\u0394}ABC\\) is similar to \\(\\text{\u0394}XYZ,\\) then their corresponding angle measure are equal and their corresponding sides have the same ratio.<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167830703946\" data-alt=\"The first figure is triangle A B C with side A B c units long, side B C a units long, and side A C b units long. The second figure is triangle X Y Z with side X Y x units long, side Y Z x units long, and side X Z y units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. a divided by x is equal to b divided by y is equal to c divided by z.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is triangle A B C with side A B c units long, side B C a units long, and side A C b units long. The second figure is triangle X Y Z with side X Y x units long, side Y Z x units long, and side X Z y units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. a divided by x is equal to b divided by y is equal to c divided by z.\"><\/span> <\/li><li><strong data-effect=\"bold\">Direct Variation<\/strong><ul id=\"fs-id1167835318947\" data-bullet-style=\"open-circle\"><li>For any two variables <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>, <em data-effect=\"italics\">y<\/em> varies directly with <em data-effect=\"italics\">x<\/em> if \\(y=kx,\\) where \\(k\\ne 0.\\) The constant <em data-effect=\"italics\">k<\/em> is called the constant of variation.<\/li><li>How to solve direct variation problems.<div data-type=\"newline\"><br><\/div> <ol id=\"fs-id1167835354882\" type=\"1\" class=\"stepwise\"><li>Write the formula for direct variation.<\/li><li>Substitute the given values for the variables.<\/li><li>Solve for the constant of variation.<\/li><li>Write the equation that relates \\(x\\) and \\(y.\\)<\/li><\/ol><\/li><\/ul><\/li><li><strong data-effect=\"bold\">Inverse Variation<\/strong><ul id=\"fs-id1167834423074\" data-bullet-style=\"open-circle\"><li>For any two variables <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>, <em data-effect=\"italics\">y<\/em> varies inversely with <em data-effect=\"italics\">x<\/em> if \\(y=\\frac{k}{x},\\) where \\(k\\ne 0.\\) The constant <em data-effect=\"italics\">k<\/em> is called the constant of variation.<\/li><li>How to solve inverse variation problems. <ol type=\"1\" class=\"stepwise\"><li>Write the formula for inverse variation.<\/li><li>Substitute the given values for the variables.<\/li><li>Solve for the constant of variation.<\/li><li>Write the equation that relates <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/li><\/ol><\/li><\/ul><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167826857403\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834423748\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167826880194\"><strong data-effect=\"bold\">Solve Proportions<\/strong><\/p><p id=\"fs-id1167834252400\">In the following exercises, solve each proportion.<\/p><div data-type=\"exercise\" id=\"fs-id1167834252404\"><div data-type=\"problem\" id=\"fs-id1167835238764\"><p id=\"fs-id1167835238766\">\\(\\frac{x}{56}=\\frac{7}{8}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830698082\"><p>\\(x=49\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835215793\"><div data-type=\"problem\" id=\"fs-id1167826874529\"><p id=\"fs-id1167826874532\">\\(\\frac{56}{72}=\\frac{y}{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835356568\"><div data-type=\"problem\" id=\"fs-id1167835356570\"><p id=\"fs-id1167835356573\">\\(\\frac{98}{154}=\\frac{-7}{p}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834489808\"><p id=\"fs-id1167834489810\">\\(p=-11\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827988171\"><div data-type=\"problem\" id=\"fs-id1167827988173\"><p id=\"fs-id1167835268055\">\\(\\frac{72}{156}=\\frac{-6}{q}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834192953\"><div data-type=\"problem\" id=\"fs-id1167835420385\"><p id=\"fs-id1167835420387\">\\(\\frac{a}{a+12}=\\frac{4}{7}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831891006\"><p id=\"fs-id1167831891008\">\\(a=16\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831919497\"><div data-type=\"problem\" id=\"fs-id1167831919499\"><p>\\(\\frac{b}{b-16}=\\frac{11}{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835509144\"><div data-type=\"problem\" id=\"fs-id1167835509146\"><p id=\"fs-id1167835509148\">\\(\\frac{m+90}{25}=\\frac{m+30}{15}\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167835369531\">\\(m=60\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835335653\"><div data-type=\"problem\" id=\"fs-id1167835335655\"><p id=\"fs-id1167835335657\">\\(\\frac{n+10}{4}=\\frac{40-n}{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835307608\"><div data-type=\"problem\" id=\"fs-id1167835307610\"><p id=\"fs-id1167832058806\">\\(\\frac{2p+4}{8}=\\frac{p+18}{6}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826801678\"><p id=\"fs-id1167826801680\">\\(p=30\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834065787\"><div data-type=\"problem\" id=\"fs-id1167834065789\"><p id=\"fs-id1167835346269\">\\(\\frac{q-2}{2}=\\frac{2q-7}{18}\\)<\/p><\/div><\/div><p id=\"fs-id1167834472847\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167834472850\"><div data-type=\"problem\" id=\"fs-id1167834472852\"><p id=\"fs-id1167828426705\">Kevin wants to keep his heart rate at 160 beats per minute while training. During his workout he counts 27 beats in 10 seconds.<\/p><p id=\"fs-id1167828426709\"><span class=\"token\">\u24d0<\/span> How many beats per minute is this?<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> Has Kevin met his target heart rate?<\/div><div data-type=\"solution\" id=\"fs-id1167834537746\"><p id=\"fs-id1167834537748\"><span class=\"token\">\u24d0<\/span> 162 beats per minute<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> yes<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834413589\"><div data-type=\"problem\" id=\"fs-id1167832060560\"><p id=\"fs-id1167832060562\">Jesse\u2019s car gets 30 miles per gallon of gas.<\/p><p id=\"fs-id1167835253852\"><span class=\"token\">\u24d0<\/span> If Las Vegas is 285 miles away, how many gallons of gas are needed to get there and then home?<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> If gas is ?3.09 per gallon, what is the total cost of the gas for the trip?<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835346436\"><div data-type=\"problem\" id=\"fs-id1167835346438\"><p id=\"fs-id1167835346440\">Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child\u2019s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs 45 pounds?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831186069\"><p id=\"fs-id1167831186072\">\\(9\\) ml<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831970353\"><div data-type=\"problem\" id=\"fs-id1167831970355\"><p id=\"fs-id1167831970357\">A veterinarian prescribed Sunny, a 65-pound dog, an antibacterial medicine in case an infection emerges after her teeth were cleaned. If the dosage is 5 mg for every pound, how much medicine was Sunny given?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835358031\"><div data-type=\"problem\" id=\"fs-id1167835358033\"><p id=\"fs-id1167834120894\">A new energy drink advertises 106 calories for 8 ounces. How many calories are in 12 ounces of the drink?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835318972\"><p id=\"fs-id1167835318974\">\\(159\\) calories<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835329467\"><div data-type=\"problem\" id=\"fs-id1167835364886\"><p id=\"fs-id1167835364888\">One 12-ounce can of soda has 150 calories. If Josiah drinks the big 32-ounce size from the local mini-mart, how many calories does he get?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834428677\"><div data-type=\"problem\" id=\"fs-id1167834428679\"><p id=\"fs-id1167835349788\">Kyra is traveling to Canada and will change ?250 US dollars into Canadian dollars. At the current exchange rate, ?1 US is equal to ?1.3 Canadian. How many Canadian dollars will she get for her trip?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834280014\"><p id=\"fs-id1167834280017\">\\(325\\) Canadian dollars<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835621559\"><div data-type=\"problem\" id=\"fs-id1167835621562\"><p id=\"fs-id1167835621564\">Maurice is traveling to Mexico and needs to exchange ?450 into Mexican pesos. If each dollar is worth 12.29 pesos, how many pesos will he get for his trip?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834063471\"><div data-type=\"problem\" id=\"fs-id1167834063473\"><p id=\"fs-id1167835334339\">Ronald needs a morning breakfast drink that will give him at least 390 calories. Orange juice has 130 calories in one cup. How many cups does he need to drink to reach his calorie goal?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832068368\"><p id=\"fs-id1167832068370\">\\(3\\) cups<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835421080\"><div data-type=\"problem\" id=\"fs-id1167830757221\"><p id=\"fs-id1167830757223\">Sonya drinks a 32-ounce energy drink containing 80 calories per 12 ounce. How many calories did she drink?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830866157\"><div data-type=\"problem\" id=\"fs-id1167830866159\"><p id=\"fs-id1167835283628\">Phil wants to fertilize his lawn. Each bag of fertilizer covers about 4,000 square feet of lawn. Phil\u2019s lawn is approximately 13,500 square feet. How many bags of fertilizer will he have to buy?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834190288\"><p id=\"fs-id1167834190290\">4 bags<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835237671\"><div data-type=\"problem\" id=\"fs-id1167835237673\"><p id=\"fs-id1167834195103\">An oatmeal cookie recipe calls for \\(\\frac{1}{2}\\) cup of butter to make 4 dozen cookies. Hilda needs to make 10 dozen cookies for the bake sale. How many cups of butter will she need?<\/p><\/div><\/div><p id=\"fs-id1167835283974\"><strong data-effect=\"bold\">Solve Similar Figure Applications<\/strong><\/p><p id=\"fs-id1167835513374\">In the following exercises, the triangles are similar. Find the length of the indicated side.<\/p><div data-type=\"exercise\" id=\"fs-id1167832076332\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832076334\"><p id=\"fs-id1167834049061\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" data-alt=\"The first figure is triangle A B C with side A B 15 units long, side B C 9 units long, and side A C b units long. The second figure is triangle X Y Z with side X Y 10 units long, side Y Z x units long, and side X Z 8 units long.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is triangle A B C with side A B 15 units long, side B C 9 units long, and side A C b units long. The second figure is triangle X Y Z with side X Y 10 units long, side Y Z x units long, and side X Z 8 units long.\"><\/span><p id=\"fs-id1167832015935\"><span class=\"token\">\u24d0<\/span> side <em data-effect=\"italics\">x<\/em><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> side <em data-effect=\"italics\">b<\/em><\/div><div data-type=\"solution\" id=\"fs-id1167835382034\"><p id=\"fs-id1167835382036\"><span class=\"token\">\u24d0<\/span> 6 <span class=\"token\">\u24d1<\/span> \\(12\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834193222\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834193224\"><p id=\"fs-id1167831922554\"><\/p><div data-type=\"newline\"><br><\/div><span data-type=\"media\" id=\"fs-id1167831922555\" data-alt=\"The first figure is triangle DEF with side D E 5 halves units long, side E F d units long, and side D F 1 unit long. The second figure is triangle N P Q with side N P q units long, side P Q 11 halves units long, and side N Q 9 units long.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is triangle DEF with side D E 5 halves units long, side E F d units long, and side D F 1 unit long. The second figure is triangle N P Q with side N P q units long, side P Q 11 halves units long, and side N Q 9 units long.\"><\/span><p id=\"fs-id1167831076642\"><span class=\"token\">\u24d0<\/span> side <em data-effect=\"italics\">d<\/em><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> side <em data-effect=\"italics\">q<\/em><\/div><\/div><p id=\"fs-id1167835622015\">In the following exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle. The actual distance from New York to Chicago is 800 miles.<\/p><span data-type=\"media\" id=\"fs-id1167826997052\" data-alt=\"The figure is a triangle formed by Memphis, Chicago, and New York. The distance between Memphis and Chicago is 5.4 inches. The distance between Chicago and New York is 8 inches. The distance between New York and Memphis is 9.5 inches.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a triangle formed by Memphis, Chicago, and New York. The distance between Memphis and Chicago is 5.4 inches. The distance between Chicago and New York is 8 inches. The distance between New York and Memphis is 9.5 inches.\"><\/span><div data-type=\"exercise\" id=\"fs-id1167835307544\" class=\"material-set-2\"><div data-type=\"problem\"><p id=\"fs-id1167835240602\">Find the actual distance from New York to Memphis.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835377085\"><p id=\"fs-id1167835377087\">950 miles<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835379084\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835379086\"><p id=\"fs-id1167835379089\">Find the actual distance from Chicago to Memphis.<\/p><\/div><\/div><p id=\"fs-id1167834185889\">In the following exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle. The actual distance from Atlanta to New Orleans is 420 miles.<\/p><span data-type=\"media\" id=\"fs-id1167832058318\" data-alt=\"The figure is a triangle formed by New Orleans, Atlanta, and Miami. The distance between New Orleans and Atlanta is 2.1 inches. The distance between Atlanta and Miami is 3 inches. The distance between Miami and New Orleans is 3.4 inches.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a triangle formed by New Orleans, Atlanta, and Miami. The distance between New Orleans and Atlanta is 2.1 inches. The distance between Atlanta and Miami is 3 inches. The distance between Miami and New Orleans is 3.4 inches.\"><\/span><div data-type=\"exercise\" id=\"fs-id1167834439125\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831882913\"><p id=\"fs-id1167831882915\">Find the actual distance from New Orleans to Miami.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835622515\"><p id=\"fs-id1167835622517\">680 miles<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835217022\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835217024\"><p id=\"fs-id1167835217027\">Find the actual distance from Atlanta to Miami.<\/p><\/div><\/div><p id=\"fs-id1167835498842\">In the following exercises, answer each question.<\/p><div data-type=\"exercise\" id=\"fs-id1167835225788\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835225791\"><p id=\"fs-id1167835225793\">A 2-foot-tall dog casts a 3-foot shadow at the same time a cat casts a one foot shadow. How tall is the cat ?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834489708\"><p id=\"fs-id1167835352988\">\\(\\frac{2}{3}\\) foot (\\(8\\) in.)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831957047\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834555112\"><p id=\"fs-id1167834555114\">Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry\u2019s shadow was 8 feet and Tom\u2019s was 7.75 feet long. What is Tom\u2019s height?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835262108\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835262110\"><p id=\"fs-id1167835366866\">The tower portion of a windmill is 212 feet tall. A six foot tall person standing next to the tower casts a seven-foot shadow. How long is the windmill\u2019s shadow?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831031248\"><p id=\"fs-id1167831031250\">\\(247.3\\) feet<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832076492\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832076494\"><p id=\"fs-id1167832076496\">The height of the Statue of Liberty is 305 feet. Nikia, who is standing next to the statue, casts a 6-foot shadow and she is 5 feet tall. How long should the shadow of the statue be?<\/p><\/div><\/div><p id=\"fs-id1167835303952\"><strong data-effect=\"bold\">Solve Uniform Motion Applications<\/strong><\/p><p id=\"fs-id1167834432142\">In the following exercises, solve the application problem provided.<\/p><div data-type=\"exercise\" id=\"fs-id1167830702465\"><div data-type=\"problem\" id=\"fs-id1167830702467\"><p id=\"fs-id1167830702469\">Mary takes a sightseeing tour on a helicopter that can fly 450 miles against a 35-mph headwind in the same amount of time it can travel 702 miles with a 35-mph tailwind. Find the speed of the helicopter.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834517568\"><p id=\"fs-id1167835510196\">\\(160\\) mph<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835240350\"><p id=\"fs-id1167834431101\">A private jet can fly 1,210 miles against a 25-mph headwind in the same amount of time it can fly 1694 miles with a 25-mph tailwind. Find the speed of the jet.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834184544\"><div data-type=\"problem\" id=\"fs-id1167834184546\"><p id=\"fs-id1167834184548\">A boat travels 140 miles downstream in the same time as it travels 92 miles upstream. The speed of the current is 6mph. What is the speed of the boat?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826978746\"><p id=\"fs-id1167835344947\">\\(29\\) mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835387124\"><div data-type=\"problem\" id=\"fs-id1167835387126\"><p id=\"fs-id1167832152880\">Darrin can skateboard 2 miles against a 4-mph wind in the same amount of time he skateboards 6 miles with a 4-mph wind. Find the speed Darrin skateboards with no wind.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832055442\"><div data-type=\"problem\"><p id=\"fs-id1167834432149\">Jane spent 2 hours exploring a mountain with a dirt bike. First, she rode 40 miles uphill. After she reached the peak she rode for 12 miles along the summit. While going uphill, she went 5 mph slower than when she was on the summit. What was her rate along the summit?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834279319\"><p id=\"fs-id1167834131066\">\\(30\\) mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831881826\"><div data-type=\"problem\" id=\"fs-id1167831881828\"><p id=\"fs-id1167832198573\">Laney wanted to lose some weight so she planned a day of exercising. She spent a total of 2 hours riding her bike and jogging. She biked for 12 miles and jogged for 6 miles. Her rate for jogging was 10 mph less than biking rate. What was her rate when jogging?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835326552\"><div data-type=\"problem\" id=\"fs-id1167834539256\"><p id=\"fs-id1167834539258\">Byron wanted to try out different water craft. He went 62 miles downstream in a motor boat and 27 miles downstream on a jet ski. His speed on the jet ski was 10 mph faster than in the motor boat. Bill spent a total of 4 hours on the water. What was his rate of speed in the motor boat?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167827987903\"><p id=\"fs-id1167831920754\">\\(20\\) mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835283707\"><div data-type=\"problem\" id=\"fs-id1167835283710\"><p id=\"fs-id1167834131729\">Nancy took a 3-hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835374050\"><div data-type=\"problem\" id=\"fs-id1167835374052\"><p id=\"fs-id1167835374055\">Chester rode his bike uphill 24 miles and then back downhill at 2 mph faster than his uphill. If it took him 2 hours longer to ride uphill than downhill, what was his uphill rate?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834195866\"><p id=\"fs-id1167835308775\">\\(4\\) mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831882440\"><div data-type=\"problem\" id=\"fs-id1167831882442\"><p id=\"fs-id1167835511581\">Matthew jogged to his friend\u2019s house 12 miles away and then got a ride back home. It took him 2 hours longer to jog there than ride back. His jogging rate was 25 mph slower than the rate when he was riding. What was his jogging rate?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826802097\"><div data-type=\"problem\" id=\"fs-id1167826802099\"><p id=\"fs-id1167826802102\">Hudson travels 1080 miles in a jet and then 240 miles by car to get to a business meeting. The jet goes 300 mph faster than the rate of the car, and the car ride takes 1 hour longer than the jet. What is the speed of the car?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826997726\"><p id=\"fs-id1167835422406\">\\(60\\) mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835513046\"><div data-type=\"problem\" id=\"fs-id1167835534047\"><p id=\"fs-id1167835534049\">Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834228325\"><div data-type=\"problem\" id=\"fs-id1167834228327\"><p id=\"fs-id1167834228330\">John can fly his airplane 2800 miles with a wind speed of 50 mph in the same time he can travel 2400 miles against the wind. If the speed of the wind is 50 mph, find the speed of his airplane.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835238517\"><p id=\"fs-id1167834066283\">\\(650\\) mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831892512\"><div data-type=\"problem\"><p id=\"fs-id1167826986754\">Jim\u2019s speedboat can travel 20 miles upstream against a 3-mph current in the same amount of time it travels 22 miles downstream with a 3-mph current speed . Find the speed of the Jim\u2019s boat.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835283953\"><div data-type=\"problem\" id=\"fs-id1167835283955\"><p id=\"fs-id1167834448720\">Hazel needs to get to her granddaughter\u2019s house by taking an airplane and a rental car. She travels 900 miles by plane and 250 miles by car. The plane travels 250 mph faster than the car. If she drives the rental car for 2 hours more than she rode the plane, find the speed of the car.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834448724\"><p id=\"fs-id1167835363625\">\\(50\\) mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831896687\"><div data-type=\"problem\" id=\"fs-id1167831896689\"><p id=\"fs-id1167835356344\">Stu trained for 3 hours yesterday. He ran 14 miles and then biked 40 miles. His biking speed is 6 mph faster than his running speed. What is his running speed?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835240369\"><div data-type=\"problem\" id=\"fs-id1167835240371\"><p id=\"fs-id1167835240374\">When driving the 9-hour trip home, Sharon drove 390 miles on the interstate and 150 miles on country roads. Her speed on the interstate was 15 more than on country roads. What was her speed on country roads?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835343339\"><p id=\"fs-id1167835257646\">\\(50\\) mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834196066\"><div data-type=\"problem\" id=\"fs-id1167834196068\"><p id=\"fs-id1167831919456\">Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835515481\"><div data-type=\"problem\" id=\"fs-id1167835515483\"><p id=\"fs-id1167835515485\">Dana enjoys taking her dog for a walk, but sometimes her dog gets away, and she has to run after him. Dana walked her dog for 7 miles but then had to run for 1 mile, spending a total time of 2.5 hours with her dog. Her running speed was 3 mph faster than her walking speed. Find her walking speed.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834138038\"><p id=\"fs-id1167830894100\">4.2 mph<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831893592\"><div data-type=\"problem\" id=\"fs-id1167831893595\"><p id=\"fs-id1167831893597\">Ken and Joe leave their apartment to go to a football game 45 miles away. Ken drives his car 30 mph faster Joe can ride his bike. If it takes Joe 2 hours longer than Ken to get to the game, what is Joe\u2019s speed?<\/p><\/div><\/div><p id=\"fs-id1167835320259\"><strong data-effect=\"bold\">Solve Work Applications<\/strong><\/p><div data-type=\"exercise\" id=\"fs-id1167834111772\"><div data-type=\"problem\" id=\"fs-id1167834111774\"><p id=\"fs-id1167834111776\">Mike, an experienced bricklayer, can build a wall in 3 hours, while his son, who is learning, can do the job in 6 hours. How long does it take for them to build a wall together?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832058504\"><p>\\(2\\) hours<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826808725\"><div data-type=\"problem\" id=\"fs-id1167826808727\"><p id=\"fs-id1167834395184\">It takes Sam 4 hours to rake the front lawn while his brother, Dave, can rake the lawn in 2 hours. How long will it take them to rake the lawn working together?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834299718\"><div data-type=\"problem\" id=\"fs-id1167834299720\"><p id=\"fs-id1167835496364\">Mia can clean her apartment in 6 hours while her roommate can clean the apartment in 5 hours. If they work together, how long would it take them to clean the apartment?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834228222\"><p id=\"fs-id1167834228224\">\\(2\\) hours and \\(44\\) minutes <\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835235908\"><div data-type=\"problem\" id=\"fs-id1167835218029\"><p id=\"fs-id1167835218031\">Brian can lay a slab of concrete in 6 hours, while Greg can do it in 4 hours. If Brian and Greg work together, how long will it take?