{"id":911,"date":"2018-12-11T13:22:43","date_gmt":"2018-12-11T18:22:43","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/fractions-2\/"},"modified":"2018-12-11T13:22:43","modified_gmt":"2018-12-11T18:22:43","slug":"fractions-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/fractions-2\/","title":{"raw":"Fractions","rendered":"Fractions"},"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>Simplify fractions<\/li><li>Multiply and divide fractions<\/li><li>Add and subtract fractions<\/li><li>Use the order of operations to simplify fractions<\/li><li>Evaluate variable expressions with fractions<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167836516264\" class=\"be-prepared\"><p id=\"fs-id1167836311643\">A more thorough introduction to the topics covered in this section can be found in the <em data-effect=\"italics\">Elementary Algebra<\/em> chapter, Foundations.<\/p><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829716499\"><h3 data-type=\"title\">Simplify Fractions<\/h3><p id=\"fs-id1167833058122\">A <span data-type=\"term\">fraction<\/span> is a way to represent parts of a whole. The fraction \\(\\frac{2}{3}\\) represents two of three equal parts. See <a href=\"#CNX_IntAlg_Figure_01_03_001\" class=\"autogenerated-content\">(Figure)<\/a>. In the fraction \\(\\frac{2}{3},\\) the 2 is called the <span data-type=\"term\">numerator<\/span> and the 3 is called the <span data-type=\"term\">denominator<\/span>. The line is called the fraction bar.<\/p><div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_03_001\"><div class=\"bc-figcaption figcaption\">In the circle, \\(\\frac{2}{3}\\) of the circle is shaded\u20142 of the 3 equal parts.<\/div><span data-type=\"media\" id=\"fs-id1167836629535\" data-alt=\"Figure shows a circle divided in three equal parts. 2 of these are shaded.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_001_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a circle divided in three equal parts. 2 of these are shaded.\"><\/span><\/div><div data-type=\"note\" id=\"fs-id1167833412814\"><div data-type=\"title\">Fraction<\/div><p id=\"fs-id1167836688384\">A <strong data-effect=\"bold\">fraction<\/strong> is written \\(\\frac{a}{b},\\) where \\(b\\ne 0\\) and<\/p><p id=\"fs-id1167836408815\"><em data-effect=\"italics\">a<\/em> is the <strong data-effect=\"bold\">numerator<\/strong> and <em data-effect=\"italics\">b<\/em> is the <strong data-effect=\"bold\">denominator<\/strong>.<\/p><p id=\"fs-id1167836688080\">A fraction represents parts of a whole. The denominator \\(b\\) is the number of equal parts the whole has been divided into, and the numerator \\(a\\) indicates how many parts are included.<\/p><\/div><p id=\"fs-id1167836521454\">Fractions that have the same value are <span data-type=\"term\">equivalent fractions<\/span>. The Equivalent Fractions<\/p><p id=\"fs-id1167829755870\">Property allows us to find equivalent fractions and also simplify fractions.<\/p><div data-type=\"note\" id=\"fs-id1167829596336\"><div data-type=\"title\">Equivalent Fractions Property<\/div><p id=\"fs-id1167832940622\">If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are numbers where \\(b\\ne 0,c\\ne 0,\\)<\/p><p id=\"fs-id1167836550529\">then \\(\\frac{a}{b}=\\frac{a\u00b7c}{b\u00b7c}\\) and \\(\\frac{a\u00b7c}{b\u00b7c}=\\frac{a}{b}.\\)<\/p><\/div><p id=\"fs-id1167836327003\">A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.<\/p><p id=\"fs-id1167836447306\">For example,<\/p><p id=\"fs-id1167836388086\">\u2003\u2003\\(\\frac{2}{3}\\) is simplified because there are no common factors of 2 and \\(3.\\)<\/p><p id=\"fs-id1167829695451\">\u2003\u2003\\(\\frac{10}{15}\\) is not simplified because 5 is a common factor of 10 and \\(15.\\)<\/p><p id=\"fs-id1167836377781\">We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.<\/p><p id=\"fs-id1167836493293\">Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the Equivalent Fractions Property.<\/p><div data-type=\"example\" id=\"fs-id1167836620030\" class=\"textbox textbox--examples\"><div data-type=\"title\">How To Simplify a Fraction<\/div><div data-type=\"exercise\" id=\"fs-id1167836627469\"><div data-type=\"problem\" id=\"fs-id1167836340855\"><p id=\"fs-id1167836526280\">Simplify: \\(-\\frac{315}{770}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836527773\"><span data-type=\"media\" id=\"fs-id1167836514898\" data-alt=\"Step 1 is to rewrite the numerator and denominator to show the common factors. If needed, use a factor tree. Here, we rewrite 315 and 770 as the product of the primes. Starting with minus 315 divided by 770, we get, minus 3 times 3 time 5 times 7 divided by 2 times 5 times 7 times 11.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_002a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to rewrite the numerator and denominator to show the common factors. If needed, use a factor tree. Here, we rewrite 315 and 770 as the product of the primes. Starting with minus 315 divided by 770, we get, minus 3 times 3 time 5 times 7 divided by 2 times 5 times 7 times 11.\"><\/span><span data-type=\"media\" id=\"fs-id1167836391701\" data-alt=\"Step 2 is to simplify using the Equivalent Fractions Property by dividing out common factors. We first mark out the common factors 5 and 7 and then divide them out. This leaves minus 3 times 3 divided by 2 times 11.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_002b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to simplify using the Equivalent Fractions Property by dividing out common factors. We first mark out the common factors 5 and 7 and then divide them out. This leaves minus 3 times 3 divided by 2 times 11.\"><\/span><span data-type=\"media\" id=\"fs-id1167836516667\" data-alt=\"Step 3 is to multiply the remaining factors, if necessary. We get minus 9 by 22.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_002c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to multiply the remaining factors, if necessary. We get minus 9 by 22.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836297017\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836318031\"><div data-type=\"problem\" id=\"fs-id1167836550421\"><p id=\"fs-id1167836321058\">Simplify: \\(-\\frac{69}{120}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836362368\"><p id=\"fs-id1167836356125\">\\(-\\frac{23}{40}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833047887\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836363380\"><div data-type=\"problem\"><p id=\"fs-id1167836525114\">Simplify: \\(-\\frac{120}{192}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836320202\"><p id=\"fs-id1167829717832\">\\(-\\frac{5}{8}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836624000\">We now summarize the steps you should follow to simplify fractions.<\/p><div data-type=\"note\" id=\"fs-id1167836282262\" class=\"howto\"><div data-type=\"title\">Simplify a fraction.<\/div><ol id=\"fs-id1167836693896\" type=\"1\" class=\"stepwise\"><li>Rewrite the numerator and denominator to show the common factors.<div data-type=\"newline\"><br><\/div> If needed, factor the numerator and denominator into prime numbers first.<\/li><li>Simplify using the Equivalent Fractions Property by dividing out common factors.<\/li><li>Multiply any remaining factors.<\/li><\/ol><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836486885\"><h3 data-type=\"title\">Multiply and Divide Fractions<\/h3><p id=\"fs-id1167836523502\">Many people find multiplying and dividing fractions easier than adding and subtracting fractions.<\/p><p id=\"fs-id1167829750700\">To multiply fractions, we multiply the numerators and multiply the denominators.<\/p><div data-type=\"note\" id=\"fs-id1167836537188\"><div data-type=\"title\">Fraction Multiplication<\/div><p id=\"fs-id1167836546369\">If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>, and <em data-effect=\"italics\">d<\/em> are numbers where \\(b\\ne 0,\\) and \\(d\\ne 0,\\) then<\/p><div data-type=\"equation\" id=\"fs-id1167836493747\" class=\"unnumbered\" data-label=\"\">\\(\\frac{a}{b}\u00b7\\frac{c}{d}=\\frac{ac}{bd}\\)<\/div><p id=\"fs-id1167836612729\">To multiply fractions, multiply the numerators and multiply the denominators.<\/p><\/div><p id=\"fs-id1167836706036\">When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In <a href=\"#fs-id1167836390100\" class=\"autogenerated-content\">(Figure)<\/a>, we will multiply negative and a positive, so the product will be negative.<\/p><p id=\"fs-id1167836511224\">When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, <em data-effect=\"italics\">a<\/em>, can be written as \\(\\frac{a}{1}.\\) So, for example, \\(3=\\frac{3}{1}.\\)<\/p><div data-type=\"example\" id=\"fs-id1167836390100\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829598267\"><div data-type=\"problem\" id=\"fs-id1167836701672\"><p id=\"fs-id1167836391386\">Multiply: \\(-\\frac{12}{5}\\left(-20x\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836688426\"><p id=\"fs-id1167836391689\">The first step is to find the sign of the product. Since the signs are the same, the product is positive.<\/p><table id=\"fs-id1167836521020\" class=\"unnumbered unstyled\" summary=\"The expression is minus 12 divided by 5 into minus 20 x. We first determine the sign of the product. The signs are the same, so the product is positive. Now we have 12 by 5 into 20 x. Now we write 20 x as a fraction. Hence, we get 12 by 5 open parentheses 20 x by 1 close parentheses. We multiply 12 times 20 x in the numerator and 5 times 1 in the denominator. We then rewrite 20 to show the common factor 5 and divide it out. We simplify to get 48 x.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836544266\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_003a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Determine the sign of the product. The signs\u2003\u2003\u2003\u2003\u2003\u2003<div data-type=\"newline\"><br><\/div> are the same, so the product is positive.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836318933\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_003b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write 20<em data-effect=\"italics\">x<\/em> as a fraction.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836547838\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_003c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836342022\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_003d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite 20 to show the common factor 5<div data-type=\"newline\"><br><\/div> and divide it out.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833055928\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_003e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836547919\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_003f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833024061\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836352569\"><div data-type=\"problem\" id=\"fs-id1167836533364\"><p id=\"fs-id1167836518690\">Multiply: \\(\\frac{11}{3}\\left(-9a\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829693025\"><p id=\"fs-id1167836523426\">\\(-33a\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836285673\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836538840\"><div data-type=\"problem\" id=\"fs-id1167836627735\"><p id=\"fs-id1167836418131\">Multiply: \\(\\frac{13}{7}\\left(-14b\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836613391\"><p id=\"fs-id1167833055111\">\\(-26b\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836622994\">Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, we need some vocabulary. The <span data-type=\"term\">reciprocal<\/span> of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of \\(\\frac{2}{3}\\) is \\(\\frac{3}{2}.\\) Since 4 is written in fraction form as \\(\\frac{4}{1},\\) the reciprocal of 4 is \\(\\frac{1}{4}.\\)<\/p><p id=\"fs-id1167836335184\">To divide fractions, we multiply the first fraction by the reciprocal of the second.<\/p><div data-type=\"note\" id=\"fs-id1167829693065\"><div data-type=\"title\">Fraction Division<\/div><p id=\"fs-id1167836545781\">If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>, and <em data-effect=\"italics\">d<\/em> are numbers where \\(b\\ne 0,c\\ne 0,\\) and \\(d\\ne 0,\\) then<\/p><div data-type=\"equation\" id=\"fs-id1167829695610\" class=\"unnumbered\" data-label=\"\">\\(\\frac{a}{b}\u00f7\\frac{c}{d}=\\frac{a}{b}\u00b7\\frac{d}{c}\\)<\/div><p id=\"fs-id1167836706779\">To divide fractions, we multiply the first fraction by the <strong data-effect=\"bold\">reciprocal<\/strong> of the second.<\/p><\/div><p id=\"fs-id1167836544502\">We need to say \\(b\\ne 0,\\) \\(c\\ne 0,\\) and \\(d\\ne 0,\\) to be sure we don\u2019t divide by zero!<\/p><div data-type=\"example\" id=\"fs-id1167836356409\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836554684\"><div data-type=\"problem\" id=\"fs-id1167836349206\"><p id=\"fs-id1167836448362\">Find the quotient: \\(-\\frac{7}{18}\u00f7\\left(-\\frac{14}{27}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836512548\"><table id=\"fs-id1167833021584\" class=\"unnumbered unstyled\" summary=\"The expression is minus 7 by 8 divided by minus 14 by 27. To divide, multiply the first fraction by the reciprocal of the second. We get minus 7 by 8 multiplied by minus 27 by 14. Determine the sign of the product, and then multiply. We get 7 times 27 divided by 18 times 14. We rewrite showing common factors to get 7 times 9 times 3 divided by 9 times 2 times 7 times 2. We remove the common factors between numerator and denominator. We get 3 in the numerator and 2 times 2 in the denominator. We simplify to get 3 by 4.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836386433\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_004a_img_Errata-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To divide, multiply the first fraction by the\u2003\u2003\u2003\u2003\u2003\u2003<div data-type=\"newline\"><br><\/div>reciprocal of the second.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836378177\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_004b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Determine the sign of the product, and<div data-type=\"newline\"><br><\/div>then multiply.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836295914\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_004c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Rewrite showing common factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836289041\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_004d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Remove common factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836622702\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_004e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836545813\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_004f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836595955\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829690872\"><div data-type=\"problem\" id=\"fs-id1167836362778\"><p id=\"fs-id1167836398875\">Divide: \\(-\\frac{7}{27}\u00f7\\left(-\\frac{35}{36}\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836362322\"><p id=\"fs-id1167836697174\">\\(\\frac{4}{15}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836293918\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836378913\"><div data-type=\"problem\" id=\"fs-id1167829597190\"><p id=\"fs-id1167836417232\">Divide: \\(-\\frac{5}{14}\u00f7\\left(-\\frac{15}{28}\\right).\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167829580094\">\\(\\frac{2}{3}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836700387\">The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a <span data-type=\"term\">complex fraction<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167833018456\"><div data-type=\"title\">Complex Fraction<\/div><p id=\"fs-id1167836611505\">A <strong data-effect=\"bold\">complex fraction<\/strong> is a fraction in which the numerator or the denominator contains a fraction.<\/p><\/div><p id=\"fs-id1167829719773\">Some examples of complex fractions are:<\/p><div data-type=\"equation\" id=\"fs-id1167836706809\" class=\"unnumbered\" data-label=\"\">\\(\\frac{\\frac{6}{7}}{3}\\phantom{\\rule{5em}{0ex}}\\frac{\\frac{3}{4}}{\\frac{5}{8}}\\phantom{\\rule{5em}{0ex}}\\frac{\\frac{x}{2}}{\\frac{5}{6}}\\)<\/div><p id=\"fs-id1167829936808\">To simplify a complex fraction, remember that the fraction bar means division. For example, the complex fraction \\(\\frac{\\frac{3}{4}}{\\frac{5}{8}}\\) means \\(\\frac{3}{4}\u00f7\\frac{5}{8}.\\)<\/p><div data-type=\"example\" id=\"fs-id1167829590640\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836596927\"><div data-type=\"problem\" id=\"fs-id1167836688056\"><p id=\"fs-id1167836652786\">Simplify: \\(\\frac{\\frac{x}{2}}{\\frac{xy}{6}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829594563\"><p id=\"fs-id1167836552744\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\frac{x}{2}}{\\frac{xy}{6}}\\hfill \\\\ \\text{Rewrite as division}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{x}{2}\u00f7\\frac{xy}{6}\\hfill \\\\ \\text{Multiply the first fraction by the reciprocal of the second}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{x}{2}\u00b7\\frac{6}{xy}\\hfill \\\\ \\text{Multiply}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{x\u00b76}{2\u00b7xy}\\hfill \\\\ \\text{Look for common factors}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\overline{)x}\u00b73\u00b7\\overline{)2}}{\\overline{)2}\u00b7\\overline{)x}\u00b7y}\\hfill \\\\ \\text{Divide common factors and simplify}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{3}{y}\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836621356\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836689418\"><div data-type=\"problem\" id=\"fs-id1167836521629\"><p id=\"fs-id1167836521631\">Simplify: \\(\\frac{\\frac{a}{8}}{\\frac{ab}{6}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836522187\"><p id=\"fs-id1167829750716\">\\(\\frac{3}{4b}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829716940\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836613207\"><div data-type=\"problem\" id=\"fs-id1167836520162\"><p id=\"fs-id1167836520164\">Simplify: \\(\\frac{\\frac{p}{2}}{\\frac{pq}{8}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833224683\"><p id=\"fs-id1167836551420\">\\(\\frac{4}{q}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829720923\"><h3 data-type=\"title\">Add and Subtract Fractions<\/h3><p id=\"fs-id1167836585263\">When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.