<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835282827\"><p id=\"fs-id1167835282829\">Josephine can correct her students test papers in 5 hours, but if her teacher\u2019s assistant helps, it would take them 3 hours. How long would it take the assistant to do it alone?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835282832\"><p id=\"fs-id1167831884906\">\\(7\\) hours and \\(30\\) minutes <\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835301974\"><div data-type=\"problem\"><p id=\"fs-id1167832087118\">Washing his dad\u2019s car alone, eight year old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take Levi\u2019s dad to wash the car by himself?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834063286\"><div data-type=\"problem\" id=\"fs-id1167835384857\"><p id=\"fs-id1167835384859\">At the end of the day Dodie can clean her hair salon in 15 minutes. Ann, who works with her, can clean the salon in 30 minutes. How long would it take them to clean the shop if they work together?<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832066033\"><p id=\"fs-id1167832066035\">\\(10\\) min<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835343355\"><div data-type=\"problem\" id=\"fs-id1167835343357\"><p id=\"fs-id1167835343359\">Ronald can shovel the driveway in 4 hours, but if his brother Donald helps it would take 2 hours. How long would it take Donald to shovel the driveway alone?<\/p><\/div><\/div><p id=\"fs-id1167826996824\"><strong data-effect=\"bold\">Solve Direct Variation Problems<\/strong><\/p><p id=\"fs-id1167830866117\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167830866120\"><div data-type=\"problem\" id=\"fs-id1167834061688\"><p id=\"fs-id1167834061690\">If \\(y\\) varies directly as \\(x\\) and \\(y=14,\\text{\u200b}\\) when \\(x=3.\\) find the equation that relates \\(x\\) and \\(y.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830705509\"><p id=\"fs-id1167830705511\">\\(y=\\frac{14}{3}x\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835243929\"><div data-type=\"problem\" id=\"fs-id1167835243931\"><p id=\"fs-id1167826996702\">If \\(a\\) varies directly as \\(b\\) and \\(a=16,\\text{\u200b}\\text{\u200b}\\) when \\(b=4.\\) find the equation that relates \\(a\\) and \\(b.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834562571\"><div data-type=\"problem\" id=\"fs-id1167835422662\"><p id=\"fs-id1167835422664\">If \\(p\\) varies directly as \\(q\\) and \\(p=9.6,\\text{\u200b}\\text{\u200b}\\) when \\(q=3.\\) find the equation that relates \\(p\\) and \\(q.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831922015\"><p id=\"fs-id1167831922017\">\\(p=3.2q\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834053691\"><div data-type=\"problem\" id=\"fs-id1167834053693\"><p id=\"fs-id1167834053695\">If \\(v\\) varies directly as \\(w\\) and \\(v=8,\\text{\u200b}\\) when \\(\\text{\u200b}w=\\frac{1}{2}.\\) find the equation that relates \\(v\\) and \\(w.\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167834535299\">The price, \\(P,\\) that Eric pays for gas varies directly with the number of gallons, \\(g,\\) he buys. It costs him ?50 to buy 20 gallons of gas.<\/p><p id=\"fs-id1167835368004\"><span class=\"token\">\u24d0<\/span> Write the equation that relates \\(P\\) and \\(g.\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> How much would 33 gallons cost Eric?<\/div><div data-type=\"solution\" id=\"fs-id1167835414664\"><p id=\"fs-id1167835414667\"><span class=\"token\">\u24d0<\/span>\\(P=2.5g\\)<span class=\"token\">\u24d1<\/span>\\(\\text{?}82.50\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834222099\"><div data-type=\"problem\" id=\"fs-id1167834222101\"><p id=\"fs-id1167835513096\">Joseph is traveling on a road trip. The distance, \\(d,\\) he travels before stopping for lunch varies directly with the speed, \\(v,\\) he travels. He can travel 120 miles at a speed of 60 mph.<\/p><p id=\"fs-id1167834464336\"><span class=\"token\">\u24d0<\/span> Write the equation that relates \\(d\\) and \\(v.\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> How far would he travel before stopping for lunch at a rate of 65 mph?<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835532705\"><div data-type=\"problem\" id=\"fs-id1167835230446\"><p id=\"fs-id1167835230448\">The mass of a liquid varies directly with its volume. A liquid with mass 16 kilograms has a volume of 2 liters.<\/p><p id=\"fs-id1167834587806\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the mass to the volume.<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> What is the volume of this liquid if its mass is 128 kilograms?<\/div><div data-type=\"solution\" id=\"fs-id1167835284798\"><p id=\"fs-id1167835284800\"><span class=\"token\">\u24d0<\/span>\\(m=8v\\)<span class=\"token\">\u24d1<\/span>\\(16\\) liters<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830757576\"><div data-type=\"problem\" id=\"fs-id1167830757579\"><p id=\"fs-id1167835166911\">The length that a spring stretches varies directly with a weight placed at the end of the spring. When Sarah placed a 10-pound watermelon on a hanging scale, the spring stretched 5 inches.<\/p><p id=\"fs-id1167835370110\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the length of the spring to the weight.<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> What weight of watermelon would stretch the spring 6 inches?<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835180625\"><div data-type=\"problem\" id=\"fs-id1167832054992\"><p id=\"fs-id1167832054994\">The maximum load a beam will support varies directly with the square of the diagonal of the beam\u2019s cross-section. A beam with diagonal 6 inch will support a maximum load of 108 pounds.<\/p><p id=\"fs-id1167835237348\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the load to the diagonal of the cross-section.<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> What load will a beam with a 10-inch diagonal support?<\/div><div data-type=\"solution\" id=\"fs-id1167831884883\"><p id=\"fs-id1167831884886\"><span class=\"token\">\u24d0<\/span>\\(L=3{d}^{2}\\)<span class=\"token\">\u24d1<\/span>\\(300\\) pounds<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831880549\"><div data-type=\"problem\" id=\"fs-id1167831880551\"><p id=\"fs-id1167831880553\">The area of a circle varies directly as the square of the radius. A circular pizza with a radius of 6 inches has an area of 113.04 square inches.<\/p><p id=\"fs-id1167835594886\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the area to the radius.<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> What is the area of a personal pizza with a radius 4 inches?<\/div><\/div><p id=\"fs-id1167834094671\"><strong data-effect=\"bold\">Solve Inverse Variation Problems<\/strong><\/p><p id=\"fs-id1167830698200\">In the following exercises, solve.<\/p><div data-type=\"exercise\" id=\"fs-id1167835267805\"><div data-type=\"problem\" id=\"fs-id1167835267807\"><p id=\"fs-id1167835267809\">If \\(y\\) varies inversely with \\(x\\) and \\(y=5\\) when \\(x=4,\\) find the equation that relates \\(x\\) and \\(y.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826993939\"><p id=\"fs-id1167834229238\">\\(y=\\frac{20}{x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834510611\"><div data-type=\"problem\" id=\"fs-id1167831871826\"><p id=\"fs-id1167831871828\">If \\(p\\) varies inversely with \\(q\\) and \\(p=2\\) when \\(q=1\\), find the equation that relates \\(p\\) and \\(q.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835341860\"><div data-type=\"problem\" id=\"fs-id1167835341862\"><p id=\"fs-id1167835341864\">If \\(v\\) varies inversely with \\(w\\) and \\(v=6\\) when \\(w=\\frac{1}{2},\\) find the equation that relates \\(v\\) and \\(w.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835283409\"><p id=\"fs-id1167835283411\">\\(v=\\frac{3}{w}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835343539\"><div data-type=\"problem\" id=\"fs-id1167835343542\"><p id=\"fs-id1167835343544\">If \\(a\\) varies inversely with \\(b\\) and \\(a=12\\) when \\(b=\\frac{1}{3},\\) find the equation that relates \\(a\\) and \\(b.\\)<\/p><\/div><\/div><p id=\"fs-id1167835346236\">In the following exercises, write an inverse variation equation to solve the following problems.<\/p><div data-type=\"exercise\" id=\"fs-id1167835346241\"><div data-type=\"problem\"><p>The fuel consumption (mpg) of a car varies inversely with its weight. A Toyota Corolla weighs 2800 pounds getting 33 mpg on the highway.<\/p><p id=\"fs-id1167832153050\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the mpg to the car\u2019s weight.<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> What would the fuel consumption be for a Toyota Sequoia that weighs 5500 pounds?<\/div><div data-type=\"solution\" id=\"fs-id1167835367134\"><p id=\"fs-id1167835367137\"><span class=\"token\">\u24d0<\/span>\\(g=\\frac{92,400}{w}\\)<span class=\"token\">\u24d1<\/span> 16.8 mpg<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835353322\"><div data-type=\"problem\" id=\"fs-id1167835353324\"><p id=\"fs-id1167835353326\">A car\u2019s value varies inversely with its age. Jackie bought a 10-year-old car for ?2,400.<\/p><p id=\"fs-id1167834284490\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the car\u2019s value to its age.<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> What will be the value of Jackie\u2019s car when it is 15 years old?<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834300577\"><div data-type=\"problem\" id=\"fs-id1167835191230\"><p id=\"fs-id1167835191232\">The time required to empty a tank varies inversely as the rate of pumping. It took Ada 5 hours to pump her flooded basement using a pump that was rated at 200 gpm (gallons per minute).<\/p><p id=\"fs-id1167834300868\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the number of hours to the pump rate.<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> How long would it take Ada to pump her basement if she used a pump rated at 400 gpm?<\/div><div data-type=\"solution\" id=\"fs-id1167834066023\"><p id=\"fs-id1167834066025\"><span class=\"token\">\u24d0<\/span>\\(t=\\frac{1000}{r}\\)<span class=\"token\">\u24d1<\/span>\\(2.5\\) hours<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835238240\"><div data-type=\"problem\" id=\"fs-id1167835238242\"><p id=\"fs-id1167835238245\">On a string instrument, the length of a string varies inversely as the frequency of its vibrations. An 11-inch string on a violin has a frequency of 400 cycles per second.<\/p><p id=\"fs-id1167835338961\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the string length to its frequency. <span class=\"token\">\u24d1<\/span> What is the frequency of a 10 inch string?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835318274\"><div data-type=\"problem\" id=\"fs-id1167835318276\"><p id=\"fs-id1167831040509\">Paul, a dentist, determined that the number of cavities that develops in his patient\u2019s mouth each year varies inversely to the number of minutes spent brushing each night. His patient, Lori, had four cavities when brushing her teeth 30 seconds (0.5 minutes) each night.<\/p><p id=\"fs-id1167831040512\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the number of cavities to the time spent brushing.<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> How many cavities would Paul expect Lori to have if she had brushed her teeth for 2 minutes each night?<\/div><div data-type=\"solution\" id=\"fs-id1167831880396\"><p id=\"fs-id1167831880399\"><span class=\"token\">\u24d0<\/span>\\(c=\\frac{2}{t}\\)<span class=\"token\">\u24d1<\/span>\\(1\\) cavity<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835514476\"><div data-type=\"problem\" id=\"fs-id1167835514478\"><p id=\"fs-id1167835303202\">Boyle\u2019s law states that if the temperature of a gas stays constant, then the pressure varies inversely to the volume of the gas. Braydon, a scuba diver, has a tank that holds 6 liters of air under a pressure of 220 psi.<\/p><p><span class=\"token\">\u24d0<\/span> Write the equation that relates pressure to volume.<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> If the pressure increases to 330 psi, how much air can Braydon\u2019s tank hold?<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834387485\"><div data-type=\"problem\" id=\"fs-id1167834387487\"><p id=\"fs-id1167834387489\">The cost of a ride service varies directly with the distance traveled. It costs ?35 for a ride from the city center to the airport, 14 miles away.<\/p><p id=\"fs-id1167835414006\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the cost, <em data-effect=\"italics\">c<\/em>, with the number of miles, \\(m.\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> What would it cost to travel 22 miles with this service?<\/div><div data-type=\"solution\" id=\"fs-id1167835343158\"><p id=\"fs-id1167835331448\"><span class=\"token\">\u24d0<\/span>\\(c=2.5m\\)<span class=\"token\">\u24d1<\/span> ?55<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835240050\"><div data-type=\"problem\" id=\"fs-id1167835240052\"><p id=\"fs-id1167835351291\">The number of hours it takes Jack to drive from Boston to Bangor is inversely proportional to his average driving speed. When he drives at an average speed of 40 miles per hour, it takes him 6 hours for the trip.<\/p><p id=\"fs-id1167835351296\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the number of hours, \\(h,\\) with the speed, \\(s.\\)<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span> How long would the trip take if his average speed was 75 miles per hour?<\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167832151081\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167835353400\"><div data-type=\"problem\" id=\"fs-id1167835353402\"><p id=\"fs-id1167835353404\">Marisol solves the proportion \\(\\frac{144}{a}=\\frac{9}{4}\\) by \u2018cross multiplying,\u2019 so her first step looks like \\(4\u00b7144=9\u00b7a.\\) Explain how this differs from the method of solution shown in <a href=\"#fs-id1167835341039\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835420340\"><p id=\"fs-id1167831025321\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831025326\"><div data-type=\"problem\" id=\"fs-id1167835364640\"><p id=\"fs-id1167835364643\">Paula and Yuki are roommates. It takes Paula 3 hours to clean their apartment. It takes Yuki 4 hours to clean the apartment. The equation \\(\\frac{1}{3}+\\frac{1}{4}=\\frac{1}{t}\\) can be used to find <em data-effect=\"italics\">t<\/em>, the number of hours it would take both of them, working together, to clean their apartment. Explain how this equation models the situation.<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167832074381\"><p id=\"fs-id1167835356060\">In your own words, explain the difference between direct variation and inverse variation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835356064\"><p id=\"fs-id1167830836880\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830836886\"><div data-type=\"problem\" id=\"fs-id1167835341031\"><p id=\"fs-id1167835341034\">Make up an example from your life experience of inverse variation.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835346512\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167835240157\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167835376880\" data-alt=\"This table has four columns and seven rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was solve proportions. In row 3, the I can was solve similar figure applications. In row 4, the I can was solve uniform motion applications. In row 5, the I can was solve work applications. In row 6, the I can was solve direct variation problems. In row 7, the I can was solve inverse variation problems.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and seven rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was solve proportions. In row 3, the I can was solve similar figure applications. In row 4, the I can was solve uniform motion applications. In row 5, the I can was solve work applications. In row 6, the I can was solve direct variation problems. In row 7, the I can was solve inverse variation problems.\"><\/span><p id=\"fs-id1167835503869\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167834523835\"><dt>proportion<\/dt><dd id=\"fs-id1167834191234\">When two rational expressions are equal, the equation relating them is called a proportion.<\/dd><\/dl><dl id=\"fs-id1167831923341\"><dt>similar figures<\/dt><dd id=\"fs-id1167831908814\">Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides have the same ratio.<\/dd><\/dl><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Solve proportions<\/li>\n<li>Solve similar figure applications<\/li>\n<li>Solve uniform motion applications<\/li>\n<li>Solve work applications<\/li>\n<li>Solve direct variation problems<\/li>\n<li>Solve inverse variation problems<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835419617\" class=\"be-prepared\">\n<p id=\"fs-id1167832053944\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167834526371\" type=\"1\">\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b8a186a927ac371b864910595823bef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#52;&#61;&#45;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"157\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167836399284\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>An express train and a charter bus leave Chicago to travel to Champaign. The express train can make the trip in two hours and the bus takes five hours for the trip. The speed of the express train is 42 miles per hour faster than the speed of the bus. Find the speed of the bus.\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/36adea73-2201-46d3-b9b6-d13ef7df78b2#fs-id1167836309166\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7915b570a7888fdb6e9f5d8dc8332252_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#120;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"100\" style=\"vertical-align: -6px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/9f100e8f-6d15-4cae-bc22-c306e9d7d55c#fs-id1167833239741\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834162026\">\n<h3 data-type=\"title\">Solve Proportions<\/h3>\n<p>When two rational expressions are equal, the equation relating them is called a <span data-type=\"term\">proportion<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835165941\">\n<div data-type=\"title\">Proportion<\/div>\n<p id=\"fs-id1167826873771\">A <strong data-effect=\"bold\">proportion<\/strong> is an equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60c029a08a2eb9a150ad607019e89782_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -6px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e327228415f3008762bc19707f9a109d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#100;&#92;&#110;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"94\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167830908992\">The proportion is read \u201c<em data-effect=\"italics\">a<\/em> is to <em data-effect=\"italics\">b<\/em> as <em data-effect=\"italics\">c<\/em> is to <em data-effect=\"italics\">d.<\/em>\u201d<\/p>\n<\/div>\n<p id=\"fs-id1167834137555\">The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07b9374c4ae0e62e6b79f800bf9bb51e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"41\" style=\"vertical-align: -6px;\" \/> is a proportion because the two fractions are equal. The proportion <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07b9374c4ae0e62e6b79f800bf9bb51e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"41\" style=\"vertical-align: -6px;\" \/> is read \u201c1 is to 2 as 4 is to 8.\u201d<\/p>\n<p>Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved rational equations. We\u2019ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831112213\">\n<div data-type=\"problem\" id=\"fs-id1167835234960\">\n<p id=\"fs-id1167835339103\">Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3130c57275334b639df34a3192268619_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#110;&#43;&#49;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#55;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"73\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835330048\">\n<table class=\"unnumbered unstyled can-break\" summary=\"Solve n divided by the quantity n plus 14 is equal to 5 divided by 7. Notice that the least common denominator is the product of 7 and the quantity n plus 14. Multiply each side of the equation by the least common denominator and remove the common factors on each side. The result is 7 n is equal to 5 times the quantity n plus 14. Simplify the equation on the right side. The result is 7 n is equal to 5 n plus 70. By further simplifying, the equation becomes 2 n is equal to 70. When the equation is solved, the result is n is equal to 35. Check the solution by substituting into the original equation. The result is 35 divided by the sum of 35 and 14 is equal to 5 divided by 7. Determine whether the equation is true by simplifying. Is 35 divided by 49 is equal to 5 divided by 7 a true equation? Show the common factors on the right side. Is the quantity 5 times 7 divided by the quantity 7 times 7 is equal to 5 divided by 7 true? 5 divided by 7 is equal to 5 divided by 7 is a true equation.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-align=\"left\">Multiply both sides by LCD.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835335325\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-align=\"left\">Remove common factors on each side.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831910788\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-align=\"left\">Solve for <em data-effect=\"italics\">n<\/em>.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835533903\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835188971\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"3\" data-align=\"left\">Check.<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835237211\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831116538\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835345556\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Show common factors.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834422480\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_001f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834138284\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835241089\">Solve the proportion: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-033207c0c5a7355dbc96eae3cabf9d9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#121;&#43;&#53;&#53;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"72\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831880440\">\n<p id=\"fs-id1167835345046\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ed3c46d82892240ea2c73c501eda962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#51;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826996879\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835367206\">Solve the proportion: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5aa8bda4cb76087e787801d11eae912_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#122;&#125;&#123;&#122;&#45;&#56;&#52;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"85\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167834430708\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b32c2c12cf5c045f18f940c6afbfa7c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#122;&#61;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835309313\">Notice in the last example that when we cleared the fractions by multiplying by the LCD, the result is the same as if we had cross-multiplied.<\/p>\n<p><span data-type=\"media\" data-alt=\"The equation n divided by the quantity n plus 14 is equal to 5 divided by 7 can be solved by multiplying each side by the least common denominator, 7 times the quantity n plus 14. Multiplying by the least common denominator is a way to clear the fractions. The result is 7 n is equal to 5 times the quantity n plus 14. The equation n divided by the quantity n plus 14 is equal to 5 divided by 7 can also be solved using cross multiplication. Multiply n and 7. Multiply the quantity n plus 14 and 5. The result is also 7 n is equal to 5 times the quantity n plus 14. Cross multiplication also clears fractions.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_018_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equation n divided by the quantity n plus 14 is equal to 5 divided by 7 can be solved by multiplying each side by the least common denominator, 7 times the quantity n plus 14. Multiplying by the least common denominator is a way to clear the fractions. The result is 7 n is equal to 5 times the quantity n plus 14. The equation n divided by the quantity n plus 14 is equal to 5 divided by 7 can also be solved using cross multiplication. Multiply n and 7. Multiply the quantity n plus 14 and 5. The result is also 7 n is equal to 5 times the quantity n plus 14. Cross multiplication also clears fractions.\" \/><\/span><\/p>\n<p id=\"fs-id1167835309022\">For any proportion, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60c029a08a2eb9a150ad607019e89782_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -6px;\" \/> we get the same result when we clear the fractions by multiplying by the LCD as when we cross-multiply.