<\/p><div data-type=\"note\" id=\"fs-id1167829744486\"><div data-type=\"title\">Fraction Addition and Subtraction<\/div><p id=\"fs-id1167836493307\">If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are numbers where \\(c\\ne 0,\\) then<\/p><div data-type=\"equation\" id=\"fs-id1171791493356\" class=\"unnumbered\" data-label=\"\">\\(\\frac{a}{c}+\\frac{b}{c}=\\frac{a+b}{c}\\phantom{\\rule{0.5em}{0ex}}\\text{and}\\phantom{\\rule{0.5em}{0ex}}\\frac{a}{c}-\\frac{b}{c}=\\frac{a-b}{c}\\)<\/div><p id=\"fs-id1167833024310\">To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.<\/p><\/div><p id=\"fs-id1167829717535\">The <span data-type=\"term\">least common denominator<\/span> (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.<\/p><div data-type=\"note\" id=\"fs-id1167829714220\"><div data-type=\"title\">Least Common Denominator<\/div><p id=\"fs-id1167836546986\">The <strong data-effect=\"bold\">least common denominator<\/strong> (LCD) of two fractions is the least common multiple (LCM) of their denominators.<\/p><\/div><p id=\"fs-id1167829718952\">After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!<\/p><div data-type=\"example\" id=\"fs-id1167836518722\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Add or Subtract Fractions<\/div><div data-type=\"exercise\" id=\"fs-id1167836518880\"><div data-type=\"problem\" id=\"fs-id1167836550915\"><p id=\"fs-id1167836550917\">Add: \\(\\frac{7}{12}+\\frac{5}{18}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836522037\"><span data-type=\"media\" id=\"fs-id1167836522040\" data-alt=\"The expression is 7 by 12 plus 5 by 18. Step 1 is to check if the two numbers have a common denominator. Since they do not, rewrite each fraction with the LCD (least common denominator). For finding the LCD, we write the factors of 12 as 2 times 2 times 2 and the factors of 18 as 2 times 3 times 3. The LCD is 2 times 2 times 3 times 3, which is equal to 36.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_005a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The expression is 7 by 12 plus 5 by 18. Step 1 is to check if the two numbers have a common denominator. Since they do not, rewrite each fraction with the LCD (least common denominator). For finding the LCD, we write the factors of 12 as 2 times 2 times 2 and the factors of 18 as 2 times 3 times 3. The LCD is 2 times 2 times 3 times 3, which is equal to 36.\"><\/span><span data-type=\"media\" id=\"fs-id1167829721204\" data-alt=\"Step 2 is to add or subtract the fractions. We multiply the numerator and denominator of each fraction by the factor needed to get the denominator to be 36. Do not simplify the equivalent fractions. If you do, you\u2019ll get back to the original fractions and lose the common denominator. We multiply the numerator and denominator of 7 divided by 12, by 3 times. We multiply numerator and denominator of 5 divided by 18 by 2 times. We get the expression 21 by 36 plus 10 by 36.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_005b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to add or subtract the fractions. We multiply the numerator and denominator of each fraction by the factor needed to get the denominator to be 36. Do not simplify the equivalent fractions. If you do, you\u2019ll get back to the original fractions and lose the common denominator. We multiply the numerator and denominator of 7 divided by 12, by 3 times. We multiply numerator and denominator of 5 divided by 18 by 2 times. We get the expression 21 by 36 plus 10 by 36.\"><\/span><span data-type=\"media\" id=\"fs-id1167836481443\" data-alt=\"Step 3 is to simplify is possible. Since 31 is prime, its only factors are 1and 31. Since 31 does not go into 36, the answer is simplified.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_005c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to simplify is possible. Since 31 is prime, its only factors are 1and 31. Since 31 does not go into 36, the answer is simplified.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836558604\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829692339\"><div data-type=\"problem\" id=\"fs-id1167829692341\"><p id=\"fs-id1167836537182\">Add: \\(\\frac{7}{12}+\\frac{11}{15}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829715998\"><p id=\"fs-id1167829716000\">\\(\\frac{79}{60}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836684166\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836684169\"><div data-type=\"problem\" id=\"fs-id1167829580090\"><p id=\"fs-id1167829580092\">Add: \\(\\frac{13}{15}+\\frac{17}{20}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829614382\"><p id=\"fs-id1167829691275\">\\(\\frac{103}{60}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836610898\" class=\"howto\"><div data-type=\"title\">Add or subtract fractions.<\/div><ol id=\"fs-id1167829715974\" type=\"1\" class=\"stepwise\"><li>Do they have a common denominator? <ul id=\"fs-id1167836546938\" data-bullet-style=\"bullet\"><li>Yes\u2014go to step 2.<\/li><li>No\u2014rewrite each fraction with the LCD (least common denominator). <ul id=\"fs-id1167833053829\" data-bullet-style=\"bullet\"><li>Find the LCD.<\/li><li>Change each fraction into an equivalent fraction with the LCD as its denominator.<\/li><\/ul><\/li><\/ul><\/li><li>Add or subtract the fractions.<\/li><li>Simplify, if possible.<\/li><\/ol><\/div><p id=\"fs-id1167829744496\">We now have all four operations for fractions. <a href=\"#fs-id1167829751987\" class=\"autogenerated-content\">(Figure)<\/a> summarizes fraction operations.<\/p><table id=\"fs-id1167829751987\" summary=\"This table gives notes on fraction multiplication, division, addition and subtraction. For fraction multiplication, multiply the numerators and multiply the denominators. Hence, a by b times c by d is ac by bd. For fraction division, multiply the first fraction by the reciprocal of the second. Hence, a by b divided by c by d is ad by bc. For fraction addition, add the numerators and place the sum over the common denominator. Hence, a by c plus b by c is open parentheses a plus b close parentheses by c. For fraction subtraction, subtract the numerators and place the difference over the common denominator. Hence, a by c minus b by c is open parentheses a minus b close parentheses by c. To multiply or divide fractions, an LCD is NOT needed. To add or subtract fractions, an LCD is needed.\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Fraction Multiplication<\/strong><\/td><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Fraction Division<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\frac{a}{b}\u00b7\\frac{c}{d}=\\frac{ac}{bd}\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\frac{a}{b}\u00f7\\frac{c}{d}=\\frac{a}{b}\u00b7\\frac{d}{c}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply the numerators and multiply the denominators<\/td><td data-valign=\"top\" data-align=\"left\">Multiply the first fraction by the reciprocal of the second.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Fraction Addition<\/strong><\/td><td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Fraction Subtraction<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">\\(\\frac{a}{c}+\\frac{b}{c}=\\frac{a+b}{c}\\)<\/td><td data-valign=\"top\" data-align=\"left\">\\(\\frac{a}{c}-\\frac{b}{c}=\\frac{a-b}{c}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add the numerators and place the sum over the common denominator.<\/td><td data-valign=\"top\" data-align=\"left\">Subtract the numerators and place the difference over the common denominator.<\/td><\/tr><tr valign=\"top\"><td colspan=\"2\" data-valign=\"top\" data-align=\"center\">To multiply or divide fractions, an LCD is NOT needed.<div data-type=\"newline\"><br><\/div>To add or subtract fractions, an LCD is needed.<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836367514\">When starting an exercise, always identify the operation and then recall the methods needed for that operation.<\/p><div data-type=\"example\" id=\"fs-id1167836553755\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836553757\"><div data-type=\"problem\" id=\"fs-id1167836553759\"><p id=\"fs-id1167833240062\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{5x}{6}-\\frac{3}{10}\\) <span class=\"token\">\u24d1<\/span> \\(\\frac{5x}{6}\u00b7\\frac{3}{10}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836530461\"><p id=\"fs-id1167829754896\">First ask, \u201cWhat is the operation?\u201d Identifying the operation will determine whether or not we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.<\/p><p id=\"fs-id1167829754898\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{ccc}\\text{What is the operation? The operation is subtraction}.\\hfill &amp; &amp; \\\\ \\text{Do the fractions have a common denominator? No}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}\\frac{5x}{6}-\\frac{3}{10}\\hfill \\\\ \\text{Find the LCD of}\\phantom{\\rule{0.2em}{0ex}}6\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}10\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}\\text{The LCD is 30.}\\hfill \\\\ \\begin{array}{ccccccc}&amp; &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{1.5em}{0ex}}6=2\u00b73\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; &amp; \\underset{___________}{\\phantom{\\rule{0.4em}{0ex}}10=2\u00b75}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; &amp; \\text{LCD}=2\u00b73\u00b75\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; &amp; \\text{LCD}=30\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\\\ \\\\ \\\\ \\text{Rewrite each fraction as an equivalent fraction with the LCD}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}\\frac{5x\u00b75}{6\u00b75}-\\frac{3\u00b73}{10\u00b73}\\hfill \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}\\frac{25x}{30}-\\frac{9}{30}\\hfill \\\\ \\begin{array}{c}\\text{Subtract the numerators and place the}\\hfill \\\\ \\text{difference over the common denominators}.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}\\frac{25x-9}{30}\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\text{Simplify},\\text{if possible}.\\phantom{\\rule{0.2em}{0ex}}\\text{There are no common factors}.\\hfill \\\\ \\text{The fraction is simplified}.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\end{array}\\)<p id=\"fs-id1167833340094\"><span class=\"token\">\u24d1<\/span><\/p><div data-type=\"newline\"><br><\/div>\\(\\begin{array}{cccc}\\text{What is the operation? Multiplication}.\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{7em}{0ex}}\\frac{25x}{6}\u00b7\\frac{3}{10}\\hfill \\\\ \\begin{array}{c}\\text{To multiply fractions},\\text{multiply the numerators}\\hfill \\\\ \\text{and multiply the denominators}.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{7em}{0ex}}\\frac{25x\u00b73}{6\u00b710}\\hfill \\\\ \\begin{array}{c}\\text{Rewrite, showing common factors.}\\hfill \\\\ \\text{Remove common factors.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{7em}{0ex}}\\frac{\\overline{)5}x\u00b7\\overline{)3}}{2\u00b7\\overline{)3}\u00b72\u00b7\\overline{)5}}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{7em}{0ex}}\\frac{x}{4}\\hfill \\end{array}\\)<p id=\"fs-id1167836692915\">Notice, we needed an LCD to add \\(\\frac{25x}{6}-\\frac{3}{10},\\) but not to multiply \\(\\frac{25x}{6}\u00b7\\frac{3}{10}.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836698608\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836698612\"><div data-type=\"problem\" id=\"fs-id1167836684139\"><p id=\"fs-id1167836684141\">Simplify: <span class=\"token\">\u24d0<\/span> \\(\\frac{3a}{4}-\\frac{8}{9}\\) <span class=\"token\">\u24d1<\/span> \\(\\frac{3a}{4}\u00b7\\frac{8}{9}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836684124\"><p id=\"fs-id1167836684126\"><span class=\"token\">\u24d0<\/span>\\(\\frac{27a-32}{36}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{2a}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829744140\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829744143\"><div data-type=\"problem\" id=\"fs-id1167836595910\"><p id=\"fs-id1167836595912\">Simplify: <span class=\"token\">\u24d0<\/span>\\(\\frac{4k}{5}-\\frac{1}{6}\\) <span class=\"token\">\u24d1<\/span> \\(\\frac{4k}{5}\u00b7\\frac{1}{6}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829589696\"><p id=\"fs-id1167829589698\"><span class=\"token\">\u24d0<\/span>\\(\\frac{24k-5}{30}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{2k}{15}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829720090\"><h3 data-type=\"title\">Use the Order of Operations to Simplify Fractions<\/h3><p id=\"fs-id1167836688781\">The fraction bar in a fraction acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.<\/p><div data-type=\"note\" id=\"fs-id1167836688785\" class=\"howto\"><div data-type=\"title\">Simplify an expression with a fraction bar.<\/div><ol id=\"fs-id1167833412506\" type=\"1\" class=\"stepwise\"><li>Simplify the expression in the numerator. Simplify the expression in the denominator.<\/li><li>Simplify the fraction.<\/li><\/ol><\/div><p id=\"fs-id1167829744159\">Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.<\/p><div data-type=\"equation\" id=\"fs-id1167829754611\" class=\"unnumbered\" data-label=\"\">\\(\\frac{-1}{3}=-\\frac{1}{3}\\phantom{\\rule{3em}{0ex}}\\frac{\\text{negative}}{\\text{positive}}=\\text{negative}\\)<\/div><div data-type=\"equation\" id=\"fs-id1167833050784\" class=\"unnumbered\" data-label=\"\">\\(\\frac{1}{-3}=-\\frac{1}{3}\\phantom{\\rule{3em}{0ex}}\\frac{\\text{positive}}{\\text{negative}}=\\text{negative}\\)<\/div><div data-type=\"note\" id=\"fs-id1167829742311\"><div data-type=\"title\">Placement of Negative Sign in a Fraction<\/div><p id=\"fs-id1167829741605\">For any positive numbers <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>,<\/p><div data-type=\"equation\" id=\"fs-id1167829754672\" class=\"unnumbered\" data-label=\"\">\\(\\frac{\\text{\u2212}a}{b}=\\frac{a}{\\text{\u2212}b}=-\\frac{a}{b}\\)<\/div><\/div><div data-type=\"example\" id=\"fs-id1167829755877\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829755879\"><div data-type=\"problem\" id=\"fs-id1167829755881\"><p id=\"fs-id1167836484393\">Simplify: \\(\\frac{4\\left(-3\\right)+6\\left(-2\\right)}{-3\\left(2\\right)-2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829750490\"><p id=\"fs-id1167829750492\">The fraction bar acts like a grouping symbol. So completely simplify the numerator and the denominator separately.<\/p><p id=\"fs-id1167829750496\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{4\\left(-3\\right)+6\\left(-2\\right)}{-3\\left(2\\right)-2}\\hfill \\\\ \\text{Multiply.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{-12+\\left(-12\\right)}{-6-2}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{-24}{-8}\\hfill \\\\ \\text{Divide.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}3\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833412533\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829627923\"><div data-type=\"problem\" id=\"fs-id1167829627925\"><p id=\"fs-id1167829627927\">Simplify: \\(\\frac{8\\left(-2\\right)+4\\left(-3\\right)}{-5\\left(2\\right)+3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829753437\"><p id=\"fs-id1167829753439\">4<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829743958\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829743962\"><div data-type=\"problem\" id=\"fs-id1167829743964\"><p id=\"fs-id1167829691090\">Simplify: \\(\\frac{7\\left(-1\\right)+9\\left(-3\\right)}{-5\\left(3\\right)-2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829754202\"><p id=\"fs-id1167829754205\">2<\/p><\/div><\/div><\/div><p id=\"fs-id1167829754211\">Now we\u2019ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator as the fraction bar means division.<\/p><div data-type=\"example\" id=\"fs-id1167829627815\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Simplify Complex Fractions<\/div><div data-type=\"exercise\" id=\"fs-id1167829627820\"><div data-type=\"problem\" id=\"fs-id1167829754213\"><p id=\"fs-id1167829752824\">Simplify: \\(\\frac{{\\left(\\frac{1}{2}\\right)}^{2}}{4+{3}^{2}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829754739\"><span data-type=\"media\" id=\"fs-id1167829754742\" data-alt=\"The expression is 1 by 2 the whole squared divided by 4 plus 3 squared. Step 1 is to simplify the numerator, which becomes 1 by 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_006a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The expression is 1 by 2 the whole squared divided by 4 plus 3 squared. Step 1 is to simplify the numerator, which becomes 1 by 4.\"><\/span><span data-type=\"media\" id=\"fs-id1167829754924\" data-alt=\"Step 2 is to simplify the denominator, which becomes 4 plus 9 equals 13.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_006b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to simplify the denominator, which becomes 4 plus 9 equals 13.