<\/p>\n<p><span data-type=\"media\" data-alt=\"Multiply each side of a proportion a divided by b is equal to c divided by d by the least common denominator, b d, to clear the fractions. The result is a d is equal to b c. Cross multiply to clear the fractions in the proportion a divided by b is equal to c divided by d. The cross products are a times d and b times c. The result is also a d is equal to b c.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_019_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Multiply each side of a proportion a divided by b is equal to c divided by d by the least common denominator, b d, to clear the fractions. The result is a d is equal to b c. Cross multiply to clear the fractions in the proportion a divided by b is equal to c divided by d. The cross products are a times d and b times c. The result is also a d is equal to b c.\" \/><\/span><\/p>\n<p id=\"fs-id1167835335405\">To solve applications with proportions, we will follow our usual strategy for solving applications But when we set up the proportion, we must make sure to have the units correct\u2014the units in the numerators must match each other and the units in the denominators must also match each other.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835304047\">\n<div data-type=\"problem\" id=\"fs-id1167835305367\">\n<p id=\"fs-id1167835343502\">When pediatricians prescribe acetaminophen to children, they prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of the child\u2019s weight. If Zoe weighs 80 pounds, how many milliliters of acetaminophen will her doctor prescribe?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834300223\">\n<table id=\"fs-id1167835377332\" class=\"unnumbered unstyled\" summary=\"Identify what must be found and a variable to represent it. The number of milliliters of acetaminophen the doctor will prescribe must be found. Let a represent the number of milliliters of acetaminophen. Write a sentence that gives the information to find a. If 5 milliliters are prescribed for every 25 pounds, find how much will be prescribed for 80 pounds. Translate the sentence into a proportion. Be careful of the units, making sure the proportion shows the number of milliliters divided by the number of pounds for the rational expressions on each side. The proportion to find how much acetaminophen is prescribed for 80 pounds is 5 divided by 25 is equal to a divided by 80. Multiply each side of the proportion by the least common denominator, which is 400. The result is 400 times the quantity 5 divided by 25 is equal to 400 times the quantity a divided by 80. Remove the common factors on each side. The common factor on the left side is 25. The common factor on the right side is 8. The equation that result is 16 times 5 is equal to 5 a. Do not simplify on the left side of the equation. Think about the next step. Solve for a by dividing each side by 5. The result is 16 is equal to a. Check that the answer is reasonable. Notice that 80 is about 3 times 25. The amount of acetaminophen should be about 3 times 5. So, 16 milliliters makes sense. To be sure, substitute a is equal to 16 in the original proportion. Is 5 divided by 25 equal to 16 divided by 80? The proportion simplifies to 1 divided by 5 is equal to 1 divided by 5, which is true. The solution, 16 is equal to a, checks. Write a complete sentence. The pediatrician would prescribe 16 milliliters of acetaminophen to Zoe.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-align=\"left\">Identify what we are asked to find,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>and choose a variable to represent it.<\/td>\n<td data-align=\"left\">How many ml of acetaminophen will the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>doctor prescribe?<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\">Let <em data-effect=\"italics\">a<\/em> = ml of acetaminophen.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write a sentence that gives the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>information to find it.<\/td>\n<td data-align=\"left\">If 5 ml is prescribed for every<\/p>\n<div data-type=\"newline\"><\/div>\n<p>25 pounds, how much will be<\/p>\n<div data-type=\"newline\"><\/div>\n<p>prescribed for 80 pounds?<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Translate into a proportion\u2014be<\/p>\n<div data-type=\"newline\"><\/div>\n<p>careful of the units.<\/td>\n<td data-align=\"left\"><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834403463\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Multiply both sides by the LCD, 400.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Remove common factors on each side.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831880394\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify, but don\u2019t multiply on the left. Notice<\/p>\n<div data-type=\"newline\"><\/div>\n<p>what the next step will be.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835226082\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Solve for <em data-effect=\"italics\">a<\/em>.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Check.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Is the answer reasonable?<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_002b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write a complete sentence.<\/td>\n<td data-align=\"left\">The pediatrician would prescribe 16 ml of<\/p>\n<div data-type=\"newline\"><\/div>\n<p>acetaminophen to Zoe.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835478978\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835233288\">\n<div data-type=\"problem\" id=\"fs-id1167835371012\">\n<p id=\"fs-id1167830959928\">Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child\u2019s weight. How many milliliters of acetaminophen will the doctor prescribe for Emilia, who weighs 60 pounds?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835346003\">\n<p>The pediatrician will prescribe 12 ml of acetaminophen to Emilia.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835336774\">\n<div data-type=\"problem\" id=\"fs-id1167832031142\">\n<p id=\"fs-id1167834330002\">For every 1 kilogram (kg) of a child\u2019s weight, pediatricians prescribe 15 milligrams (mg) of a fever reducer. If Isabella weighs 12 kg, how many milligrams of the fever reducer will the pediatrician prescribe?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835335689\">\n<p id=\"fs-id1167834191169\">The pediatrician will prescribe 180 mg of fever reducer to Isabella.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\">Solve similar figure applications<\/h3>\n<p id=\"fs-id1167835237568\">When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with <span data-type=\"term\">similar figures<\/span>. If two figures have exactly the same shape, but different sizes, they are said to be similar. One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides have the same ratio.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834439091\">\n<div data-type=\"title\">Similar Figures<\/div>\n<p id=\"fs-id1167835368059\">Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides have the same ratio.<\/p>\n<\/div>\n<p>For example, the two triangles in <a href=\"#CNX_IntAlg_Figure_07_05_003\" class=\"autogenerated-content\">(Figure)<\/a> are similar. Each side of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91acd2d3a8297e2d0caa8e3d4bfae3ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> is four times the length of the corresponding side of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32a93b0a8112ce101de969b1d2e44893_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#88;&#89;&#90;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/><\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_07_05_003\"><span data-type=\"media\" id=\"fs-id1167826869931\" data-alt=\"The first figure is triangle A B C with side A B 12 units long, side B C 16 units long, and side A C 20 units long. The second figure is triangle X Y Z with side X Y 3 units long, side Y X 4 units long, and side X Z is 5 units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. 16 divided by 4 is equal to 20 divided 5 is equal to 12 divided by 3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_003_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is triangle A B C with side A B 12 units long, side B C 16 units long, and side A C 20 units long. The second figure is triangle X Y Z with side X Y 3 units long, side Y X 4 units long, and side X Z is 5 units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. 16 divided by 4 is equal to 20 divided 5 is equal to 12 divided by 3.\" \/><\/span><\/div>\n<p>This is summed up in the Property of Similar Triangles.<\/p>\n<div data-type=\"note\">\n<div data-type=\"title\">Property of Similar Triangles<\/div>\n<p id=\"fs-id1167831928919\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91acd2d3a8297e2d0caa8e3d4bfae3ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> is similar to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8491fe25e881cb20249417d7fc85afbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#88;&#89;&#90;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> then their corresponding angle measure are equal and their corresponding sides have the same ratio.<\/p>\n<p><span data-type=\"media\" data-alt=\"The first figure is triangle A B C with side A B c units long, side B C a units long, and side A C b units long. The second figure is triangle X Y Z with side X Y x units long, side Y Z x units long, and side X Z y units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. a divided by x is equal to b divided by y is equal to c divided by z.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is triangle A B C with side A B c units long, side B C a units long, and side A C b units long. The second figure is triangle X Y Z with side X Y x units long, side Y Z x units long, and side X Z y units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. a divided by x is equal to b divided by y is equal to c divided by z.\" \/><\/span><\/div>\n<p id=\"fs-id1167834335084\">To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.<\/p>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p>On a map, San Francisco, Las Vegas, and Los Angeles form a triangle. The distance between the cities is measured in inches. The figure on the left below represents the triangle formed by the cities on the map. If the actual distance from Los Angeles to Las Vegas is 270 miles, find the distance from Los Angeles to San Francisco.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167832057115\" data-alt=\"The first figure is a triangle labeled \u201cDistances on map.\u201d The triangle is formed by San Francisco, Las Vegas, and Los Angeles. The distance between San Francisco and Las Vegas is 2.1 inches. The distance between Las Vegas and Los Angeles is 1 inch. The distance between Los Angeles and San Francisco is 1.3 inches. The second figure is a triangle labeled \u201cActual Distances.\u201d The triangle is formed by San Francisco, Las Vegas, and Los Angeles. The distance between Las Vegas and Los Angeles is 270 miles. The distance between Los Angeles and San Francisco is labeled x.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is a triangle labeled \u201cDistances on map.\u201d The triangle is formed by San Francisco, Las Vegas, and Los Angeles. The distance between San Francisco and Las Vegas is 2.1 inches. The distance between Las Vegas and Los Angeles is 1 inch. The distance between Los Angeles and San Francisco is 1.3 inches. The second figure is a triangle labeled \u201cActual Distances.\u201d The triangle is formed by San Francisco, Las Vegas, and Los Angeles. The distance between Las Vegas and Los Angeles is 270 miles. The distance between Los Angeles and San Francisco is labeled x.\" \/><\/span><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834397757\">\n<p>Since the triangles are similar, the corresponding sides are proportional.<\/p>\n<table class=\"unnumbered unstyled\" summary=\"Read the problem. Draw the figures and label it with the given information. Recall that the figures are shown above. Identify what we are looking for. It is the actual distance from Los Angeles to San Francisco. Name the variable to represent the distance. Let x be equal to the distance from Los Angeles to San Francisco. Translate the problem into an equation. Since the triangles are similar, the corresponding sides are proportional. Make the numerators miles and the numerator inches. The equation is x miles divided by 1.3 inches is equal to 270 miles divided by 1 inch. Solve the equation, 1.3 times the quantity x divided by 1.3 is equal to 1.3 times the quantity 270 divided by 1. The solution is x is equal to 351. Check the solution. On the map, the distance from Los Angeles to San Francisco is more than the distance from Los Angeles to Las Vegas. Since 351 is more than 270, the answer makes sense. Now check x is equal to 351 in the original proportion. Use a calculator. Is 351 miles divided by 1.3 inches equal to 270 miles divided by 1 inch? When simplified, the equation becomes 270 miles divided by 1 inch is equal to 270 miles divided by 1 inch, which is true. Answer the question. The distance from Los Angeles to San Francisco is 351 miles.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-align=\"left\"><strong data-effect=\"bold\">Read<\/strong> the problem. Draw the figures and label<\/p>\n<div data-type=\"newline\"><\/div>\n<p>it with the given information.<\/td>\n<td data-align=\"left\">The figures are shown above.<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Identify<\/strong> what we are looking for.<\/td>\n<td data-align=\"left\">the actual distance from Los Angeles<\/p>\n<div data-type=\"newline\"><\/div>\n<p>to San Francisco<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Name<\/strong> the variables.<\/td>\n<td data-align=\"left\">Let <em data-effect=\"italics\">x<\/em> = distance from Los Angeles<\/p>\n<div data-type=\"newline\"><\/div>\n<p>to San Francisco.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\"><strong data-effect=\"bold\">Translate<\/strong> into an equation.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Since the triangles are similar, the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>corresponding sides are proportional. We\u2019ll<\/p>\n<div data-type=\"newline\"><\/div>\n<p>make the numerators \u201cmiles\u201d and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the denominators \u201cinches\u201d.<\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835186937\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\"><strong data-effect=\"bold\">Solve<\/strong> the equation.<\/td>\n<td data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835347899\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\"><strong data-effect=\"bold\">Check.<\/strong><\/p>\n<div data-type=\"newline\"><\/div>\n<p>On the map, the distance from Los Angeles<\/p>\n<div data-type=\"newline\"><\/div>\n<p>to San Francisco is more than<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the distance from Los Angeles to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Las Vegas. Since 351 is more than 270<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the answer makes sense.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835307534\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Answer<\/strong> the question.<\/td>\n<td data-align=\"left\">The distance from Los Angeles to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>San Francisco is 351 miles.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835362231\">On the map, Seattle, Portland, and Boise form a triangle. The distance between the cities is measured in inches. The figure on the left below represents the triangle formed by the cities on the map. The actual distance from Seattle to Boise is 400 miles.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835288353\" data-alt=\"The figure is a triangle formed by Portland, Seattle, and Boise. The distance between Portland and Seattle is 1.5 inches. The distance between Seattle and Boise is 4 inches. The distance between Boise and Portland is 3.5 inches.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_007_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a triangle formed by Portland, Seattle, and Boise. The distance between Portland and Seattle is 1.5 inches. The distance between Seattle and Boise is 4 inches. The distance between Boise and Portland is 3.5 inches.\" \/><\/span><\/p>\n<div data-type=\"note\" id=\"fs-id1167835361550\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835283341\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834161709\">Find the actual distance from Seattle to Portland.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835370464\">\n<p id=\"fs-id1167835305350\">The distance is 150 miles.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834137694\">\n<div data-type=\"problem\" id=\"fs-id1167835226378\">\n<p id=\"fs-id1167834190063\">Find the actual distance from Portland to Boise.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835410965\">\n<p id=\"fs-id1167835365483\">The distance is 350 miles.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835347011\">We can use similar figures to find heights that we cannot directly measure.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835369388\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167831106741\">\n<p id=\"fs-id1167835381415\">Tyler is 6 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a tree was 24 feet long. Find the height of the tree.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832053612\">\n<table id=\"fs-id1167827987927\" class=\"unnumbered unstyled\" summary=\"Read the problem and draw a figure. The figure is a triangle formed by the height of a tree labeled h and the shadow of the tree, which is 24 feet long. Within the triangle is a smaller triangle formed by the height of Tyler, which is 6 feet and Tyler\u2019s shadow, which is 8 feet. We are looking for h, the height of the tree. We will use similar triangles to write an equation. The equation is h divided by 24 is equal to 6 divided by 8. The small triangle is similar to the large triangle. Solve the proportion by multiplying each side by the least common denominator, 24. 24 times the quantity 6 divided by 8 is equal to 24 times the quantity h divided by 24. Simplify the equation. The result is 18 is equal to h. Check the answer. Tyler\u2019s height is less than his shadow\u2019s, so it makes sense that the tree\u2019s height is less than the length of its shadow. Check that h is equal to 18 in the original equation. Is 6 divided by 8 equal to 18 divided by 24? When simplified, the equation becomes 3 divided by 4 is equal to 3 divided by 4, which checks.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-align=\"left\">Read the problem and draw a figure.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">We are looking for <em data-effect=\"italics\">h<\/em>, the height of the tree.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We will use similar triangles to write an equation.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The small triangle is similar to the large triangle.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834300681\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Solve the proportion.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834422775\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830960983\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Check.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835338242\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835348414\">\n<div data-type=\"problem\" id=\"fs-id1167835417710\">\n<p id=\"fs-id1167835181752\">A telephone pole casts a shadow that is 50 feet long. Nearby, an 8 foot tall traffic sign casts a shadow that is 10 feet long. How tall is the telephone pole?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832051470\">\n<p>The telephone pole is 40 feet tall.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835280790\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167826993906\">A pine tree casts a shadow of 80 feet next to a 30 foot tall building which casts a 40 feet shadow. How tall is the pine tree?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831881158\">\n<p id=\"fs-id1167830964488\">The pine tree is 60 feet tall.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167832042392\">\n<h3 data-type=\"title\">Solve Uniform Motion Applications<\/h3>\n<p id=\"fs-id1167835339943\">We have solved uniform motion problems using the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-164fcc881021ce79be78a8adbd303299_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#114;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"53\" style=\"vertical-align: 0px;\" \/> in previous chapters. We used a table like the one below to organize the information and lead us to the equation.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834194145\" data-alt=\"This chart has two columns and three rows. The first row is a header row and the second column is labeled \u201cRate times Time is equal to Distance.\u201d There is nothing in the rest of the chart.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This chart has two columns and three rows. The first row is a header row and the second column is labeled \u201cRate times Time is equal to Distance.\u201d There is nothing in the rest of the chart.\" \/><\/span><\/p>\n<p id=\"fs-id1167835335379\">The formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-164fcc881021ce79be78a8adbd303299_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#114;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"53\" style=\"vertical-align: 0px;\" \/> assumes we know <em data-effect=\"italics\">r<\/em> and <em data-effect=\"italics\">t<\/em> and use them to find <em data-effect=\"italics\">D<\/em>. If we know <em data-effect=\"italics\">D<\/em> and <em data-effect=\"italics\">r<\/em> and need to find <em data-effect=\"italics\">t<\/em>, we would solve the equation for <em data-effect=\"italics\">t<\/em> and get the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dcba036d9b04d8b3561ef96c22b0e42f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#68;&#125;&#123;&#114;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/p>\n<p id=\"fs-id1167834095294\">We have also explained how flying with or against the wind affects the speed of a plane. We will revisit that idea in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835274969\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835337384\">\n<div data-type=\"problem\" id=\"fs-id1167834587428\">\n<p>An airplane can fly 200 miles into a 30 mph headwind in the same amount of time it takes to fly 300 miles with a 30 mph tailwind. What is the speed of the airplane?<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167830865629\">This is a uniform motion situation. A diagram will help us visualize the situation.<\/p>\n<table id=\"fs-id1167831920753\" class=\"unnumbered unstyled can-break\" summary=\"A figure shows the uniform motion of an airplane. An arrow labeled \u201c300 miles with the wind; r plus 30\u201d runs parallel to the wind which is labeled \u201cWind; 30 miles per hour.\u201d A second arrow also runs parallel to the wind, is labeled \u201c200 miles against the wind; r minus 30,\u201d and points opposite the first arrow. We fill in the chart to organize the information. The chart has two columns and three rows. The first row is a header row and the second column is labeled \u201cRate times Time is equal to Distance.\u201d The second column is subdivided into three columns for \u201cRate,\u201d \u201cTime,\u201d and \u201cDistance.\u201d The first column is a header column with the first row labeled \u201cHeadwind\u201d and the second row labeled \u201cTailwind.\u201d We are looking for the speed of the airplane. Let r be equal to the speed of the airplane. When the plane flies with the wind, the wind increases its speed and so the rate is r plus 30. When the plane flies against the wind, the wind decreases its speed and the rate is r minus 30. Write the headwind speed, r minus 30, and the tailwind speed r plus 30 in the \u201cRate\u201d column of the chart. Write the distances 200 and 300 in the \u201cDistance\u201d column of the chart. Since D is equal to r times t, we solve for t and get t is equal to D divided by r. We divided the distance by the rate in each row, and place the expressions 200 divided by the quantity r minus 30 for the headwind and 300 divided by the quantity r plus 30 in the \u201cTime\u201d column. We know the times are equal and so we write our equation, 200 divided by the quantity r minus 30 is equal to 300 divided by the quantity r plus 30. We multiply both sides of the equation by the least common denominator, the quantity r plus 30 times the quantity r minus 30. The result is 200 times the quantity r plus 30 is equal to 300 times the quantity r minus 30. Simplify. The equation becomes 200 r plus 6,000 is equal to 300 r minus 9,000, which becomes 15,000 is equal to 100 r. The result is 150 is equal to r. Check. Is 150 miles per hour a reasonable speed for an airplane? Yes. If the plane is traveling 150 miles per hour and the wind is 30 miles per hour, the tailwind is 150 plus 30, which is equal to 180 miles per hour and 300 divided by 180 is equal to five-thirds hours. Also, the headwind is 150 minus 30, which is equal to 120 miles per hour and 200 divided by 120 is equal to Five-thirds hours. The times are equal, so the plane was traveling 150 miles per hour.\" data-label=\"\">\n<tbody>\n<tr>\n<td colspan=\"2\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835331004\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-align=\"left\">We fill in the chart to organize the information.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We are looking for the speed of the airplane.<\/td>\n<td data-align=\"left\">Let <em data-effect=\"italics\">r<\/em> = the speed of the airplane.<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-align=\"left\">When the plane flies with the wind,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the wind increases its speed and so the rate is <em data-effect=\"italics\">r<\/em> + 30.<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-align=\"left\">When the plane flies against the wind,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the wind decreases its speed and the rate is <em data-effect=\"italics\">r<\/em> \u2212 30.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write in the rates.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Write in the distances.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66202b61b641e007970e1b69bff10ee0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#114;&middot;&#116;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"57\" style=\"vertical-align: -4px;\" \/> we solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> and get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dcba036d9b04d8b3561ef96c22b0e42f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#68;&#125;&#123;&#114;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>We divide the distance by the rate in each row, and place the expression in the time column.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831884957\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We know the times are equal and so we write<\/p>\n<div data-type=\"newline\"><\/div>\n<p>our equation.<\/td>\n<td data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0c58628a43740c82eebfbd203287942_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#48;&#125;&#123;&#114;&#45;&#51;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#48;&#125;&#123;&#114;&#43;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"90\" style=\"vertical-align: -8px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We multiply both sides by the LCD.<\/td>\n<td data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a93fa7d1e77005dc196a1c688d948970_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#51;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#51;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#48;&#125;&#123;&#114;&#45;&#51;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#51;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#51;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#48;&#125;&#123;&#114;&#43;&#51;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"391\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6056cb66a2e2a6817a5b71aaad023dc2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#52;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#51;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#48;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#51;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#51;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"219\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2bb3cb67128cabf293d87d5da0130fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#48;&#48;&#114;&#43;&#54;&#48;&#48;&#48;&#61;&#51;&#48;&#48;&#114;&#45;&#57;&#48;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"209\" style=\"vertical-align: -2px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Solve.<\/td>\n<td data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81c5991f9ead643519f6a60787395ca8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#46;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#53;&#48;&#48;&#48;&#61;&#49;&#48;&#48;&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"102\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\"><strong data-effect=\"bold\">Check.<\/strong><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Is 150 mph a reasonable speed for an airplane? Yes.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>If the plane is traveling 150 mph and the wind is 30 mph,<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f32ea0a71687c2a79baeb7650d54b083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#97;&#105;&#108;&#119;&#105;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#49;&#53;&#48;&#43;&#51;&#48;&#61;&#49;&#56;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#112;&#104;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#48;&#125;&#123;&#49;&#56;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#111;&#117;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#101;&#97;&#100;&#119;&#105;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#49;&#53;&#48;&#45;&#51;&#48;&#61;&#49;&#50;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#112;&#104;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#48;&#125;&#123;&#49;&#50;&#48;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#111;&#117;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"386\" style=\"vertical-align: -18px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The times are equal, so it checks.<\/td>\n<td data-align=\"left\">The plane was traveling 150 mph.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826927156\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831880816\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167831919776\">Link can ride his bike 20 miles into a 3 mph headwind in the same amount of time he can ride 30 miles with a 3 mph tailwind. What is Link\u2019s biking speed?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830699550\">\n<p id=\"fs-id1167835530027\">Link\u2019s biking speed is 15 mph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835420100\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167826813961\">\n<div data-type=\"problem\" id=\"fs-id1167835365574\">\n<p id=\"fs-id1167831923144\">Danica can sail her boat 5 miles into a 7 mph headwind in the same amount of time she can sail 12 miles with a 7 mph tailwind. What is the speed of Danica\u2019s boat without a wind?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835188060\">\n<p id=\"fs-id1167835346588\">The speed of Danica\u2019s boat is 17 mph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831823931\">In the next example, we will know the total time resulting from travelling different distances at different speeds.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834593545\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167830757319\">\n<div data-type=\"problem\" id=\"fs-id1167834228747\">\n<p id=\"fs-id1167834146995\">Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835333772\">\n<p id=\"fs-id1167834367171\">This is a uniform motion situation. A diagram will help us visualize the situation.<\/p>\n<table id=\"fs-id1167835348572\" class=\"unnumbered unstyled can-break\" summary=\"A figure shows the uniform motion of an airplane. An arrow labeled \u201c300 miles with the wind; r plus 30\u201d runs parallel to the wind which is labeled \u201cWind; 30 miles per hour.\u201d A second arrow also runs parallel to the wind, is labeled \u201c200 miles against the wind; r minus 30,\u201d and points opposite the first arrow. We fill in the chart to organize the information. The chart has two columns and three rows. The first row is a header row and the second column is labeled \u201cRate times Time is equal to Distance.\u201d The second column is subdivided into three columns for \u201cRate,\u201d \u201cTime,\u201d and \u201cDistance.\u201d The first column is a header column with the first row labeled \u201cHeadwind\u201d and the second row labeled \u201cTailwind.\u201d We are looking for the speed of the airplane. Let r be equal to the speed of the airplane. When the plane flies with the wind, the wind increases its speed and so the rate is r plus 30. When the plane flies against the wind, the wind decreases its speed and the rate is r minus 30. Write the headwind speed, r minus 30, and the tailwind speed r plus 30 in the \u201cRate\u201d column of the chart. Write the distances 200 and 300 in the \u201cDistance\u201d column of the chart. Since D is equal to r times t, we solve for t and get t is equal to D divided by r. We divided the distance by the rate in each row, and place the expressions 200 divided by the quantity r minus 30 for the headwind and 300 divided by the quantity r plus 30 in the \u201cTime\u201d column. We know the times are equal and so we write our equation, 200 divided by the quantity r minus 30 is equal to 300 divided by the quantity r plus 30. We multiply both sides of the equation by the least common denominator, the quantity r plus 30 times the quantity r minus 30. The result is 200 times the quantity r plus 30 is equal to 300 times the quantity r minus 30. Simplify. The equation becomes 200 r plus 6,000 is equal to 300 r minus 9,000, which becomes 15,000 is equal to 100 r. The result is 150 is equal to r. Check. Is 150 miles per hour a reasonable speed for an airplane? Yes. If the plane is traveling 150 miles per hour and the wind is 30 miles per hour, the tailwind is 150 plus 30, which is equal to 180 miles per hour and 300 divided by 180 is equal to five-thirds hours. Also, the headwind is 150 minus 30, which is equal to 120 miles per hour and 200 divided by 120 is equal to Five-thirds hours. The times are equal, so the plane was traveling 150 miles per hour.\" data-label=\"\">\n<tbody>\n<tr>\n<td colspan=\"2\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835269037\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_011a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-align=\"left\">We fill in the chart to organize the information.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We are looking for Jazmine\u2019s running speed.<\/td>\n<td data-align=\"left\">Let <em data-effect=\"italics\">r<\/em> = Jazmine\u2019s running speed.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Her biking speed is 4 miles faster than her<\/p>\n<div data-type=\"newline\"><\/div>\n<p>running speed.<\/td>\n<td data-align=\"left\"><em data-effect=\"italics\">r<\/em> + 4 = her biking speed<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The distances are given, enter them into the chart.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66202b61b641e007970e1b69bff10ee0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#114;&middot;&#116;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"57\" style=\"vertical-align: -4px;\" \/> we solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> and get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dcba036d9b04d8b3561ef96c22b0e42f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#68;&#125;&#123;&#114;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>We divide the distance by the rate in each row, and place the expression in the time column.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835234049\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_011b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write a word sentence.<\/td>\n<td data-align=\"left\">Her time plus the time biking is 3 hours.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Translate the sentence to get the equation.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4bc212587051de15aad699ee7b78dff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#114;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#52;&#125;&#123;&#114;&#43;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"109\" style=\"vertical-align: -8px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8607109cee4a00110f2c0a3d1a644bf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#114;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#114;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#52;&#125;&#123;&#114;&#43;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&middot;&#114;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#52;&#114;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#114;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#114;&#43;&#51;&#50;&#43;&#50;&#52;&#114;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#114;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#50;&#43;&#51;&#50;&#114;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#50;&#114;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#51;&#123;&#114;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#48;&#114;&#45;&#51;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#114;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"140\" width=\"312\" style=\"vertical-align: -64px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e10d760b2c3d6ba05b0d857289758125_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#114;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#114;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"203\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4b4d9290724a268c919956ddfd4caaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#114;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#114;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"126\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Check.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>A negative speed does not make sense in this problem,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca3085d981ac492e6a62aabf76e4de69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/> is the solution.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Is 8 mph a reasonable running speed? Yes.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>If Jazmine\u2019s running rate is 4, then her biking rate,<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51782f2ac3758ffc5e2dfc804edc0136_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#43;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"43\" style=\"vertical-align: -4px;\" \/> which is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da04056064d4915c566feeca00c6a462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#43;&#52;&#61;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"85\" style=\"vertical-align: -2px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84b0f11b64848c0dc71ce93bfa91b05f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#117;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#112;&#104;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#108;&#101;&#115;&#125;&#125;&#123;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#112;&#104;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#111;&#117;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#66;&#105;&#107;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#112;&#104;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#105;&#108;&#101;&#115;&#125;&#125;&#123;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#112;&#104;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#111;&#117;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"51\" width=\"263\" style=\"vertical-align: -21px;\" \/><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8ae7c56823cf6d7cf81b9df6828e6e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>Total 3 hours.<\/td>\n<td data-align=\"left\">Jazmine\u2019s running speed is 8 mph.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835254604\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835377806\">\n<div data-type=\"problem\" id=\"fs-id1167835300559\">\n<p id=\"fs-id1167834377240\">Dennis went cross-country skiing for 6 hours on Saturday. He skied 20 mile uphill and then 20 miles back downhill, returning to his starting point. His uphill speed was 5 mph slower than his downhill speed. What was Dennis\u2019 speed going uphill and his speed going downhill?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835225817\">\n<p id=\"fs-id1167826937400\">Dennis\u2019s uphill speed was 10 mph and his downhill speed was 5 mph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835420116\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835332653\">\n<div data-type=\"problem\" id=\"fs-id1167834135092\">\n<p id=\"fs-id1167834064113\">Joon drove 4 hours to his home, driving 208 miles on the interstate and 40 miles on country roads. If he drove 15 mph faster on the interstate than on the country roads, what was his rate on the country roads?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834462842\">\n<p id=\"fs-id1167834533387\">Joon\u2019s rate on the country roads is 50 mph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835339905\">Once again, we will use the uniform motion formula solved for the variable <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4998f61a094184afa02f41dd4ab518c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"example\" id=\"fs-id1167831823993\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834219719\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167832057221\">Hamilton rode his bike downhill 12 miles on the river trail from his house to the ocean and then rode uphill to return home. His uphill speed was 8 miles per hour slower than his downhill speed. It took him 2 hours longer to get home than it took him to get to the ocean. Find Hamilton\u2019s downhill speed.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835306833\">\n<p id=\"fs-id1167835302060\">This is a uniform motion situation. A diagram will help us visualize the situation.<\/p>\n<table id=\"fs-id1167834184998\" class=\"unnumbered unstyled can-break\" summary=\"A figure shows the uniform motion of a bike. An arrow is labeled \u201c12 miles.\u201d A second arrow in the opposite direction is labeled \u201c8 miles per hour slower; 2 hours longer.\u201d We fill in the chart to organize the information. The first row is a header row and the second column is labeled \u201cRate times Time is equal to Distance.\u201d The second column is subdivided into three columns for \u201cRate,\u201d \u201cTime,\u201d and \u201cDistance.\u201d The first column is a header column with the first row labeled \u201cDownhill\u201d and the second row labeled \u201cUphill.\u201d We are looking for Hamilton\u2019s downhill speed. Let h be equal to Hamilton\u2019s downhill speed. His uphill speed is 8 miles per hour slower. Let h minus 8 be equal to Hamilton\u2019s uphill speed. Write the rates, h and the quantity h minus 8, in the \u201cRate\u201d column. The distance is the same in both directions, 12 miles. Since D is equal to r times t, we solve for t and get t is equal to D divided by r. We divide the distance by the rate in each row and place the expressions in the \u201ctime\u201d column. The downhill time is 12 divided by h. The uphill time is 12 divided by the quantity h minus 8. Write a word sentence about the time. He took 2 hours longer uphill than downhill. The uphill time is 2 more than the downhill time. Translate the sentence to get the equation. The equation is 12 divided by the quantity h minus 8 is equal to the sum of 12 divided by h and 2. Solve the equation by multiplying each side by the least common denominator h times the quantity h minus 8. The result is h times the quantity h minus 8 times 12 divided by the quantity h minus 8 is equal to h times the quantity h minus 8 times the sum of 12 divided by h and 2. When completely simplified, the result is 0 is 0 is equal to 2 h squared minus 16 h minus 96. Notice that the factor 2 can be removed on the right side. The equation becomes 0 is equal to 2 times the quantity h squared minus 8 h minus 48. Factoring the right side, the equation becomes 0 is equal to 2 times the quantity h minus 12 times the quantity h plus 4, which means h minus 12 is equal to 0 or h plus 4 is equal to 0. h is equal to 12 can be a solution, but h is equal to negative 4 cannot. Check. Is 12 miles per hour a s reasonable speed for biking downhill. Yes. The downhill speed is 12 miles per hour, so the time is 12 miles divided by 12 miles per hour is equal to 1 hour. The uphill speed is 12 minus 8 is equal to 4 miles per hour, so the time is 12 miles divided by 4 miles per hour is equal to 3 hours. The uphill time is 2 hours more than the downhill time. Hamilton\u2019s downhill speed is 12 miles per hour.\" data-label=\"\">\n<tbody>\n<tr>\n<td colspan=\"2\" data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831071365\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_012a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\" data-align=\"left\">We fill in the chart to organize the information.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We are looking for Hamilton\u2019s downhill speed.<\/td>\n<td data-align=\"left\">Let <em data-effect=\"italics\">h<\/em> = Hamilton\u2019s downhill speed.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">His uphill speed is 8 miles per hour slower.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Enter the rates into the chart.<\/td>\n<td data-align=\"left\"><em data-effect=\"italics\">h<\/em> \u2212 8 = Hamilton\u2019s uphill speed<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The distance is the same in both directions.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>12 miles.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66202b61b641e007970e1b69bff10ee0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#114;&middot;&#116;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"57\" style=\"vertical-align: -4px;\" \/> we solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\" \/> and get <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dcba036d9b04d8b3561ef96c22b0e42f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#68;&#125;&#123;&#114;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>We divide the distance by the rate in each row, and place the expression in the time column.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835376704\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write a word sentence about the line.<\/td>\n<td data-align=\"left\">He took 2 hours longer uphill than downhill.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>The uphill time is 2 more than the downhill time.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Translate the sentence to get the equation.<\/td>\n<td data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c5b54df0640d391f2b365cc470e4405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#104;&#45;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#104;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"106\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve.<\/td>\n<td data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2628a50dbd00809e1c4036a90ecac30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#104;&#45;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#104;&#125;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#50;&#104;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#50;&#104;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#50;&#104;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#50;&#104;&#45;&#57;&#54;&#43;&#50;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#104;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#104;&#45;&#57;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#104;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#104;&#45;&#52;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#57;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#45;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#43;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#104;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#104;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"229\" width=\"340\" style=\"vertical-align: -109px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Check.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Is 12 mph a reasonable speed for biking downhill? Yes.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-450e278eda3ac1625ec7df95309bab6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#111;&#119;&#110;&#104;&#105;&#108;&#108;&#125;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#32;&#109;&#112;&#104;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#32;&#109;&#105;&#108;&#101;&#115;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#32;&#109;&#112;&#104;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#104;&#111;&#117;&#114;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#112;&#104;&#105;&#108;&#108;&#125;&#38;&#32;&#49;&#50;&#45;&#56;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#32;&#109;&#112;&#104;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#32;&#109;&#105;&#108;&#101;&#115;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#32;&#109;&#112;&#104;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#32;&#104;&#111;&#117;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"51\" width=\"384\" style=\"vertical-align: -21px;\" \/><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The uphill time is 2 hours more that the downhill time.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\">Hamilton\u2019s downhill speed is 12 mph.