\"><\/span><span data-type=\"media\" id=\"fs-id1167829720638\" data-alt=\"Step 3 is to divide the numerator by the denominator and simplify if possible. Now the expression becomes 1 by 4 divided by 13 by 1, which equals 1 by 4 multiplied by 1 by 13, which equals 1 by 52\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_006c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to divide the numerator by the denominator and simplify if possible. Now the expression becomes 1 by 4 divided by 13 by 1, which equals 1 by 4 multiplied by 1 by 13, which equals 1 by 52\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829753146\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829753149\"><div data-type=\"problem\" id=\"fs-id1167829753151\"><p id=\"fs-id1167829753154\">Simplify: \\(\\frac{{\\left(\\frac{1}{3}\\right)}^{2}}{{2}^{3}+2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833025479\"><p id=\"fs-id1167833025481\">\\(\\frac{1}{90}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836477915\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836477918\"><div data-type=\"problem\" id=\"fs-id1167836477920\"><p id=\"fs-id1167836580186\">Simplify: \\(\\frac{1+{4}^{2}}{{\\left(\\frac{1}{4}\\right)}^{2}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829751585\"><p id=\"fs-id1167829751588\">272<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829753394\" class=\"howto\"><div data-type=\"title\">Simplify complex fractions.<\/div><ol id=\"fs-id1167829753400\" type=\"1\" class=\"stepwise\"><li>Simplify the numerator.<\/li><li>Simplify the denominator.<\/li><li>Divide the numerator by the denominator. Simplify if possible.<\/li><\/ol><\/div><div data-type=\"example\" id=\"fs-id1167829750481\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829750483\"><div data-type=\"problem\" id=\"fs-id1167829750485\"><p id=\"fs-id1167829755820\">Simplify: \\(\\frac{\\frac{1}{2}+\\frac{2}{3}}{\\frac{3}{4}-\\frac{1}{6}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829750436\"><p id=\"fs-id1167829750438\">It may help to put parentheses around the numerator and the denominator.<\/p><p id=\"fs-id1167829750442\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\left(\\frac{1}{2}+\\frac{2}{3}\\right)}{\\left(\\frac{3}{4}-\\frac{1}{6}\\right)}\\hfill \\\\ \\begin{array}{c}\\text{Simplify the numerator}\\phantom{\\rule{0.2em}{0ex}}\\left(\\text{LCD}=6\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\hfill \\\\ \\text{simplify the denominator}\\phantom{\\rule{0.2em}{0ex}}\\left(\\text{LCD}=12\\right).\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\left(\\frac{3}{6}+\\frac{4}{6}\\right)}{\\left(\\frac{9}{12}-\\frac{2}{12}\\right)}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\left(\\frac{7}{6}\\right)}{\\left(\\frac{7}{12}\\right)}\\hfill \\\\ \\text{Divide the numerator by the denominator.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{7}{6}\u00f7\\frac{7}{12}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{7}{6}\\cdot \\frac{12}{7}\\hfill \\\\ \\text{Divide out common factors.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\frac{\\overline{)7}\\cdot \\overline{)6}\\cdot 2}{\\overline{)6}\\cdot \\overline{)7}\\cdot 1}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}2\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836407228\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836407232\"><div data-type=\"problem\" id=\"fs-id1167836407234\"><p id=\"fs-id1167836407236\">Simplify: \\(\\frac{\\frac{1}{3}+\\frac{1}{2}}{\\frac{3}{4}-\\frac{1}{3}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829754361\"><p id=\"fs-id1167829754363\">2<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836585276\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836585279\"><div data-type=\"problem\" id=\"fs-id1167836585281\"><p id=\"fs-id1167836585283\">Simplify: \\(\\frac{\\frac{2}{3}-\\frac{1}{2}}{\\frac{1}{4}+\\frac{1}{3}}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829750741\"><p id=\"fs-id1167829750743\">\\(\\frac{2}{7}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829754323\"><h3 data-type=\"title\">Evaluate Variable Expressions with Fractions<\/h3><p id=\"fs-id1167829754328\">We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.<\/p><div data-type=\"example\" id=\"fs-id1167829754333\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829754335\"><div data-type=\"problem\" id=\"fs-id1167829754337\"><p id=\"fs-id1167836477223\">Evaluate \\(2{x}^{2}y\\) when \\(x=\\frac{1}{4}\\) and \\(y=-\\frac{2}{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829741825\"><p id=\"fs-id1167829741827\">Substitute the values into the expression.<\/p><table id=\"fs-id1167829741831\" class=\"unnumbered unstyled\" summary=\"The expression is 2 x squared y. Substitute 1 by 4 for x and minus 2 by 3 for y. We now have 2 into open parentheses 1 by 4 close parentheses squared open parentheses minus 2 by 3 close parentheses. Simplifying the exponents first, we get, 2 into open parentheses 1 by 16 close parentheses open parentheses minus 2 by 3 close parentheses. We multiply to get minus 2 times 1 times 2 divided by 2 times 2 times 4 times 3. We write 16 as 2 times 2 times 4 to make it easy to remove common factors. Now divide out the common factors. We are left with minus 1 in the numerator and 4 times 3 in the denominator. We simplify to get minus 1 by 12.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829756310\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_007b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829756324\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_007a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829756334\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_007c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify exponents first.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836409091\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_007d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Multiply; divide out the common factors.<div data-type=\"newline\"><br><\/div>Notice we write 16 as \\(2\u00b72\u00b74\\) to make it easy to<div data-type=\"newline\"><br><\/div>remove common factors.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829749843\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_007e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829756113\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_007f_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829756127\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829756130\"><div data-type=\"problem\" id=\"fs-id1167829756133\"><p id=\"fs-id1167829756135\">Evaluate \\(3a{b}^{2}\\) when \\(a=-\\frac{2}{3}\\) and \\(b=-\\frac{1}{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829743935\"><p id=\"fs-id1167829743938\">\\(-\\frac{1}{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829741554\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829741557\"><div data-type=\"problem\" id=\"fs-id1167829741560\"><p id=\"fs-id1167829741562\">Evaluate \\(4{c}^{3}d\\) when \\(c=-\\frac{1}{2}\\) and \\(d=-\\frac{4}{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829742275\"><p id=\"fs-id1167829742277\">\\(\\frac{2}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829742290\" class=\"media-2\"><p id=\"fs-id1167829742293\">Access this online resource for additional instruction and practice with fractions.<\/p><ul id=\"fs-id1167829742297\" data-bullet-style=\"bullet\"><li><a href=\"https:\/\/openstax.org\/l\/25addfractions\">Adding Fractions with Unlike Denominators<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836700134\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167835361771\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Equivalent Fractions Property<\/strong><div data-type=\"newline\"><br><\/div> If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are numbers where \\(b\\ne 0,c\\ne 0,\\) then<div data-type=\"newline\"><br><\/div> \\(\\phantom{\\rule{2em}{0ex}}\\frac{a}{b}=\\frac{a\u00b7c}{b\u00b7c}\\phantom{\\rule{0.5em}{0ex}}\\text{and}\\phantom{\\rule{0.5em}{0ex}}\\frac{a\u00b7c}{b\u00b7c}=\\frac{a}{b}.\\)<\/li><li><strong data-effect=\"bold\">How to simplify a fraction.<\/strong><ol id=\"fs-id1167829690649\" type=\"1\" class=\"stepwise\"><li>Rewrite the numerator and denominator to show the common factors.<div data-type=\"newline\"><br><\/div> If needed, factor the numerator and denominator into prime numbers first.<\/li><li>Simplify using the Equivalent Fractions Property by dividing out common factors.<\/li><li>Multiply any remaining factors.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">Fraction Multiplication<\/strong><div data-type=\"newline\"><br><\/div> If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>, and <em data-effect=\"italics\">d<\/em> are numbers where \\(b\\ne 0,\\) and \\(d\\ne 0,\\) then<div data-type=\"newline\"><br><\/div> \\(\\phantom{\\rule{2em}{0ex}}\\frac{a}{b}\u00b7\\frac{c}{d}=\\frac{ac}{bd}.\\)<div data-type=\"newline\"><br><\/div> To multiply fractions, multiply the numerators and multiply the denominators.<\/li><li><strong data-effect=\"bold\">Fraction Division<\/strong><div data-type=\"newline\"><br><\/div> If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>, and <em data-effect=\"italics\">d<\/em> are numbers where \\(b\\ne 0,c\\ne 0,\\) and \\(d\\ne 0,\\) then<div data-type=\"newline\"><br><\/div> \\(\\phantom{\\rule{2em}{0ex}}\\frac{a}{b}\u00f7\\frac{c}{d}=\\frac{a}{b}\u00b7\\frac{d}{c}.\\)<div data-type=\"newline\"><br><\/div> To divide fractions, we multiply the first fraction by the reciprocal of the second.