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167830699110\">\n<div data-type=\"problem\" id=\"fs-id1167826874280\">\n<p id=\"fs-id1167832066052\">Kayla rode her bike 75 miles home from college one weekend and then rode the bus back to college. It took her 2 hours less to ride back to college on the bus than it took her to ride home on her bike, and the average speed of the bus was 10 miles per hour faster than Kayla\u2019s biking speed. Find Kayla\u2019s biking speed.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831923485\">\n<p id=\"fs-id1167835381684\">Kayla\u2019s biking speed was 15 mph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826880259\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835334970\">\n<div data-type=\"problem\" id=\"fs-id1167834084978\">\n<p id=\"fs-id1167830914996\">Victoria jogs 12 miles to the park along a flat trail and then returns by jogging on an 20 mile hilly trail. She jogs 1 mile per hour slower on the hilly trail than on the flat trail, and her return trip takes her two hours longer. Find her rate of jogging on the flat trail.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826782738\">\n<p id=\"fs-id1167835479367\">Victoria jogged 6 mph on the flat trail.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835497506\">\n<h3 data-type=\"title\">Solve Work Applications<\/h3>\n<p id=\"fs-id1167835305013\">The weekly gossip magazine has a big story about the Princess\u2019 baby and the editor wants the magazine to be printed as soon as possible. She has asked the printer to run an extra printing press to get the printing done more quickly. Press #1 takes 6 hours to do the job and Press #2 takes 12 hours to do the job. How long will it take the printer to get the magazine printed with both presses running together?<\/p>\n<p id=\"fs-id1167832060221\">This is a typical \u2018work\u2019 application. There are three quantities involved here\u2014the time it would take each of the two presses to do the job alone and the time it would take for them to do the job together.<\/p>\n<p id=\"fs-id1165927750470\">If Press #1 can complete the job in 6 hours, in one hour it would complete <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c27e6c52f9cb0bf70516c13f69448e9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> of the job.<\/p>\n<p id=\"fs-id1165928009731\">If Press #2 can complete the job in 12 hours, in one hour it would complete <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e697137c2fc49524ec52982e9a9dc34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/> of the job.<\/p>\n<p id=\"fs-id1167834138137\">We will let <em data-effect=\"italics\">t<\/em> be the number of hours it would take the presses to print the magazines with both presses running together. So in 1 hour working together they have completed <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f281d05f2a22766256542610870950ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> of the job.<\/p>\n<p id=\"fs-id1167832134011\">We can model this with the word equation and then translate to a rational equation. To find the time it would take the presses to complete the job if they worked together, we solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4998f61a094184afa02f41dd4ab518c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167834376982\">A chart will help us organize the information. We are looking for how many hours it would take to complete the job with both presses running together.<\/p>\n<table id=\"fs-id1167835569736\" class=\"unnumbered unstyled can-break\" summary=\"A figure shows the uniform motion of a bike. An arrow is labeled \u201c12 miles.\u201d A second arrow in the opposite direction is labeled \u201c8 miles per hour slower; 2 hours longer.\u201d We fill in the chart to organize the information. The first row is a header row and the second column is labeled \u201cRate times Time is equal to Distance.\u201d The second column is subdivided into three columns for \u201cRate,\u201d \u201cTime,\u201d and \u201cDistance.\u201d The first column is a header column with the first row labeled \u201cDownhill\u201d and the second row labeled \u201cUphill.\u201d We are looking for Hamilton\u2019s downhill speed. Let h be equal to Hamilton\u2019s downhill speed. His uphill speed is 8 miles per hour slower. Let h minus 8 be equal to Hamilton\u2019s uphill speed. Write the rates, h and the quantity h minus 8, in the \u201cRate\u201d column. The distance is the same in both directions, 12 miles. Since D is equal to r times t, we solve for t and get t is equal to D divided by r. We divide the distance by the rate in each row and place the expressions in the \u201ctime\u201d column. The downhill time is 12 divided by h. The uphill time is 12 divided by the quantity h minus 8. Write a word sentence about the time. He took 2 hours longer uphill than downhill. The uphill time is 2 more than the downhill time. Translate the sentence to get the equation. The equation is 12 divided by the quantity h minus 8 is equal to the sum of 12 divided by h and 2. Solve the equation by multiplying each side by the least common denominator h times the quantity h minus 8. The result is h times the quantity h minus 8 times 12 divided by the quantity h minus 8 is equal to h times the quantity h minus 8 times the sum of 12 divided by h and 2. When completely simplified, the result is 0 is 0 is equal to 2 h squared minus 16 h minus 96. Notice that the factor 2 can be removed on the right side. The equation becomes 0 is equal to 2 times the quantity h squared minus 8 h minus 48. Factoring the right side, the equation becomes 0 is equal to 2 times the quantity h minus 12 times the quantity h plus 4, which means h minus 12 is equal to 0 or h plus 4 is equal to 0. h is equal to 12 can be a solution, but h is equal to negative 4 cannot. Check. Is 12 miles per hour a s reasonable speed for biking downhill. Yes. The downhill speed is 12 miles per hour, so the time is 12 miles divided by 12 miles per hour is equal to 1 hour. The uphill speed is 12 minus 8 is equal to 4 miles per hour, so the time is 12 miles divided by 4 miles per hour is equal to 3 hours. The uphill time is 2 hours more than the downhill time. Hamilton\u2019s downhill speed is 12 miles per hour.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-align=\"left\">Let <em data-effect=\"italics\">t<\/em> = the number of hours needed to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>complete the job together.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\" data-valign=\"top\">Enter the hours per job for Press #1,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Press #2, and when they work together.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>If a job on Press #1 takes 6 hours, then in 1 hour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c27e6c52f9cb0bf70516c13f69448e9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> of the job is completed.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Similarly find the part of the job completed\/hours for Press #2 and when thet both together.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>Write a word sentence.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832134116\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_014a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\">The part completed by Press #1 plus the part completed by Press #2 equals the amount completed together.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Translate into an equation.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834308133\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_014b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Solve.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826798768\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_014c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Mutiply by the LCD, 12<em data-effect=\"italics\">t<\/em><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832056805\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_014d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835418136\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_014e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\">When both presses are running it<\/p>\n<div data-type=\"newline\"><\/div>\n<p>takes 4 hours to do the job.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Keep in mind, it should take less time for two presses to complete a job working together than for either press to do it alone.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835325647\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835287848\">\n<div data-type=\"problem\" id=\"fs-id1167826996733\">\n<p>Suppose Pete can paint a room in 10 hours. If he works at a steady pace, in 1 hour he would paint <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9948ce98461082e7c8558675198f9ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/> of the room. If Alicia would take 8 hours to paint the same room, then in 1 hour she would paint <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9de931d332a83dd3da0c1ba21347996c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> of the room. How long would it take Pete and Alicia to paint the room if they worked together (and didn\u2019t interfere with each other\u2019s progress)?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834432456\">\n<p id=\"fs-id1167835530396\">This is a \u2018work\u2019 application. A chart will help us organize the information. We are looking for the numbers of hours it will take them to paint the room together.<\/p>\n<p id=\"fs-id1167834282673\">In one hour Pete did <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9948ce98461082e7c8558675198f9ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/> of the job. Alicia did <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9de931d332a83dd3da0c1ba21347996c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> of the job. And together they did <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f281d05f2a22766256542610870950ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> of the job.<\/p>\n<table id=\"fs-id1167835370917\" class=\"unnumbered unstyled can-break\" summary=\"Let t be equal to the number of hours needed to paint the room together. The chart has three columns and four rows. The first row is a header row and it labels the first column \u201cNumber of hours to complete the job.\u201d and the second column \u201cPart of job completed or hour\u201d. The first column is a header column and it labels the first row \u201cPete\u201d, the second row \u201cAlicia\u201d, and the third row \u201cTogether\u201d. Enter the hours per job for Pete, Alicia, and when they work together. They are 10, 8, and t. In 1 hour of working together, they have completed 1 divided by t of the job. Similarly find the part of the job completed by Pete, and then by Alicia. They are one-tenth of the job and one-eighth of the job. Write a word sentence. The work completed by Pete plus the work completed by Alicia equals the total work completed. When the sentence is translated into an equation, the result is one-tenth plus one-eighth is equal to 1 divided by t. To solve the equation, multiply by the least common denominator, 40 t. The result is 40 t times the sum of one-tenth and one-eighth is equal to 40 t times the quantity 1 divided by t. Distribute 40 t. When simplified, the equation becomes 4 t plus 5 t is equal to 40. The solution is t is equal to 40 divided by 9. We\u2019ll write it as a mixed number so that we can convert it to hours and minutes. The solution is t is equal to 4 and four-ninths hours. Remember 1 hour is equal to 60 minutes. Multiply, and then round to the nearest minute to convert the fraction. The result is t is equal to 4 hours plus four-ninths times 60 minutes. So, t is equal to 4 hours plus 27 minutes. It would take Pete and Alicia about 4 hours and 27 minutes to paint the room.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-align=\"left\">Let <em data-effect=\"italics\">t<\/em> be the number of hours needed<\/p>\n<div data-type=\"newline\"><\/div>\n<p>to paint the room together.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Enter the hours per job for Pete, Alicia, and when they work together.<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p>In 1 hour working together, they have completed <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f281d05f2a22766256542610870950ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> of the job.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Similarly, find the part of the job<\/p>\n<div data-type=\"newline\"><\/div>\n<p>completed\/hour by Pete and then by Alicia.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835390377\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Write a word sentence.<\/td>\n<td data-align=\"left\">The work completed by Pete plus the work<\/p>\n<div data-type=\"newline\"><\/div>\n<p>completed by Alicia equals the total<\/p>\n<div data-type=\"newline\"><\/div>\n<p>work completed.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b0c01809498b952e2c488a85b3a8da6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>Work completed by:<\/p>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167835319199\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <span data-type=\"media\" id=\"fs-id1167835300385\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span> <\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Multiply by the LCD, 40<em data-effect=\"italics\">t<\/em>.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835284770\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Distribute.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835301933\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify and solve.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835417811\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We\u2019ll write as a mixed number<\/p>\n<div data-type=\"newline\"><\/div>\n<p>so that we can convert it to hours<\/p>\n<div data-type=\"newline\"><\/div>\n<p>and minutes.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Remember, 1 hour = 60 minutes.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830704125\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Multiply, and then round to the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>nearest minute.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834196502\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_013i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"center\">It would take Pete and Alica about<\/p>\n<div data-type=\"newline\"><\/div>\n<p>4 hours and 27 minutes to paint the room.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832065583\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831107028\">\n<div data-type=\"problem\" id=\"fs-id1167831107030\">\n<p id=\"fs-id1167835511112\">One gardener can mow a golf course in 4 hours, while another gardener can mow the same golf course in 6 hours. How long would it take if the two gardeners worked together to mow the golf course?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835530131\">\n<p id=\"fs-id1167834131823\">When the two gardeners work together it takes 2 hours and 24 minutes.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831908325\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831959454\">\n<div data-type=\"problem\" id=\"fs-id1167831959457\">\n<p id=\"fs-id1167831040305\">Daria can weed the garden in 7 hours, while her mother can do it in 3. How long will it take the two of them working together?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834219754\">\n<p id=\"fs-id1167834527691\">When Daria and her mother work together it takes 2 hours and 6 minutes.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167834196208\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835186400\">\n<div data-type=\"problem\" id=\"fs-id1167835186402\">\n<p id=\"fs-id1167831955876\">Ra\u2019shon can clean the house in 7 hours. When his sister helps him it takes 3 hours. How long does it take his sister when she cleans the house alone?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831955879\">\n<p id=\"fs-id1167835326432\">This is a work problem. A chart will help us organize the information.<\/p>\n<p id=\"fs-id1167835254281\">We are looking for how many hours it would take Ra\u2019shon\u2019s sister to complete the job by herself.<\/p>\n<table id=\"fs-id1167826978190\" class=\"unnumbered unstyled can-break\" summary=\"Let s be equal to the number of hours Ra\u2019shon\u2019s sister takes to clean the house alone. The chart has three columns and four rows. The first row is a header row and it labels the first column \u201cNumber of hours to clean the house\u201d and the second column \u201cPart of job completed or hour\u201d. The first column is a header column and it labels the first row \u201cRa\u2019shon\u201d, the second row \u201cHis sister\u201d, and the third row \u201cTogether\u201d. Enter the hours per job for Ra\u2019shon, his sister, and when they work together. They are 7, s, and 3. If Ras\u2019shon takes 7 hours, then in 1 hour, one-seventh of the job is completed. If Ra\u2019shon\u2019s sister takes s hours, then in 1 hour, 1 divided by s of the job is completed. Write a word sentence. The part completed by Ras\u2019shon plus the part completed by his sister equals the amount completed together. Translate to an equation. One-seventh is equal to 1 divided by s is equal to one-third. Solve the equation. Multiply by the least common denominator, 21 s. The result is 21 s times sum of one-seventh and one divided by s is equal to 21 s times one-third. When simplified, the equation becomes 3 s plus 21 is equal to 7 s. When solved for s, the result is s is equal to 5 and one-fourth hours. There are 60 minutes in 1 hour, so s is equal to 5 hours plus one-fourth times 60 minutes. That means s is equal to 5 hours plus 15 minutes. It would take Ra\u2019Shon\u2019s sister 5 hours and 15 minutes to clean the house alone.\" data-label=\"\">\n<tbody>\n<tr>\n<td data-align=\"left\">Let <em data-effect=\"italics\">s<\/em> be the number of hours Ra\u2019shon\u2019s<\/p>\n<div data-type=\"newline\"><\/div>\n<p>sister takes to clean the house alone.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Enter the hours per job for Ra\u2019shon, his<\/p>\n<div data-type=\"newline\"><\/div>\n<p>sister, and when they work together.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>If Ra\u2019shon takes 7 hours, then in 1 hour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad9e98aa7250277761bf01899f916f02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>of the job is completed.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>If Ra\u2019shon\u2019s sister takes <em data-effect=\"italics\">s<\/em> hours, then in<\/p>\n<div data-type=\"newline\"><\/div>\n<p>1 hour <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56f3096c4cfcac2ff29e4aab6a43dcbe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> of the job is completed.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835198865\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write a word sentence.<\/td>\n<td data-align=\"left\">The part completed by Ra\u2019shon plus the part<\/p>\n<div data-type=\"newline\"><\/div>\n<p>by his sister equals the amount completed together.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Translate to an equation.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835280078\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Solve.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Multiply by the LCD, 21<em data-effect=\"italics\">s<\/em>.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835368800\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835509964\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write as a mixed number to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>convert it to hours and minutes.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834191154\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">There are 60 minutes in 1 hour.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835309911\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_020g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\">It would take Ra\u2019shon\u2019s sister 5 hours and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>15 minutes to clean the house alone.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831025386\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835343869\">\n<div data-type=\"problem\" id=\"fs-id1167835343871\">\n<p>Alice can paint a room in 6 hours. If Kristina helps her it takes them 4 hours to paint the room. How long would it take Kristina to paint the room by herself?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835354966\">\n<p id=\"fs-id1167835340158\">Kristina can paint the room in 12 hours.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835422085\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834257815\">\n<div data-type=\"problem\" id=\"fs-id1167834196340\">\n<p id=\"fs-id1167834196342\">Tracy can lay a slab of concrete in 3 hours, with Jordan\u2019s help they can do it in 2 hours. If Jordan works alone, how long will it take?<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p>It will take Jordan 6 hours.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835351315\">\n<h3 data-type=\"title\">Solve Direct Variation Problems<\/h3>\n<p id=\"fs-id1167831106792\">When two quantities are related by a proportion, we say they are <em data-effect=\"italics\">proportional<\/em> to each other. Another way to express this relation is to talk about the <em data-effect=\"italics\">variation<\/em> of the two quantities. We will discuss direct variation and inverse variation in this section.<\/p>\n<p id=\"fs-id1167826880152\">Lindsay gets paid ?15 per hour at her job. If we let <em data-effect=\"italics\">s<\/em> be her salary and <em data-effect=\"italics\">h<\/em> be the number of hours she has worked, we could model this situation with the equation<\/p>\n<div data-type=\"equation\" id=\"fs-id1167830836766\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2a373cac1b3db063e888e7dc49b26366_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#49;&#53;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"60\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"fs-id1167835344749\">Lindsay\u2019s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay\u2019s salary <em data-effect=\"italics\">varies directly<\/em> with the number of hours she works. Two variables vary directly if one is the product of a constant and the other.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835374810\">\n<div data-type=\"title\">Direct Variation<\/div>\n<p id=\"fs-id1167835332296\">For any two variables <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>, <em data-effect=\"italics\">y<\/em> varies directly with <em data-effect=\"italics\">x<\/em> if<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835304679\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-633b75e930b1747ca7d4de1bcc09b20c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#107;&#120;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#101;&#114;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#107;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"156\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167832058585\">The constant <em data-effect=\"italics\">k<\/em> is called the constant of variation.<\/p>\n<\/div>\n<p id=\"fs-id1167835229188\">In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y.<\/em> Then we can use that equation to find values of <em data-effect=\"italics\">y<\/em> for other values of <em data-effect=\"italics\">x<\/em>.<\/p>\n<p id=\"fs-id1167834346195\">We\u2019ll list the steps here.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834346198\" class=\"howto\">\n<div data-type=\"title\">Solve direct variation problems.<\/div>\n<ol id=\"fs-id1167835417621\" type=\"1\" class=\"stepwise\">\n<li>Write the formula for direct variation.<\/li>\n<li>Substitute the given values for the variables.<\/li>\n<li>Solve for the constant of variation.<\/li>\n<li>Write the equation that relates <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> using the constant of variation.<\/li>\n<\/ol>\n<\/div>\n<p>Now we\u2019ll solve an application of direct variation.<\/p>\n<div data-type=\"example\" id=\"fs-id1167832150888\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832150890\">\n<div data-type=\"problem\" id=\"fs-id1167835330703\">\n<p id=\"fs-id1167835330705\">When Raoul runs on the treadmill at the gym, the number of calories, <em data-effect=\"italics\">c<\/em>, he burns varies directly with the number of minutes, <em data-effect=\"italics\">m<\/em>, he uses the treadmill. He burned 315 calories when he used the treadmill for 18 minutes.<\/p>\n<p id=\"fs-id1167834186254\"><span class=\"token\">\u24d0<\/span> Write the equation that relates <em data-effect=\"italics\">c<\/em> and <em data-effect=\"italics\">m<\/em>. <span class=\"token\">\u24d1<\/span> How many calories would he burn if he ran on the treadmill for 25 minutes?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835354913\">\n<p id=\"fs-id1167835354915\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834279347\" class=\"unnumbered unstyled can-break\" summary=\"The number of calories, c, varies directly with the number of minutes, m, on a treadmill, and c is equal to 315 and m is equal to 18. Write the formula for direct variation, y is equal to k times x. We will use c in place of y and m in place of x. So, the equation is c is equal to k times m instead. Substitute the given values for the variables. The result is 315 is equal to k times 18. Solve for the constant of variation by dividing each side of the equation by 18. The result is 17.5 is equal to k. Write the equation that relates c and m. The equation is c is equal to k times m. Substitute the constant of variation. The result is c is equal to 17.5 times m.