<\/li><li><strong data-effect=\"bold\">Fraction Addition and Subtraction<\/strong><div data-type=\"newline\"><br><\/div> If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are numbers where \\(c\\ne 0,\\) then<div data-type=\"newline\"><br><\/div> \\(\\phantom{\\rule{2em}{0ex}}\\frac{a}{c}+\\frac{b}{c}=\\frac{a+b}{c}\\phantom{\\rule{0.5em}{0ex}}\\text{and}\\phantom{\\rule{0.5em}{0ex}}\\frac{a}{c}-\\frac{b}{c}=\\frac{a-b}{c}.\\)<div data-type=\"newline\"><br><\/div> To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.<\/li><li><strong data-effect=\"bold\">How to add or subtract fractions.<\/strong><div data-type=\"newline\"><br><\/div><ol id=\"fs-id1167829751330\" type=\"1\" class=\"stepwise\"><li>Do they have a common denominator? <ul id=\"fs-id1167829751341\" data-bullet-style=\"open-circle\"><li>Yes\u2014go to step 2.<\/li><li>No\u2014rewrite each fraction with the LCD (least common denominator). <ul id=\"fs-id1167829751354\" data-bullet-style=\"bullet\"><li>Find the LCD.<\/li><li>Change each fraction into an equivalent fraction with the LCD as its denominator.<\/li><\/ul><\/li><\/ul><\/li><li>Add or subtract the fractions.<\/li><li>Simplify, if possible.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">How to simplify an expression with a fraction bar.<\/strong><ol id=\"fs-id1167829751381\" type=\"1\" class=\"stepwise\"><li>Simplify the expression in the numerator. Simplify the expression in the denominator.<\/li><li>Simplify the fraction.<\/li><\/ol><\/li><li><strong data-effect=\"bold\">Placement of Negative Sign in a Fraction<\/strong><div data-type=\"newline\"><br><\/div> For any positive numbers <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>,<div data-type=\"newline\"><br><\/div> \\(\\phantom{\\rule{2em}{0ex}}\\frac{\\text{\u2212}a}{b}=\\frac{a}{\\text{\u2212}b}=-\\frac{a}{b}.\\)<\/li><li><strong data-effect=\"bold\">How to simplify complex fractions.<\/strong><div data-type=\"newline\"><br><\/div><ol id=\"fs-id1167829754290\" type=\"1\" class=\"stepwise\"><li>Simplify the numerator.<\/li><li>Simplify the denominator.<\/li><li>Divide the numerator by the denominator. Simplify if possible.<\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836597065\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836597069\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167836597076\"><strong data-effect=\"bold\">Simplify Fractions<\/strong><\/p><p id=\"fs-id1167836597082\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1167836597086\"><div data-type=\"problem\" id=\"fs-id1167836597088\"><p id=\"fs-id1167836597090\">\\(-\\frac{108}{63}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836597105\"><p id=\"fs-id1167836597107\">\\(-\\frac{12}{7}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836597122\"><div data-type=\"problem\" id=\"fs-id1167836597124\"><p id=\"fs-id1167836597126\">\\(-\\frac{104}{48}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829624446\"><div data-type=\"problem\" id=\"fs-id1167829624448\"><p id=\"fs-id1167829624450\">\\(\\frac{120}{252}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829624463\"><p id=\"fs-id1167829624465\">\\(\\frac{10}{21}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829624479\"><div data-type=\"problem\" id=\"fs-id1167829624481\"><p id=\"fs-id1167829624483\">\\(\\frac{182}{294}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829624512\"><div data-type=\"problem\" id=\"fs-id1167829624514\"><p id=\"fs-id1167829624516\">\\(\\frac{14{x}^{2}}{21y}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829717620\"><p id=\"fs-id1167829717622\">\\(\\frac{2{x}^{2}}{3y}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829717644\"><div data-type=\"problem\" id=\"fs-id1167829717646\"><p id=\"fs-id1167829717648\">\\(\\frac{24a}{32{b}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829614578\"><div data-type=\"problem\" id=\"fs-id1167829614580\"><p id=\"fs-id1167829614582\">\\(-\\frac{210{a}^{2}}{110{b}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829614608\"><p id=\"fs-id1167829614610\">\\(-\\frac{21{a}^{2}}{11{b}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829614637\"><div data-type=\"problem\" id=\"fs-id1167829614639\"><p id=\"fs-id1167829614641\">\\(-\\frac{30{x}^{2}}{105{y}^{2}}\\)<\/p><\/div><\/div><p id=\"fs-id1167836698533\"><strong data-effect=\"bold\">Multiply and Divide Fractions<\/strong><\/p><p id=\"fs-id1167836698539\">In the following exercises, perform the indicated operation.<\/p><div data-type=\"exercise\" id=\"fs-id1167836698542\"><div data-type=\"problem\" id=\"fs-id1167836698544\"><p id=\"fs-id1167836698547\">\\(-\\frac{3}{4}\\left(-\\frac{4}{9}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836698573\"><p id=\"fs-id1167836698575\">\\(\\frac{1}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829755551\"><div data-type=\"problem\" id=\"fs-id1167829755553\"><p id=\"fs-id1167829755556\">\\(-\\frac{3}{8}\u00b7\\frac{4}{15}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829755594\"><div data-type=\"problem\" id=\"fs-id1167829755596\"><p id=\"fs-id1167829755598\">\\(\\left(-\\frac{14}{15}\\right)\\left(\\frac{9}{20}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829755632\"><p id=\"fs-id1167829755634\">\\(-\\frac{21}{50}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829755734\"><div data-type=\"problem\" id=\"fs-id1167829755737\"><p id=\"fs-id1167829755739\">\\(\\left(-\\frac{9}{10}\\right)\\left(\\frac{25}{33}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829755791\"><div data-type=\"problem\" id=\"fs-id1167829755793\"><p id=\"fs-id1167829755795\">\\(\\left(-\\frac{63}{84}\\right)\\left(-\\frac{44}{90}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836541862\"><p id=\"fs-id1167836541864\">\\(\\frac{11}{30}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836541878\"><div data-type=\"problem\" id=\"fs-id1167836541880\"><p id=\"fs-id1167836541882\">\\(\\left(-\\frac{33}{60}\\right)\\left(-\\frac{40}{88}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836541934\"><div data-type=\"problem\" id=\"fs-id1167836541936\"><p id=\"fs-id1167836541938\">\\(\\frac{3}{7}\u00b721n\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829720481\"><p id=\"fs-id1167829720483\">\\(9n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829720494\"><div data-type=\"problem\" id=\"fs-id1167829720496\"><p id=\"fs-id1167829720498\">\\(\\frac{5}{6}\u00b730m\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829720528\"><div data-type=\"problem\" id=\"fs-id1167829720530\"><p id=\"fs-id1167829720532\">\\(\\frac{3}{4}\u00f7\\frac{x}{11}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829720551\"><p id=\"fs-id1167829720553\">\\(\\frac{33}{4x}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750069\"><div data-type=\"problem\" id=\"fs-id1167829750071\"><p id=\"fs-id1167829750073\">\\(\\frac{2}{5}\u00f7\\frac{y}{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750109\"><div data-type=\"problem\" id=\"fs-id1167829750111\"><p id=\"fs-id1167829750114\">\\(\\frac{5}{18}\u00f7\\left(-\\frac{15}{24}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829750143\"><p id=\"fs-id1167829750146\">\\(-\\frac{4}{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750159\"><div data-type=\"problem\" id=\"fs-id1167829750162\"><p id=\"fs-id1167829750164\">\\(\\frac{7}{18}\u00f7\\left(-\\frac{14}{27}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829741742\"><div data-type=\"problem\" id=\"fs-id1167829741745\"><p id=\"fs-id1167829741747\">\\(\\frac{8u}{15}\u00f7\\frac{12v}{25}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829741773\"><p id=\"fs-id1167829741776\">\\(\\frac{10u}{9v}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829741794\"><div data-type=\"problem\" id=\"fs-id1167829754089\"><p id=\"fs-id1167829754091\">\\(\\frac{12r}{25}\u00f7\\frac{18s}{35}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829754138\"><div data-type=\"problem\" id=\"fs-id1167829754140\"><p id=\"fs-id1167829754142\">\\(\\frac{3}{4}\u00f7\\left(-12\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829754163\"><p id=\"fs-id1167829754165\">\\(-\\frac{1}{16}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829754180\"><div data-type=\"problem\" id=\"fs-id1167829754182\"><p id=\"fs-id1167829754185\">\\(-15\u00f7\\left(-\\frac{5}{3}\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1167833350831\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1167833350834\"><div data-type=\"problem\" id=\"fs-id1167833350836\"><p id=\"fs-id1167833350838\">\\(\\frac{-\\frac{8}{21}}{\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}\\frac{12}{35}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833350868\"><p id=\"fs-id1167833350870\">\\(-\\frac{10}{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833350885\"><div data-type=\"problem\" id=\"fs-id1167833350888\"><p id=\"fs-id1167833350890\">\\(\\frac{-\\frac{9}{16}}{\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}\\frac{33}{40}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829743148\"><div data-type=\"problem\" id=\"fs-id1167829743150\"><p id=\"fs-id1167829743152\">\\(\\frac{-\\frac{4}{5}}{\\phantom{\\rule{0.2em}{0ex}}\\text{}\\phantom{\\rule{0.2em}{0ex}}2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829743178\"><p id=\"fs-id1167829743180\">\\(-\\frac{2}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829743194\"><div data-type=\"problem\" id=\"fs-id1167829743196\"><p id=\"fs-id1167829743198\">\\(\\frac{\\frac{5}{3}}{\\phantom{\\rule{0.2em}{0ex}}\\text{10}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836409504\"><div data-type=\"problem\" id=\"fs-id1167836409506\"><p id=\"fs-id1167836409508\">\\(\\frac{\\frac{m}{3}}{\\frac{n}{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836409527\"><p id=\"fs-id1167836409529\">\\(\\frac{2m}{3n}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836409548\"><div data-type=\"problem\" id=\"fs-id1167836409550\"><p id=\"fs-id1167836409552\">\\(\\frac{-\\frac{3}{8}}{-\\frac{y}{12}}\\)<\/p><\/div><\/div><p id=\"fs-id1167836409593\"><strong data-effect=\"bold\">Add and Subtract Fractions<\/strong><\/p><p id=\"fs-id1167836409599\">In the following exercises, add or subtract.