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\">The number of calories, <em data-effect=\"italics\">c<\/em>, varies directly with<\/p>\n<div data-type=\"newline\"><\/div>\n<p>the number of minutes, <em data-effect=\"italics\">m<\/em>, on the treadmill,<\/p>\n<div data-type=\"newline\"><\/div>\n<p>and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-176f1672af90697069ffb74747f11bc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#51;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"57\" style=\"vertical-align: -1px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2369a6b292f4d48382c8eec6a71d0e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: -1px;\" \/>.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write the formula for direct variation.<\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167830700596\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We will use <em data-effect=\"italics\">c<\/em> in place of <em data-effect=\"italics\">y<\/em> and <em data-effect=\"italics\">m<\/em> in place of <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167835339267\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Substitute the given values for the variables.<\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831148800\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Solve for the constant of variation.<\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167831103299\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831879987\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write the equation that relates <em data-effect=\"italics\">c<\/em> and <em data-effect=\"italics\">m<\/em>.<\/td>\n<td data-align=\"center\"><span data-type=\"media\" id=\"fs-id1167834053655\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Substitute in the constant of variation.<\/td>\n<td data-align=\"center\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_015g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167834432629\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167832081890\" class=\"unnumbered unstyled can-break\" summary=\"Find c when m is equal to 25. Write the equation that relates c to m. the equation is c is equal to 17.5 m. Substitute the given value for m. The result is c is equal to 17.5 times 25. Simplify the equation. The result is c is equal to 4375. Raoul would burn 437.5 calories if he used the treadmill for 25 minutes.\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd7685e83e17dfc4e2461b96fde6243a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#48;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>Find <em data-effect=\"italics\">c<\/em> when <em data-effect=\"italics\">m<\/em> = 25.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write the equation that relates <em data-effect=\"italics\">c<\/em> and <em data-effect=\"italics\">m<\/em>.<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eb9ee1e7091a42d510e7aaeb0bee668a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2528cba283016a959c999b0d945c9b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><span data-type=\"media\" id=\"fs-id1167835380271\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_016a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Substitute the given value for <em data-effect=\"italics\">m<\/em>.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2528cba283016a959c999b0d945c9b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><span data-type=\"media\" id=\"fs-id1167834433091\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_016b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2528cba283016a959c999b0d945c9b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><span data-type=\"media\" id=\"fs-id1167835236743\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_016c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2528cba283016a959c999b0d945c9b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>Raoul would burn 437.5 calories if<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2528cba283016a959c999b0d945c9b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>he used the treadmill for 25 minutes.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835533838\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832054682\">\n<div data-type=\"problem\" id=\"fs-id1167832054685\">\n<p id=\"fs-id1167832054687\">The number of calories, <em data-effect=\"italics\">c<\/em>, burned varies directly with the amount of time, <em data-effect=\"italics\">t<\/em>, spent exercising. Arnold burned 312 calories in 65 minutes exercising.<\/p>\n<p id=\"fs-id1167835369286\"><span class=\"token\">\u24d0<\/span> Write the equation that relates <em data-effect=\"italics\">c<\/em> and <em data-effect=\"italics\">t<\/em>. <span class=\"token\">\u24d1<\/span> How many calories would he burn if he exercises for 90 minutes?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835333023\">\n<p id=\"fs-id1167834133512\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-452c3f3f0c51962074065a7c7f2ccc9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#52;&#46;&#56;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: -1px;\" \/><span class=\"token\">\u24d1<\/span> He would burn 432 calories.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835362159\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834532325\">\n<div data-type=\"problem\" id=\"fs-id1167834532327\">\n<p id=\"fs-id1167834532329\">The distance a moving body travels, <em data-effect=\"italics\">d<\/em>, varies directly with time, <em data-effect=\"italics\">t<\/em>, it moves. A train travels 100 miles in 2 hours<\/p>\n<p id=\"fs-id1167834247063\"><span class=\"token\">\u24d0<\/span> Write the equation that relates <em data-effect=\"italics\">d<\/em> and <em data-effect=\"italics\">t<\/em>. <span class=\"token\">\u24d1<\/span> How many miles would it travel in 5 hours?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835306694\">\n<p id=\"fs-id1167831116771\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e612640c5648cac6eb2e7dfff6f806d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#53;&#48;&#116;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"57\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span> It would travel 250 miles.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835355996\">\n<h3 data-type=\"title\">Solve Inverse Variation Problems<\/h3>\n<p>Many applications involve two variable that <em data-effect=\"italics\">vary inversely<\/em>. As one variable increases, the other decreases. The equation that relates them is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ffd924ef08d74a4c7ce57f773af6c443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#107;&#125;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"49\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"note\" id=\"fs-id1167835363556\">\n<div data-type=\"title\">Inverse Variation<\/div>\n<p id=\"fs-id1167835363561\">For any two variables <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>, <em data-effect=\"italics\">y<\/em> varies inversely with <em data-effect=\"italics\">x<\/em> if<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835514630\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27b9045ba07a01e45ae57fd2dac08340_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#107;&#125;&#123;&#120;&#125;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#101;&#114;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#107;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"147\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>The constant <em data-effect=\"italics\">k<\/em> is called the constant of variation.<\/p>\n<\/div>\n<p id=\"fs-id1167835417942\">The word \u2018inverse\u2019 in inverse variation refers to the multiplicative inverse. The multiplicative inverse of <em data-effect=\"italics\">x<\/em> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c03d1daf4bbfac1ecf047599f344cc8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<p id=\"fs-id1167835320238\">We solve inverse variation problems in the same way we solved direct variation problems. Only the general form of the equation has changed. We will copy the procedure box here and just change \u2018direct\u2019 to \u2018inverse\u2019.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835489429\" class=\"howto\">\n<div data-type=\"title\">Solve inverse variation problems.<\/div>\n<ol id=\"fs-id1167831880447\" type=\"1\" class=\"stepwise\">\n<li>Write the formula for inverse variation.<\/li>\n<li>Substitute the given values for the variables.<\/li>\n<li>Solve for the constant of variation.<\/li>\n<li>Write the equation that relates <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em> using the constant of variation.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835512316\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835512318\">\n<div data-type=\"problem\" id=\"fs-id1167835512320\">\n<p id=\"fs-id1167834252390\">The frequency of a guitar string varies inversely with its length. A 26 in.-long string has a frequency of 440 vibrations per second.<\/p>\n<p id=\"fs-id1167834252394\"><span class=\"token\">\u24d0<\/span> Write the equation of variation. <span class=\"token\">\u24d1<\/span> How many vibrations per second will there be if the string\u2019s length is reduced to 20 inches by putting a finger on a fret?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834189087\">\n<p id=\"fs-id1167834189089\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167831847028\" class=\"unnumbered unstyled can-break\" summary=\"The frequency varies inversely with the length. Name the variables. Let f be equal to frequency and L be equal to length. Write the formula for inverse variation. The formula is y is equal to k divided by x. We will use f in place of y and L in place of x. The equation is f is equal to the quantity k divided by L instead. Substitute the given values for the variables. f is equal to 440 when L is equal to 26. The result is 440 is equal k divided by 26. Solve for the constant of variation. Multiplying each side by the least common denominator 26, the result is 26 times 440 is equal to 26 times the quantity k divided by 26. Solving for k, the result is 11,440 is equal to k. Write the equation that relates f and L. The equation is f is equal to the quantity k divided by L. Substitute the constant of variation. The result is f is equal to the quantity 11,440 divided by L\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\">The frequency varies<\/p>\n<div data-type=\"newline\"><\/div>\n<p>inversely with the length.<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Name the variables.<\/td>\n<td data-align=\"left\">Let <em data-effect=\"italics\">f<\/em> = frequency.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e000faf1f3ff373bd4c89a5102c331a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><em data-effect=\"italics\">L<\/em> = length<\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write the formula for inverse variation.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834556260\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We will use <em data-effect=\"italics\">f<\/em> in place of <em data-effect=\"italics\">y<\/em> and <em data-effect=\"italics\">L<\/em> in place of <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826994506\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Substitute the given values for the variables.<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c9ab22ad991fe7c719067fb4633bd8e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834534801\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835334003\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Solve for the constant of variation<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835510166\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167834539288\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Write the equation that relates <em data-effect=\"italics\">f<\/em> and <em data-effect=\"italics\">L<\/em>.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834179765\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Substitute the constant of variation<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834156945\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_017i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167834156588\"><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95d77c432269842d8a032f99021ceada_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#119;&#104;&#101;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#76;&#61;&#50;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#114;&#105;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#116;&#104;&#97;&#116;&#32;&#114;&#101;&#108;&#97;&#116;&#101;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#76;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#44;&#52;&#52;&#48;&#125;&#123;&#76;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#103;&#105;&#118;&#101;&#110;&#32;&#118;&#97;&#108;&#117;&#101;&#32;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#76;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#44;&#52;&#52;&#48;&#125;&#123;&#50;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#102;&#61;&#53;&#55;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#56;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#8243;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#103;&#117;&#105;&#116;&#97;&#114;&#32;&#115;&#116;&#114;&#105;&#110;&#103;&#32;&#104;&#97;&#115;&#32;&#102;&#114;&#101;&#113;&#117;&#101;&#110;&#99;&#121;&#32;&#53;&#55;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#118;&#105;&#98;&#114;&#97;&#116;&#105;&#111;&#110;&#115;&#32;&#112;&#101;&#114;&#32;&#115;&#101;&#99;&#111;&#110;&#100;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"128\" width=\"787\" style=\"vertical-align: -59px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834382551\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834382555\">\n<div data-type=\"problem\" id=\"fs-id1167831922745\">\n<p id=\"fs-id1167831922747\">The number of hours it takes for ice to melt varies inversely with the air temperature. Suppose a block of ice melts in 2 hours when the temperature is 65 degrees Celsius.<\/p>\n<p id=\"fs-id1167835229122\"><span class=\"token\">\u24d0<\/span> Write the equation of variation. <span class=\"token\">\u24d1<\/span> How many hours would it take for the same block of ice to melt if the temperature was 78 degrees?<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835254300\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21b352ae8745dafbffaffaa436d2414e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#48;&#125;&#123;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"57\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-595353e4bdfbcb142ab0ce88a07749e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"17\" style=\"vertical-align: -6px;\" \/> hours<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835243949\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835243953\">\n<div data-type=\"problem\" id=\"fs-id1167835288095\">\n<p id=\"fs-id1167835288097\">Xander\u2019s new business found that the daily demand for its product was inversely proportional to the price, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fa6ef6ec04c2dccd40c7f3e3be899df7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\" \/> When the price is ?5, the demand is 700 units.<\/p>\n<p id=\"fs-id1167835326113\"><span class=\"token\">\u24d0<\/span> Write the equation of variation. <span class=\"token\">\u24d1<\/span> What is the demand if the price is raised to ?7?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834185666\">\n<p id=\"fs-id1167834185668\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6aa5c5ad226f985d0486a446c1fc5db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#53;&#48;&#48;&#125;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"64\" style=\"vertical-align: -9px;\" \/><span class=\"token\">\u24d1<\/span> 500 units<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832068313\" class=\"media-2\">\n<p id=\"fs-id1167832068317\">Access this online resource for additional instruction and practice with applications of rational expressions<\/p>\n<ul id=\"fs-id1171790665329\" data-display=\"block\">\n<li id=\"fs-id1167834340056\"><a href=\"https:\/\/openstax.org\/l\/37AppRatExp\">Applications of Rational Expressions<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167834517390\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167831910687\" data-bullet-style=\"bullet\">\n<li>A proportion is an equation of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60c029a08a2eb9a150ad607019e89782_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -6px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e327228415f3008762bc19707f9a109d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#100;&#92;&#110;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"94\" style=\"vertical-align: -4px;\" \/> The proportion is read \u201c<em data-effect=\"italics\">a<\/em> is to <em data-effect=\"italics\">b<\/em> as <em data-effect=\"italics\">c<\/em> is to <em data-effect=\"italics\">d.<\/em>\u201d<\/li>\n<li><strong data-effect=\"bold\">Property of Similar Triangles<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91acd2d3a8297e2d0caa8e3d4bfae3ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#65;&#66;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> is similar to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8491fe25e881cb20249417d7fc85afbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#88;&#89;&#90;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> then their corresponding angle measure are equal and their corresponding sides have the same ratio.<\/p>\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167830703946\" data-alt=\"The first figure is triangle A B C with side A B c units long, side B C a units long, and side A C b units long. The second figure is triangle X Y Z with side X Y x units long, side Y Z x units long, and side X Z y units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. a divided by x is equal to b divided by y is equal to c divided by z.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is triangle A B C with side A B c units long, side B C a units long, and side A C b units long. The second figure is triangle X Y Z with side X Y x units long, side Y Z x units long, and side X Z y units long. The measure of angle A is equal to the measure of angle X. The measure of angle B is equal to the measure of angle Y. The measure of angle C is equal to the measure of angle Z. a divided by x is equal to b divided by y is equal to c divided by z.\" \/><\/span> <\/li>\n<li><strong data-effect=\"bold\">Direct Variation<\/strong>\n<ul id=\"fs-id1167835318947\" data-bullet-style=\"open-circle\">\n<li>For any two variables <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>, <em data-effect=\"italics\">y<\/em> varies directly with <em data-effect=\"italics\">x<\/em> if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fec5aaa7d97366666c16c7f0314bc01b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#107;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"57\" style=\"vertical-align: -4px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd59b3265c4631722a81e65130a12b62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#92;&#110;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> The constant <em data-effect=\"italics\">k<\/em> is called the constant of variation.<\/li>\n<li>How to solve direct variation problems.\n<div data-type=\"newline\"><\/div>\n<ol id=\"fs-id1167835354882\" type=\"1\" class=\"stepwise\">\n<li>Write the formula for direct variation.<\/li>\n<li>Substitute the given values for the variables.<\/li>\n<li>Solve for the constant of variation.<\/li>\n<li>Write the equation that relates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62f853fa6f372493298c507883a9f490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/><\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n<li><strong data-effect=\"bold\">Inverse Variation<\/strong>\n<ul id=\"fs-id1167834423074\" data-bullet-style=\"open-circle\">\n<li>For any two variables <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>, <em data-effect=\"italics\">y<\/em> varies inversely with <em data-effect=\"italics\">x<\/em> if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2651767021315e1ea5df8e318a62a43a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#107;&#125;&#123;&#120;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"49\" style=\"vertical-align: -6px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd59b3265c4631722a81e65130a12b62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#92;&#110;&#101;&#32;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> The constant <em data-effect=\"italics\">k<\/em> is called the constant of variation.<\/li>\n<li>How to solve inverse variation problems.\n<ol type=\"1\" class=\"stepwise\">\n<li>Write the formula for inverse variation.<\/li>\n<li>Substitute the given values for the variables.<\/li>\n<li>Solve for the constant of variation.<\/li>\n<li>Write the equation that relates <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167826857403\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167834423748\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167826880194\"><strong data-effect=\"bold\">Solve Proportions<\/strong><\/p>\n<p id=\"fs-id1167834252400\">In the following exercises, solve each proportion.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834252404\">\n<div data-type=\"problem\" id=\"fs-id1167835238764\">\n<p id=\"fs-id1167835238766\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a0edfffde78f27dc879454009b3f3d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#53;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"48\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830698082\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-74060de7edacea5476319d9be847650d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835215793\">\n<div data-type=\"problem\" id=\"fs-id1167826874529\">\n<p id=\"fs-id1167826874532\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97060b134c23a99178cab6fb6101c5dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#54;&#125;&#123;&#55;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"49\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835356568\">\n<div data-type=\"problem\" id=\"fs-id1167835356570\">\n<p id=\"fs-id1167835356573\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65224e4e35e9d53bd6dc63b5e50d4f6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#56;&#125;&#123;&#49;&#53;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#55;&#125;&#123;&#112;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"66\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834489808\">\n<p id=\"fs-id1167834489810\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a6b450c33783251326b45d4630d1db82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827988171\">\n<div data-type=\"problem\" id=\"fs-id1167827988173\">\n<p id=\"fs-id1167835268055\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5dbd314f7f595a9ee58ef37e478e6af4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#50;&#125;&#123;&#49;&#53;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#54;&#125;&#123;&#113;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"66\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834192953\">\n<div data-type=\"problem\" id=\"fs-id1167835420385\">\n<p id=\"fs-id1167835420387\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4043f8bb251246c18f7a3ee4fafeb6b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#97;&#43;&#49;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"66\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831891006\">\n<p id=\"fs-id1167831891008\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7babc4e2a5ae12ed6911d4ec333cf7fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"51\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831919497\">\n<div data-type=\"problem\" id=\"fs-id1167831919499\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8c223cc427c294ba5908df2d7a399ca7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#98;&#45;&#49;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"72\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835509144\">\n<div data-type=\"problem\" id=\"fs-id1167835509146\">\n<p id=\"fs-id1167835509148\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a81e786826827d0cf6f675977edf932b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#43;&#57;&#48;&#125;&#123;&#50;&#53;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#43;&#51;&#48;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"101\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835369531\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1d55d9487b321cf8218eba9b05255c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835335653\">\n<div data-type=\"problem\" id=\"fs-id1167835335655\">\n<p id=\"fs-id1167835335657\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e8b5d58f4a93a603c360b94c98f980a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#43;&#49;&#48;&#125;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#48;&#45;&#110;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835307608\">\n<div data-type=\"problem\" id=\"fs-id1167835307610\">\n<p id=\"fs-id1167832058806\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5a80fd5c1f8d5b764fcd05e132ab835_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#112;&#43;&#52;&#125;&#123;&#56;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#43;&#49;&#56;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"91\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826801678\">\n<p id=\"fs-id1167826801680\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16d59560d25be9f84371a79c9e23ba02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834065787\">\n<div data-type=\"problem\" id=\"fs-id1167834065789\">\n<p id=\"fs-id1167835346269\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-458a09da03ceb3ab832816aa21ebc050_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#113;&#45;&#50;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#113;&#45;&#55;&#125;&#123;&#49;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"84\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834472847\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834472850\">\n<div data-type=\"problem\" id=\"fs-id1167834472852\">\n<p id=\"fs-id1167828426705\">Kevin wants to keep his heart rate at 160 beats per minute while training. During his workout he counts 27 beats in 10 seconds.