<\/p><div data-type=\"exercise\" id=\"fs-id1167836409602\"><div data-type=\"problem\" id=\"fs-id1167833025150\"><p id=\"fs-id1167833025152\">\\(\\frac{7}{12}+\\frac{5}{8}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833025171\"><p id=\"fs-id1167833025173\">\\(\\frac{29}{24}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833025187\"><div data-type=\"problem\" id=\"fs-id1167833025189\"><p id=\"fs-id1167833025191\">\\(\\frac{5}{12}+\\frac{3}{8}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833025226\"><div data-type=\"problem\" id=\"fs-id1167833025229\"><p id=\"fs-id1167833025231\">\\(\\frac{7}{12}-\\frac{9}{16}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833025251\"><p id=\"fs-id1167833025253\">\\(\\frac{1}{48}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829753938\"><div data-type=\"problem\" id=\"fs-id1167829753940\"><p id=\"fs-id1167829753943\">\\(\\frac{7}{16}-\\frac{5}{12}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829753978\"><div data-type=\"problem\" id=\"fs-id1167829753980\"><p id=\"fs-id1167829753982\">\\(-\\frac{13}{30}+\\frac{25}{42}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829754007\"><p id=\"fs-id1167829754009\">\\(\\frac{17}{105}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829754023\"><div data-type=\"problem\" id=\"fs-id1167829754025\"><p id=\"fs-id1167829754027\">\\(-\\frac{23}{30}+\\frac{5}{48}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829756176\"><div data-type=\"problem\" id=\"fs-id1167829756178\"><p id=\"fs-id1167829756180\">\\(-\\frac{39}{56}-\\frac{22}{35}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829756204\"><p id=\"fs-id1167829756206\">\\(-\\frac{53}{40}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829756222\"><div data-type=\"problem\" id=\"fs-id1167829756225\"><p id=\"fs-id1167829756227\">\\(-\\frac{33}{49}-\\frac{18}{35}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829752841\"><div data-type=\"problem\" id=\"fs-id1167829752843\"><p id=\"fs-id1167829752845\">\\(-\\frac{2}{3}-\\left(-\\frac{3}{4}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829752874\"><p id=\"fs-id1167829752876\">\\(\\frac{1}{12}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829752889\"><div data-type=\"problem\" id=\"fs-id1167829752891\"><p id=\"fs-id1167829752893\">\\(-\\frac{3}{4}-\\left(-\\frac{4}{5}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829752937\"><div data-type=\"problem\" id=\"fs-id1167829752939\"><p id=\"fs-id1167829752941\">\\(\\frac{x}{3}+\\frac{1}{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829742577\"><p id=\"fs-id1167829742579\">\\(\\frac{4x+3}{12}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829742599\"><div data-type=\"problem\" id=\"fs-id1167829742601\"><p id=\"fs-id1167829742604\">\\(\\frac{x}{5}-\\frac{1}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829742644\"><div data-type=\"problem\" id=\"fs-id1167829742646\"><p id=\"fs-id1167829742648\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\frac{2}{3}+\\frac{1}{6}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{2}{3}\u00f7\\frac{1}{6}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167836595709\"><p id=\"fs-id1167836595711\"><span class=\"token\">\u24d0<\/span>\\(\\frac{5}{6}\\)<span class=\"token\">\u24d1<\/span>\\(4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836595737\"><div data-type=\"problem\" id=\"fs-id1167836595740\"><p id=\"fs-id1167836595742\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-\\frac{2}{5}-\\frac{1}{8}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-\\frac{2}{5}\u00b7\\frac{1}{8}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829743231\"><div data-type=\"problem\" id=\"fs-id1167829743233\"><p id=\"fs-id1167829743235\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\frac{5n}{6}\u00f7\\frac{8}{15}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{5n}{6}-\\frac{8}{15}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167829743288\"><p id=\"fs-id1167829743290\"><span class=\"token\">\u24d0<\/span>\\(\\frac{25n}{16}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{25n-16}{30}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829743334\"><div data-type=\"problem\" id=\"fs-id1167829743336\"><p id=\"fs-id1167829743338\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\frac{3a}{8}\u00f7\\frac{7}{12}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{3a}{8}-\\frac{7}{12}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829744261\"><div data-type=\"problem\" id=\"fs-id1167829744263\"><p id=\"fs-id1167829744266\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-\\frac{4x}{9}-\\frac{5}{6}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-\\frac{4k}{9}\u00b7\\frac{5}{6}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167829744812\"><p id=\"fs-id1167829744814\"><span class=\"token\">\u24d0<\/span>\\(\\frac{-8x-15}{18}\\)<span class=\"token\">\u24d1<\/span>\\(-\\frac{10k}{27}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829744860\"><div data-type=\"problem\" id=\"fs-id1167829744862\"><p id=\"fs-id1167829744864\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-\\frac{3y}{8}-\\frac{4}{3}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-\\frac{3y}{8}\u00b7\\frac{4}{3}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829744383\"><div data-type=\"problem\" id=\"fs-id1167829744385\"><p id=\"fs-id1167829744387\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(-\\frac{5a}{3}+\\left(-\\frac{10}{6}\\right)\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(-\\frac{5a}{3}\u00f7\\left(-\\frac{10}{6}\\right)\\)<\/div><div data-type=\"solution\" id=\"fs-id1167829744462\"><p id=\"fs-id1167829744464\"><span class=\"token\">\u24d0<\/span>\\(\\frac{-5\\left(a+1\\right)}{3}\\)<span class=\"token\">\u24d1<\/span>\\(a\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829753473\"><div data-type=\"problem\" id=\"fs-id1167829753475\"><p id=\"fs-id1167829753477\"><\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span>\\(\\frac{2b}{5}+\\frac{8}{15}\\)<div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span>\\(\\frac{2b}{5}\u00f7\\frac{8}{15}\\)<\/div><\/div><p id=\"fs-id1167829753575\"><strong data-effect=\"bold\">Use the Order of Operations to Simplify Fractions<\/strong><\/p><p id=\"fs-id1167829741857\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1167829741860\"><div data-type=\"problem\" id=\"fs-id1167829741862\"><p id=\"fs-id1167829741864\">\\(\\frac{5\u00b76-3\u00b74}{4\u00b75-2\u00b73}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829741903\"><p id=\"fs-id1167829741905\">\\(\\frac{9}{7}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829741916\"><div data-type=\"problem\" id=\"fs-id1167829741919\"><p id=\"fs-id1167829741921\">\\(\\frac{8\u00b79-7\u00b76}{5\u00b76-9\u00b72}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829741973\"><div data-type=\"problem\" id=\"fs-id1167829741975\"><p id=\"fs-id1167829741977\">\\(\\frac{{5}^{2}-{3}^{2}}{3-5}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829755927\"><p id=\"fs-id1167829755929\">\\(-8\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829755937\"><div data-type=\"problem\" id=\"fs-id1167829755939\"><p id=\"fs-id1167829755942\">\\(\\frac{{6}^{2}-{4}^{2}}{4-6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829755980\"><div data-type=\"problem\" id=\"fs-id1167829755982\"><p id=\"fs-id1167829755984\">\\(\\frac{7\u00b74-2\\left(8-5\\right)}{9\u00b73-3\u00b75}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829756031\"><p id=\"fs-id1167829756033\">\\(\\frac{11}{6}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829756046\"><div data-type=\"problem\" id=\"fs-id1167829756048\"><p id=\"fs-id1167829756050\">\\(\\frac{9\u00b77-3\\left(12-8\\right)}{8\u00b77-6\u00b76}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829752395\"><div data-type=\"problem\" id=\"fs-id1167829752397\"><p id=\"fs-id1167829752399\">\\(\\frac{9\\left(8-2\\right)-3\\left(15-7\\right)}{6\\left(7-1\\right)-3\\left(17-9\\right)}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829752472\"><p id=\"fs-id1167829752474\">\\(\\frac{5}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829752486\"><div