<\/p>\n<p id=\"fs-id1167828426709\"><span class=\"token\">\u24d0<\/span> How many beats per minute is this?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> Has Kevin met his target heart rate?<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834537746\">\n<p id=\"fs-id1167834537748\"><span class=\"token\">\u24d0<\/span> 162 beats per minute<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> yes<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834413589\">\n<div data-type=\"problem\" id=\"fs-id1167832060560\">\n<p id=\"fs-id1167832060562\">Jesse\u2019s car gets 30 miles per gallon of gas.<\/p>\n<p id=\"fs-id1167835253852\"><span class=\"token\">\u24d0<\/span> If Las Vegas is 285 miles away, how many gallons of gas are needed to get there and then home?<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> If gas is ?3.09 per gallon, what is the total cost of the gas for the trip?<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835346436\">\n<div data-type=\"problem\" id=\"fs-id1167835346438\">\n<p id=\"fs-id1167835346440\">Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child\u2019s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs 45 pounds?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831186069\">\n<p id=\"fs-id1167831186072\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e99b7c7ddd421682faf67e5b5f865882_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> ml<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831970353\">\n<div data-type=\"problem\" id=\"fs-id1167831970355\">\n<p id=\"fs-id1167831970357\">A veterinarian prescribed Sunny, a 65-pound dog, an antibacterial medicine in case an infection emerges after her teeth were cleaned. If the dosage is 5 mg for every pound, how much medicine was Sunny given?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835358031\">\n<div data-type=\"problem\" id=\"fs-id1167835358033\">\n<p id=\"fs-id1167834120894\">A new energy drink advertises 106 calories for 8 ounces. How many calories are in 12 ounces of the drink?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835318972\">\n<p id=\"fs-id1167835318974\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f49e1bb1ccd6e8979e56d3928f6be2b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"26\" style=\"vertical-align: -1px;\" \/> calories<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835329467\">\n<div data-type=\"problem\" id=\"fs-id1167835364886\">\n<p id=\"fs-id1167835364888\">One 12-ounce can of soda has 150 calories. If Josiah drinks the big 32-ounce size from the local mini-mart, how many calories does he get?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834428677\">\n<div data-type=\"problem\" id=\"fs-id1167834428679\">\n<p id=\"fs-id1167835349788\">Kyra is traveling to Canada and will change ?250 US dollars into Canadian dollars. At the current exchange rate, ?1 US is equal to ?1.3 Canadian. How many Canadian dollars will she get for her trip?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834280014\">\n<p id=\"fs-id1167834280017\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1af12b84aa3e35a638897c8b71fdb08a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> Canadian dollars<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835621559\">\n<div data-type=\"problem\" id=\"fs-id1167835621562\">\n<p id=\"fs-id1167835621564\">Maurice is traveling to Mexico and needs to exchange ?450 into Mexican pesos. If each dollar is worth 12.29 pesos, how many pesos will he get for his trip?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834063471\">\n<div data-type=\"problem\" id=\"fs-id1167834063473\">\n<p id=\"fs-id1167835334339\">Ronald needs a morning breakfast drink that will give him at least 390 calories. Orange juice has 130 calories in one cup. How many cups does he need to drink to reach his calorie goal?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832068368\">\n<p id=\"fs-id1167832068370\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a1d3ea4963f568cabd97329456036b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> cups<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835421080\">\n<div data-type=\"problem\" id=\"fs-id1167830757221\">\n<p id=\"fs-id1167830757223\">Sonya drinks a 32-ounce energy drink containing 80 calories per 12 ounce. How many calories did she drink?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830866157\">\n<div data-type=\"problem\" id=\"fs-id1167830866159\">\n<p id=\"fs-id1167835283628\">Phil wants to fertilize his lawn. Each bag of fertilizer covers about 4,000 square feet of lawn. Phil\u2019s lawn is approximately 13,500 square feet. How many bags of fertilizer will he have to buy?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834190288\">\n<p id=\"fs-id1167834190290\">4 bags<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835237671\">\n<div data-type=\"problem\" id=\"fs-id1167835237673\">\n<p id=\"fs-id1167834195103\">An oatmeal cookie recipe calls for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8fa03e1b526c6d07ec843385490ca4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> cup of butter to make 4 dozen cookies. Hilda needs to make 10 dozen cookies for the bake sale. How many cups of butter will she need?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835283974\"><strong data-effect=\"bold\">Solve Similar Figure Applications<\/strong><\/p>\n<p id=\"fs-id1167835513374\">In the following exercises, the triangles are similar. Find the length of the indicated side.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167832076332\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832076334\">\n<p id=\"fs-id1167834049061\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" data-alt=\"The first figure is triangle A B C with side A B 15 units long, side B C 9 units long, and side A C b units long. The second figure is triangle X Y Z with side X Y 10 units long, side Y Z x units long, and side X Z 8 units long.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is triangle A B C with side A B 15 units long, side B C 9 units long, and side A C b units long. The second figure is triangle X Y Z with side X Y 10 units long, side Y Z x units long, and side X Z 8 units long.\" \/><\/span><\/p>\n<p id=\"fs-id1167832015935\"><span class=\"token\">\u24d0<\/span> side <em data-effect=\"italics\">x<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> side <em data-effect=\"italics\">b<\/em><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835382034\">\n<p id=\"fs-id1167835382036\"><span class=\"token\">\u24d0<\/span> 6 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-40f7428479801bf4ef27a5149e9df526_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"16\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834193222\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834193224\">\n<p id=\"fs-id1167831922554\">\n<div data-type=\"newline\"><\/div>\n<p><span data-type=\"media\" id=\"fs-id1167831922555\" data-alt=\"The first figure is triangle DEF with side D E 5 halves units long, side E F d units long, and side D F 1 unit long. The second figure is triangle N P Q with side N P q units long, side P Q 11 halves units long, and side N Q 9 units long.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_202_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first figure is triangle DEF with side D E 5 halves units long, side E F d units long, and side D F 1 unit long. The second figure is triangle N P Q with side N P q units long, side P Q 11 halves units long, and side N Q 9 units long.\" \/><\/span><\/p>\n<p id=\"fs-id1167831076642\"><span class=\"token\">\u24d0<\/span> side <em data-effect=\"italics\">d<\/em><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> side <em data-effect=\"italics\">q<\/em><\/div>\n<\/div>\n<p id=\"fs-id1167835622015\">In the following exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle. The actual distance from New York to Chicago is 800 miles.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167826997052\" data-alt=\"The figure is a triangle formed by Memphis, Chicago, and New York. The distance between Memphis and Chicago is 5.4 inches. The distance between Chicago and New York is 8 inches. The distance between New York and Memphis is 9.5 inches.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_203_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a triangle formed by Memphis, Chicago, and New York. The distance between Memphis and Chicago is 5.4 inches. The distance between Chicago and New York is 8 inches. The distance between New York and Memphis is 9.5 inches.\" \/><\/span><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835307544\" class=\"material-set-2\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835240602\">Find the actual distance from New York to Memphis.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835377085\">\n<p id=\"fs-id1167835377087\">950 miles<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835379084\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835379086\">\n<p id=\"fs-id1167835379089\">Find the actual distance from Chicago to Memphis.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834185889\">In the following exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle. The actual distance from Atlanta to New Orleans is 420 miles.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167832058318\" data-alt=\"The figure is a triangle formed by New Orleans, Atlanta, and Miami. The distance between New Orleans and Atlanta is 2.1 inches. The distance between Atlanta and Miami is 3 inches. The distance between Miami and New Orleans is 3.4 inches.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_204_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure is a triangle formed by New Orleans, Atlanta, and Miami. The distance between New Orleans and Atlanta is 2.1 inches. The distance between Atlanta and Miami is 3 inches. The distance between Miami and New Orleans is 3.4 inches.\" \/><\/span><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834439125\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831882913\">\n<p id=\"fs-id1167831882915\">Find the actual distance from New Orleans to Miami.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835622515\">\n<p id=\"fs-id1167835622517\">680 miles<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835217022\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835217024\">\n<p id=\"fs-id1167835217027\">Find the actual distance from Atlanta to Miami.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835498842\">In the following exercises, answer each question.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835225788\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835225791\">\n<p id=\"fs-id1167835225793\">A 2-foot-tall dog casts a 3-foot shadow at the same time a cat casts a one foot shadow. How tall is the cat ?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834489708\">\n<p id=\"fs-id1167835352988\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/> foot (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23212a120eb92d226d44696a0b80bead_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> in.)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831957047\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834555112\">\n<p id=\"fs-id1167834555114\">Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry\u2019s shadow was 8 feet and Tom\u2019s was 7.75 feet long. What is Tom\u2019s height?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835262108\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835262110\">\n<p id=\"fs-id1167835366866\">The tower portion of a windmill is 212 feet tall. A six foot tall person standing next to the tower casts a seven-foot shadow. How long is the windmill\u2019s shadow?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831031248\">\n<p id=\"fs-id1167831031250\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc63d611e2ad37cbab097396b2db1016_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#52;&#55;&#46;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"41\" style=\"vertical-align: -1px;\" \/> feet<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832076492\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832076494\">\n<p id=\"fs-id1167832076496\">The height of the Statue of Liberty is 305 feet. Nikia, who is standing next to the statue, casts a 6-foot shadow and she is 5 feet tall. How long should the shadow of the statue be?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835303952\"><strong data-effect=\"bold\">Solve Uniform Motion Applications<\/strong><\/p>\n<p id=\"fs-id1167834432142\">In the following exercises, solve the application problem provided.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830702465\">\n<div data-type=\"problem\" id=\"fs-id1167830702467\">\n<p id=\"fs-id1167830702469\">Mary takes a sightseeing tour on a helicopter that can fly 450 miles against a 35-mph headwind in the same amount of time it can travel 702 miles with a 35-mph tailwind. Find the speed of the helicopter.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834517568\">\n<p id=\"fs-id1167835510196\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21b40b26dcd5c46e571710b78f863a87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835240350\">\n<p id=\"fs-id1167834431101\">A private jet can fly 1,210 miles against a 25-mph headwind in the same amount of time it can fly 1694 miles with a 25-mph tailwind. Find the speed of the jet.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834184544\">\n<div data-type=\"problem\" id=\"fs-id1167834184546\">\n<p id=\"fs-id1167834184548\">A boat travels 140 miles downstream in the same time as it travels 92 miles upstream. The speed of the current is 6mph. What is the speed of the boat?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826978746\">\n<p id=\"fs-id1167835344947\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da60de369e4a2bdb1a6166bdb4b6f1c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835387124\">\n<div data-type=\"problem\" id=\"fs-id1167835387126\">\n<p id=\"fs-id1167832152880\">Darrin can skateboard 2 miles against a 4-mph wind in the same amount of time he skateboards 6 miles with a 4-mph wind. Find the speed Darrin skateboards with no wind.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832055442\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834432149\">Jane spent 2 hours exploring a mountain with a dirt bike. First, she rode 40 miles uphill. After she reached the peak she rode for 12 miles along the summit. While going uphill, she went 5 mph slower than when she was on the summit. What was her rate along the summit?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834279319\">\n<p id=\"fs-id1167834131066\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-874b63beceda768488ea76f12b9cf9e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831881826\">\n<div data-type=\"problem\" id=\"fs-id1167831881828\">\n<p id=\"fs-id1167832198573\">Laney wanted to lose some weight so she planned a day of exercising. She spent a total of 2 hours riding her bike and jogging. She biked for 12 miles and jogged for 6 miles. Her rate for jogging was 10 mph less than biking rate. What was her rate when jogging?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835326552\">\n<div data-type=\"problem\" id=\"fs-id1167834539256\">\n<p id=\"fs-id1167834539258\">Byron wanted to try out different water craft. He went 62 miles downstream in a motor boat and 27 miles downstream on a jet ski. His speed on the jet ski was 10 mph faster than in the motor boat. Bill spent a total of 4 hours on the water. What was his rate of speed in the motor boat?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167827987903\">\n<p id=\"fs-id1167831920754\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-930d01a5f622cee5952bab4752e56063_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835283707\">\n<div data-type=\"problem\" id=\"fs-id1167835283710\">\n<p id=\"fs-id1167834131729\">Nancy took a 3-hour drive. She went 50 miles before she got caught in a storm. Then she drove 68 miles at 9 mph less than she had driven when the weather was good. What was her speed driving in the storm?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835374050\">\n<div data-type=\"problem\" id=\"fs-id1167835374052\">\n<p id=\"fs-id1167835374055\">Chester rode his bike uphill 24 miles and then back downhill at 2 mph faster than his uphill. If it took him 2 hours longer to ride uphill than downhill, what was his uphill rate?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834195866\">\n<p id=\"fs-id1167835308775\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6111a899fd636b7a5238708f8679f6ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: -1px;\" \/> mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831882440\">\n<div data-type=\"problem\" id=\"fs-id1167831882442\">\n<p id=\"fs-id1167835511581\">Matthew jogged to his friend\u2019s house 12 miles away and then got a ride back home. It took him 2 hours longer to jog there than ride back. His jogging rate was 25 mph slower than the rate when he was riding. What was his jogging rate?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826802097\">\n<div data-type=\"problem\" id=\"fs-id1167826802099\">\n<p id=\"fs-id1167826802102\">Hudson travels 1080 miles in a jet and then 240 miles by car to get to a business meeting. The jet goes 300 mph faster than the rate of the car, and the car ride takes 1 hour longer than the jet. What is the speed of the car?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826997726\">\n<p id=\"fs-id1167835422406\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7029ea134aa43ac5ffd9f780e196307d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835513046\">\n<div data-type=\"problem\" id=\"fs-id1167835534047\">\n<p id=\"fs-id1167835534049\">Nathan walked on an asphalt pathway for 12 miles. He walked the 12 miles back to his car on a gravel road through the forest. On the asphalt he walked 2 miles per hour faster than on the gravel. The walk on the gravel took one hour longer than the walk on the asphalt. How fast did he walk on the gravel.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834228325\">\n<div data-type=\"problem\" id=\"fs-id1167834228327\">\n<p id=\"fs-id1167834228330\">John can fly his airplane 2800 miles with a wind speed of 50 mph in the same time he can travel 2400 miles against the wind. If the speed of the wind is 50 mph, find the speed of his airplane.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835238517\">\n<p id=\"fs-id1167834066283\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e869dd3e7ceda42d40391bcc5b258444_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: 0px;\" \/> mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831892512\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167826986754\">Jim\u2019s speedboat can travel 20 miles upstream against a 3-mph current in the same amount of time it travels 22 miles downstream with a 3-mph current speed . Find the speed of the Jim\u2019s boat.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835283953\">\n<div data-type=\"problem\" id=\"fs-id1167835283955\">\n<p id=\"fs-id1167834448720\">Hazel needs to get to her granddaughter\u2019s house by taking an airplane and a rental car. She travels 900 miles by plane and 250 miles by car. The plane travels 250 mph faster than the car. If she drives the rental car for 2 hours more than she rode the plane, find the speed of the car.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834448724\">\n<p id=\"fs-id1167835363625\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39e5271c62d377c7a347b4eb07f74ab2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/> mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831896687\">\n<div data-type=\"problem\" id=\"fs-id1167831896689\">\n<p id=\"fs-id1167835356344\">Stu trained for 3 hours yesterday. He ran 14 miles and then biked 40 miles. His biking speed is 6 mph faster than his running speed. What is his running speed?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835240369\">\n<div data-type=\"problem\" id=\"fs-id1167835240371\">\n<p id=\"fs-id1167835240374\">When driving the 9-hour trip home, Sharon drove 390 miles on the interstate and 150 miles on country roads. Her speed on the interstate was 15 more than on country roads. What was her speed on country roads?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835343339\">\n<p id=\"fs-id1167835257646\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39e5271c62d377c7a347b4eb07f74ab2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: 0px;\" \/> mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834196066\">\n<div data-type=\"problem\" id=\"fs-id1167834196068\">\n<p id=\"fs-id1167831919456\">Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835515481\">\n<div data-type=\"problem\" id=\"fs-id1167835515483\">\n<p id=\"fs-id1167835515485\">Dana enjoys taking her dog for a walk, but sometimes her dog gets away, and she has to run after him. Dana walked her dog for 7 miles but then had to run for 1 mile, spending a total time of 2.5 hours with her dog. Her running speed was 3 mph faster than her walking speed. Find her walking speed.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834138038\">\n<p id=\"fs-id1167830894100\">4.2 mph<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831893592\">\n<div data-type=\"problem\" id=\"fs-id1167831893595\">\n<p id=\"fs-id1167831893597\">Ken and Joe leave their apartment to go to a football game 45 miles away. Ken drives his car 30 mph faster Joe can ride his bike. If it takes Joe 2 hours longer than Ken to get to the game, what is Joe\u2019s speed?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835320259\"><strong data-effect=\"bold\">Solve Work Applications<\/strong><\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834111772\">\n<div data-type=\"problem\" id=\"fs-id1167834111774\">\n<p id=\"fs-id1167834111776\">Mike, an experienced bricklayer, can build a wall in 3 hours, while his son, who is learning, can do the job in 6 hours. How long does it take for them to build a wall together?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832058504\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/> hours<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826808725\">\n<div data-type=\"problem\" id=\"fs-id1167826808727\">\n<p id=\"fs-id1167834395184\">It takes Sam 4 hours to rake the front lawn while his brother, Dave, can rake the lawn in 2 hours. How long will it take them to rake the lawn working together?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834299718\">\n<div data-type=\"problem\" id=\"fs-id1167834299720\">\n<p id=\"fs-id1167835496364\">Mia can clean her apartment in 6 hours while her roommate can clean the apartment in 5 hours. If they work together, how long would it take them to clean the apartment?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834228222\">\n<p id=\"fs-id1167834228224\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e584dd0bab4e6c8efc164939c28db757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\" \/> hours and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-54130b413a9a12a930c9a8acc2a09012_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"18\" style=\"vertical-align: -1px;\" \/> minutes <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835235908\">\n<div data-type=\"problem\" id=\"fs-id1167835218029\">\n<p id=\"fs-id1167835218031\">Brian can lay a slab of concrete in 6 hours, while Greg can do it in 4 hours. If Brian and Greg work together, how long will it take?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835282827\">\n<p id=\"fs-id1167835282829\">Josephine can correct her students test papers in 5 hours, but if her teacher\u2019s assistant helps, it would take them 3 hours. How long would it take the assistant to do it alone?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835282832\">\n<p id=\"fs-id1167831884906\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aca3387e59afe477960eabc2f23b3db5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> hours and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-874b63beceda768488ea76f12b9cf9e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> minutes <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835301974\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167832087118\">Washing his dad\u2019s car alone, eight year old Levi takes 2.5 hours. If his dad helps him, then it takes 1 hour. How long does it take Levi\u2019s dad to wash the car by himself?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834063286\">\n<div data-type=\"problem\" id=\"fs-id1167835384857\">\n<p id=\"fs-id1167835384859\">At the end of the day Dodie can clean her hair salon in 15 minutes. Ann, who works with her, can clean the salon in 30 minutes. How long would it take them to clean the shop if they work together?<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832066033\">\n<p id=\"fs-id1167832066035\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5450913cc453faf132dc56e5965ca797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/> min<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835343355\">\n<div data-type=\"problem\" id=\"fs-id1167835343357\">\n<p id=\"fs-id1167835343359\">Ronald can shovel the driveway in 4 hours, but if his brother Donald helps it would take 2 hours. How long would it take Donald to shovel the driveway alone?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826996824\"><strong data-effect=\"bold\">Solve Direct Variation Problems<\/strong><\/p>\n<p id=\"fs-id1167830866117\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830866120\">\n<div data-type=\"problem\" id=\"fs-id1167834061688\">\n<p id=\"fs-id1167834061690\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> varies directly as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac88e5fed1e7891904f88c9a050febb5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#49;&#52;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf60bc9fcf312a246a055c15ee98033c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/> find the equation that relates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62f853fa6f372493298c507883a9f490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830705509\">\n<p id=\"fs-id1167830705511\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b324876a578aaac751665c5112da05a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#51;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835243929\">\n<div data-type=\"problem\" id=\"fs-id1167835243931\">\n<p id=\"fs-id1167826996702\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> varies directly as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e14c1f54addb35a80271d8be686299f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#49;&#54;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d19f9b91addd75cb3653c8a6f47058c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"44\" style=\"vertical-align: -1px;\" \/> find the equation that relates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf59029b62b61185814b66fa6004b2f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834562571\">\n<div data-type=\"problem\" id=\"fs-id1167835422662\">\n<p id=\"fs-id1167835422664\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> varies directly as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a46160825808774c0fdd8bffa2a5e48c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#57;&#46;&#54;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-79a9f40a06a53b59bb074f83333de821_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"45\" style=\"vertical-align: -4px;\" \/> find the equation that relates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-478d01cb1acedf5385b972d8095b2690_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831922015\">\n<p id=\"fs-id1167831922017\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f9063a8363bda50ad021452885b5425_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#51;&#46;&#50;&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834053691\">\n<div data-type=\"problem\" id=\"fs-id1167834053693\">\n<p id=\"fs-id1167834053695\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> varies directly as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfee5c980777976ae8cf6541893fb572_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-37202cab2f778f1ca27d8a4dbe5344d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#56;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"46\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5439e4ede593a933d2d01b6148c90136_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#8203;&#125;&#119;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"52\" style=\"vertical-align: -6px;\" \/> find the equation that relates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c885ae2387fc01f58967c91d7b5a6b5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834535299\">The price, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6afe6e8cf17f5596dfba598723950b26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#80;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"16\" style=\"vertical-align: -4px;\" \/> that Eric pays for gas varies directly with the number of gallons, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa307d44fe899099cad9fc84395f6eb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/> he buys. It costs him ?50 to buy 20 gallons of gas.<\/p>\n<p id=\"fs-id1167835368004\"><span class=\"token\">\u24d0<\/span> Write the equation that relates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#80;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dce98a79808d5694e9c7a9f4545e45f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> How much would 33 gallons cost Eric?<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835414664\">\n<p id=\"fs-id1167835414667\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4f8aa4e57dbf4246358475c3680706ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#80;&#61;&#50;&#46;&#53;&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"69\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-03a89ced8198fa863d0744a62e0c98d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#63;&#125;&#56;&#50;&#46;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834222099\">\n<div data-type=\"problem\" id=\"fs-id1167834222101\">\n<p id=\"fs-id1167835513096\">Joseph is traveling on a road trip. The distance, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bca0eeb2425a54167e5e6044c55c320b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: -4px;\" \/> he travels before stopping for lunch varies directly with the speed, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fec560ccea89a9f8adbc60a211badaed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/> he travels. He can travel 120 miles at a speed of 60 mph.<\/p>\n<p id=\"fs-id1167834464336\"><span class=\"token\">\u24d0<\/span> Write the equation that relates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e8716946f6a868f015e0d62f28bc540_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24e7ac6075439e15dbe0fec157cfd5f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> How far would he travel before stopping for lunch at a rate of 65 mph?<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835532705\">\n<div data-type=\"problem\" id=\"fs-id1167835230446\">\n<p id=\"fs-id1167835230448\">The mass of a liquid varies directly with its volume. A liquid with mass 16 kilograms has a volume of 2 liters.<\/p>\n<p id=\"fs-id1167834587806\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the mass to the volume.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> What is the volume of this liquid if its mass is 128 kilograms?<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835284798\">\n<p id=\"fs-id1167835284800\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90febfbd4841d430c6b4e918cca12fab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#61;&#56;&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8dde8d4cade60a4f3ca0e779512b974c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/> liters<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830757576\">\n<div data-type=\"problem\" id=\"fs-id1167830757579\">\n<p id=\"fs-id1167835166911\">The length that a spring stretches varies directly with a weight placed at the end of the spring. When Sarah placed a 10-pound watermelon on a hanging scale, the spring stretched 5 inches.<\/p>\n<p id=\"fs-id1167835370110\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the length of the spring to the weight.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> What weight of watermelon would stretch the spring 6 inches?<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835180625\">\n<div data-type=\"problem\" id=\"fs-id1167832054992\">\n<p id=\"fs-id1167832054994\">The maximum load a beam will support varies directly with the square of the diagonal of the beam\u2019s cross-section. A beam with diagonal 6 inch will support a maximum load of 108 pounds.<\/p>\n<p id=\"fs-id1167835237348\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the load to the diagonal of the cross-section.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> What load will a beam with a 10-inch diagonal support?<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831884883\">\n<p id=\"fs-id1167831884886\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e05e70c8f14ec834a99dac0e36c44210_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#61;&#51;&#123;&#100;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"61\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd1f95d8d580ca7e0a22581555ffd7b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"27\" style=\"vertical-align: 0px;\" \/> pounds<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831880549\">\n<div data-type=\"problem\" id=\"fs-id1167831880551\">\n<p id=\"fs-id1167831880553\">The area of a circle varies directly as the square of the radius. A circular pizza with a radius of 6 inches has an area of 113.04 square inches.<\/p>\n<p id=\"fs-id1167835594886\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the area to the radius.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> What is the area of a personal pizza with a radius 4 inches?<\/div>\n<\/div>\n<p id=\"fs-id1167834094671\"><strong data-effect=\"bold\">Solve Inverse Variation Problems<\/strong><\/p>\n<p id=\"fs-id1167830698200\">In the following exercises, solve.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835267805\">\n<div data-type=\"problem\" id=\"fs-id1167835267807\">\n<p id=\"fs-id1167835267809\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> varies inversely with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-39af43abd99adaf051fde7775af522c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"41\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ad850560a1ff3e3bb4386cf732d0220_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -4px;\" \/> find the equation that relates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62f853fa6f372493298c507883a9f490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826993939\">\n<p id=\"fs-id1167834229238\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1814c4696f36a6dd1d727f2d5541c41a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#48;&#125;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"49\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834510611\">\n<div data-type=\"problem\" id=\"fs-id1167831871826\">\n<p id=\"fs-id1167831871828\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> varies inversely with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36cf66ae876ab93b38b965cbe720697e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1a70c195dac80d6b7f395a8fb53089a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"40\" style=\"vertical-align: -4px;\" \/>, find the equation that relates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bf85f1087e9fbed3a319341134ac1a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-478d01cb1acedf5385b972d8095b2690_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835341860\">\n<div data-type=\"problem\" id=\"fs-id1167835341862\">\n<p id=\"fs-id1167835341864\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> varies inversely with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfee5c980777976ae8cf6541893fb572_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29199a21511b94c982c6a3edaf349274_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e22c06f152850937071c6765ba047152_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"52\" style=\"vertical-align: -6px;\" \/> find the equation that relates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c885ae2387fc01f58967c91d7b5a6b5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835283409\">\n<p id=\"fs-id1167835283411\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67ecb03645c8ed0037cda8bfe397fe63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#119;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835343539\">\n<div data-type=\"problem\" id=\"fs-id1167835343542\">\n<p id=\"fs-id1167835343544\">If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> varies inversely with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"8\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d3708245a21c7fef79fefd12efc69f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#49;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"50\" style=\"vertical-align: -1px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a55363f62534c0fb2a16c4be80ce5dc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" style=\"vertical-align: -6px;\" \/> find the equation that relates <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf59029b62b61185814b66fa6004b2f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835346236\">In the following exercises, write an inverse variation equation to solve the following problems.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835346241\">\n<div data-type=\"problem\">\n<p>The fuel consumption (mpg) of a car varies inversely with its weight. A Toyota Corolla weighs 2800 pounds getting 33 mpg on the highway.<\/p>\n<p id=\"fs-id1167832153050\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the mpg to the car\u2019s weight.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> What would the fuel consumption be for a Toyota Sequoia that weighs 5500 pounds?<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835367134\">\n<p id=\"fs-id1167835367137\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c545d62b59584c99c773315fa14a64d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#50;&#44;&#52;&#48;&#48;&#125;&#123;&#119;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"74\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span> 16.8 mpg<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835353322\">\n<div data-type=\"problem\" id=\"fs-id1167835353324\">\n<p id=\"fs-id1167835353326\">A car\u2019s value varies inversely with its age. Jackie bought a 10-year-old car for ?2,400.<\/p>\n<p id=\"fs-id1167834284490\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the car\u2019s value to its age.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> What will be the value of Jackie\u2019s car when it is 15 years old?<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834300577\">\n<div data-type=\"problem\" id=\"fs-id1167835191230\">\n<p id=\"fs-id1167835191232\">The time required to empty a tank varies inversely as the rate of pumping. It took Ada 5 hours to pump her flooded basement using a pump that was rated at 200 gpm (gallons per minute).<\/p>\n<p id=\"fs-id1167834300868\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the number of hours to the pump rate.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> How long would it take Ada to pump her basement if she used a pump rated at 400 gpm?<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834066023\">\n<p id=\"fs-id1167834066025\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9bf19de741b017f902c446ff323aa5aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#48;&#48;&#125;&#123;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01948ea56b434f27f412043e785fe880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#46;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/> hours<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835238240\">\n<div data-type=\"problem\" id=\"fs-id1167835238242\">\n<p id=\"fs-id1167835238245\">On a string instrument, the length of a string varies inversely as the frequency of its vibrations. An 11-inch string on a violin has a frequency of 400 cycles per second.<\/p>\n<p id=\"fs-id1167835338961\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the string length to its frequency. <span class=\"token\">\u24d1<\/span> What is the frequency of a 10 inch string?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835318274\">\n<div data-type=\"problem\" id=\"fs-id1167835318276\">\n<p id=\"fs-id1167831040509\">Paul, a dentist, determined that the number of cavities that develops in his patient\u2019s mouth each year varies inversely to the number of minutes spent brushing each night. His patient, Lori, had four cavities when brushing her teeth 30 seconds (0.5 minutes) each night.<\/p>\n<p id=\"fs-id1167831040512\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the number of cavities to the time spent brushing.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> How many cavities would Paul expect Lori to have if she had brushed her teeth for 2 minutes each night?<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831880396\">\n<p id=\"fs-id1167831880399\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-777c79586f9c55451f65d45ced616b0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"40\" style=\"vertical-align: -6px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"7\" style=\"vertical-align: -1px;\" \/> cavity<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835514476\">\n<div data-type=\"problem\" id=\"fs-id1167835514478\">\n<p id=\"fs-id1167835303202\">Boyle\u2019s law states that if the temperature of a gas stays constant, then the pressure varies inversely to the volume of the gas. Braydon, a scuba diver, has a tank that holds 6 liters of air under a pressure of 220 psi.<\/p>\n<p><span class=\"token\">\u24d0<\/span> Write the equation that relates pressure to volume.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> If the pressure increases to 330 psi, how much air can Braydon\u2019s tank hold?<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834387485\">\n<div data-type=\"problem\" id=\"fs-id1167834387487\">\n<p id=\"fs-id1167834387489\">The cost of a ride service varies directly with the distance traveled. It costs ?35 for a ride from the city center to the airport, 14 miles away.<\/p>\n<p id=\"fs-id1167835414006\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the cost, <em data-effect=\"italics\">c<\/em>, with the number of miles, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4a447f470995624a31e9fc621325af2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"20\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> What would it cost to travel 22 miles with this service?<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835343158\">\n<p id=\"fs-id1167835331448\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31b3306dd4d7f9678c160903c008cae4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#50;&#46;&#53;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"69\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span> ?55<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835240050\">\n<div data-type=\"problem\" id=\"fs-id1167835240052\">\n<p id=\"fs-id1167835351291\">The number of hours it takes Jack to drive from Boston to Bangor is inversely proportional to his average driving speed. When he drives at an average speed of 40 miles per hour, it takes him 6 hours for the trip.<\/p>\n<p id=\"fs-id1167835351296\"><span class=\"token\">\u24d0<\/span> Write the equation that relates the number of hours, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf45a181677e521d243e366fceeb09a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"14\" style=\"vertical-align: -4px;\" \/> with the speed, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-23a7daa116b8874af1538c91f8d239de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span> How long would the trip take if his average speed was 75 miles per hour?<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167832151081\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167835353400\">\n<div data-type=\"problem\" id=\"fs-id1167835353402\">\n<p id=\"fs-id1167835353404\">Marisol solves the proportion <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-010a33bcbc8dd5db93a14572df441a85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#52;&#125;&#123;&#97;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"55\" style=\"vertical-align: -6px;\" \/> by \u2018cross multiplying,\u2019 so her first step looks like <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c10aa09d39b86dce8cc39bdf95e4e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&middot;&#49;&#52;&#52;&#61;&#57;&middot;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"81\" style=\"vertical-align: -1px;\" \/> Explain how this differs from the method of solution shown in <a href=\"#fs-id1167835341039\" class=\"autogenerated-content\">(Figure)<\/a>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835420340\">\n<p id=\"fs-id1167831025321\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831025326\">\n<div data-type=\"problem\" id=\"fs-id1167835364640\">\n<p id=\"fs-id1167835364643\">Paula and Yuki are roommates. It takes Paula 3 hours to clean their apartment. It takes Yuki 4 hours to clean the apartment. The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1bfb350e1c9d56f92229895a4a278c27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#116;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"74\" style=\"vertical-align: -6px;\" \/> can be used to find <em data-effect=\"italics\">t<\/em>, the number of hours it would take both of them, working together, to clean their apartment. Explain how this equation models the situation.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167832074381\">\n<p id=\"fs-id1167835356060\">In your own words, explain the difference between direct variation and inverse variation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835356064\">\n<p id=\"fs-id1167830836880\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830836886\">\n<div data-type=\"problem\" id=\"fs-id1167835341031\">\n<p id=\"fs-id1167835341034\">Make up an example from your life experience of inverse variation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167835346512\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167835240157\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835376880\" data-alt=\"This table has four columns and seven rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was solve proportions. In row 3, the I can was solve similar figure applications. In row 4, the I can was solve uniform motion applications. In row 5, the I can was solve work applications. In row 6, the I can was solve direct variation problems. In row 7, the I can was solve inverse variation problems.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_07_05_205_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and seven rows. The first row is a header and it labels each column, \u201cI can\u2026\u201d, \u201cConfidently,\u201d \u201cWith some help,\u201d and \u201cNo-I don\u2019t get it!\u201d In row 2, the I can was solve proportions. In row 3, the I can was solve similar figure applications. In row 4, the I can was solve uniform motion applications. In row 5, the I can was solve work applications. In row 6, the I can was solve direct variation problems. In row 7, the I can was solve inverse variation problems.\" \/><\/span><\/p>\n<p id=\"fs-id1167835503869\"><span class=\"token\">\u24d1<\/span> After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167834523835\">\n<dt>proportion<\/dt>\n<dd id=\"fs-id1167834191234\">When two rational expressions are equal, the equation relating them is called a proportion.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167831923341\">\n<dt>similar figures<\/dt>\n<dd id=\"fs-id1167831908814\">Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides have the same ratio.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3451","chapter","type-chapter","status-publish","hentry"],"part":3130,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3451","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3451\/revisions"}],"predecessor-version":[{"id":15249,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3451\/revisions\/15249"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/3130"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/3451\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=3451"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=3451"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=3451"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=3451"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}