data-type=\"problem\" id=\"fs-id1167829752488\"><p id=\"fs-id1167829752490\">\\(\\frac{8\\left(9-2\\right)-4\\left(14-9\\right)}{7\\left(8-3\\right)-3\\left(16-9\\right)}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829744633\"><div data-type=\"problem\" id=\"fs-id1167829744635\"><p id=\"fs-id1167829744637\">\\(\\frac{{2}^{3}+{4}^{2}}{{\\left(\\frac{2}{3}\\right)}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829744674\"><p id=\"fs-id1167829744676\">\\(54\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829744685\"><div data-type=\"problem\" id=\"fs-id1167829744687\"><p id=\"fs-id1167829744689\">\\(\\frac{{3}^{3}-{3}^{2}}{{\\left(\\frac{3}{4}\\right)}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750519\"><div data-type=\"problem\" id=\"fs-id1167829750521\"><p id=\"fs-id1167829750523\">\\(\\frac{{\\left(\\frac{3}{5}\\right)}^{2}}{{\\left(\\frac{3}{7}\\right)}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829750564\"><p id=\"fs-id1167829750566\">\\(\\frac{49}{25}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750580\"><div data-type=\"problem\" id=\"fs-id1167829750582\"><p id=\"fs-id1167829750584\">\\(\\frac{{\\left(\\frac{3}{4}\\right)}^{2}}{{\\left(\\frac{5}{8}\\right)}^{2}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750640\"><div data-type=\"problem\" id=\"fs-id1167829750643\"><p id=\"fs-id1167829750645\">\\(\\frac{2}{\\frac{1}{3}+\\frac{1}{5}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829750667\"><p id=\"fs-id1167829750669\">\\(\\frac{15}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750682\"><div data-type=\"problem\" id=\"fs-id1167833049496\"><p id=\"fs-id1167833049498\">\\(\\frac{5}{\\frac{1}{4}+\\frac{1}{3}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833049535\"><div data-type=\"problem\" id=\"fs-id1167833049537\"><p id=\"fs-id1167833049540\">\\(\\frac{\\frac{7}{8}-\\frac{2}{3}}{\\frac{1}{2}+\\frac{3}{8}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833049574\"><p id=\"fs-id1167833049576\">\\(\\frac{5}{21}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833049589\"><div data-type=\"problem\" id=\"fs-id1167833049591\"><p id=\"fs-id1167833049593\">\\(\\frac{\\frac{3}{4}-\\frac{3}{5}}{\\frac{1}{4}+\\frac{2}{5}}\\)<\/p><\/div><\/div><p id=\"fs-id1167833049642\"><strong data-effect=\"bold\">Mixed Practice<\/strong><\/p><p id=\"fs-id1167833049648\">In the following exercises, simplify.<\/p><div data-type=\"exercise\" id=\"fs-id1167833049651\"><div data-type=\"problem\" id=\"fs-id1167833049654\"><p id=\"fs-id1167833049656\">\\(-\\frac{3}{8}\u00f7\\left(-\\frac{3}{10}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829742352\"><p id=\"fs-id1167829742354\">\\(\\frac{5}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829742366\"><div data-type=\"problem\" id=\"fs-id1167829742368\"><p id=\"fs-id1167829742370\">\\(-\\frac{3}{12}\u00f7\\left(-\\frac{5}{9}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829742414\"><div data-type=\"problem\" id=\"fs-id1167829742416\"><p id=\"fs-id1167829742418\">\\(-\\frac{3}{8}+\\frac{5}{12}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829742439\"><p id=\"fs-id1167829742442\">\\(\\frac{1}{24}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829742454\"><div data-type=\"problem\" id=\"fs-id1167829742456\"><p id=\"fs-id1167829742459\">\\(-\\frac{1}{8}+\\frac{7}{12}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829742496\"><div data-type=\"problem\" id=\"fs-id1167829742498\"><p id=\"fs-id1167829742500\">\\(-\\frac{7}{15}-\\frac{y}{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829742522\"><p id=\"fs-id1167829742524\">\\(\\frac{-28-15y}{60}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836540350\"><div data-type=\"problem\" id=\"fs-id1167836540353\"><p id=\"fs-id1167836540355\">\\(-\\frac{3}{8}-\\frac{x}{11}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836540398\"><div data-type=\"problem\" id=\"fs-id1167836540401\"><p id=\"fs-id1167836540403\">\\(\\frac{11}{12a}\u00b7\\frac{9a}{16}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836540429\"><p id=\"fs-id1167836540432\">\\(\\frac{33}{64}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836540445\"><div data-type=\"problem\" id=\"fs-id1167836540448\"><p id=\"fs-id1167836540450\">\\(\\frac{10y}{13}\u00b7\\frac{8}{15y}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836540491\"><div data-type=\"problem\" id=\"fs-id1167836540493\"><p id=\"fs-id1167836540496\">\\(\\frac{1}{2}+\\frac{2}{3}\u00b7\\frac{5}{12}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836540522\"><p id=\"fs-id1167836540524\">\\(\\frac{7}{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836540536\"><div data-type=\"problem\" id=\"fs-id1167836540538\"><p id=\"fs-id1167836540540\">\\(\\frac{1}{3}+\\frac{2}{5}\u00b7\\frac{3}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750204\"><div data-type=\"problem\" id=\"fs-id1167829750206\"><p id=\"fs-id1167829750208\">\\(1-\\frac{3}{5}\u00f7\\frac{1}{10}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829750232\"><p id=\"fs-id1167829750234\">\\(-5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750242\"><div data-type=\"problem\" id=\"fs-id1167829750244\"><p id=\"fs-id1167829750246\">\\(1-\\frac{5}{6}\u00f7\\frac{1}{12}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750281\"><div data-type=\"problem\" id=\"fs-id1167829750283\"><p id=\"fs-id1167829750285\">\\(\\frac{3}{8}-\\frac{1}{6}+\\frac{3}{4}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829750310\"><p id=\"fs-id1167829750313\">\\(\\frac{23}{24}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750326\"><div data-type=\"problem\" id=\"fs-id1167829750329\"><p id=\"fs-id1167829750331\">\\(\\frac{2}{5}+\\frac{5}{8}-\\frac{3}{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750372\"><div data-type=\"problem\" id=\"fs-id1167829750374\"><p id=\"fs-id1167829750377\">\\(12\\left(\\frac{9}{20}-\\frac{4}{15}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829750405\"><p id=\"fs-id1167829750408\">\\(\\frac{11}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829750420\"><div data-type=\"problem\" id=\"fs-id1167829750422\"><p id=\"fs-id1167829750425\">\\(8\\left(\\frac{15}{16}-\\frac{5}{6}\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829738275\"><div data-type=\"problem\" id=\"fs-id1167829738277\"><p id=\"fs-id1167829738279\">\\(\\frac{\\frac{5}{8}+\\frac{1}{6}}{\\frac{19}{24}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829738308\"><p id=\"fs-id1167829738310\">\\(1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829738318\"><div data-type=\"problem\" id=\"fs-id1167829738320\"><p id=\"fs-id1167829738322\">\\(\\frac{\\frac{1}{6}+\\frac{3}{10}}{\\frac{14}{30}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829738361\"><div data-type=\"problem\" id=\"fs-id1167829738364\"><p id=\"fs-id1167829738366\">\\(\\left(\\frac{5}{9}+\\frac{1}{6}\\right)\u00f7\\left(\\frac{2}{3}-\\frac{1}{2}\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829738412\"><p id=\"fs-id1167829738414\">\\(\\frac{13}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829738426\"><div data-type=\"problem\" id=\"fs-id1167829738429\"><p id=\"fs-id1167829738431\">\\(\\left(\\frac{3}{4}+\\frac{1}{6}\\right)\u00f7\\left(\\frac{5}{8}-\\frac{1}{3}\\right)\\)<\/p><\/div><\/div><p id=\"fs-id1167829738492\"><strong data-effect=\"bold\">Evaluate Variable Expressions with Fractions<\/strong><\/p><p id=\"fs-id1167829738497\">In the following exercises, evaluate.<\/p><div data-type=\"exercise\" id=\"fs-id1167829738500\"><div data-type=\"problem\" id=\"fs-id1167829738502\"><p id=\"fs-id1167829738504\">\\(\\frac{7}{10}-w\\) when<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(w=\\frac{1}{2}\\) <span class=\"token\">\u24d1<\/span> \\(w=-\\frac{1}{2}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167829753672\"><p id=\"fs-id1167829753674\"><span class=\"token\">\u24d0<\/span>\\(\\frac{1}{5}\\)<span class=\"token\">\u24d1<\/span>\\(\\frac{6}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829753705\"><div data-type=\"problem\" id=\"fs-id1167829753707\"><p id=\"fs-id1167829753709\">\\(\\frac{5}{12}-w\\) when<\/p><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d0<\/span> \\(w=\\frac{1}{4}\\) <span class=\"token\">\u24d1<\/span> \\(w=-\\frac{1}{4}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829753796\"><div data-type=\"problem\" id=\"fs-id1167829753798\"><p id=\"fs-id1167829753801\">\\(2{x}^{2}{y}^{3}\\) when<\/p><div data-type=\"newline\"><br><\/div>\\(x=-\\frac{2}{3}\\) and \\(y=-\\frac{1}{2}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167829753849\"><p id=\"fs-id1167829753851\">\\(-\\frac{1}{9}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829753865\"><div data-type=\"problem\" id=\"fs-id1167829753867\"><p id=\"fs-id1167829753869\">\\(8{u}^{2}{v}^{3}\\) when<\/p><div data-type=\"newline\"><br><\/div>\\(u=-\\frac{3}{4}\\) and \\(v=-\\frac{1}{2}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829751999\"><div data-type=\"problem\" id=\"fs-id1167829752001\"><p id=\"fs-id1167829752003\">\\(\\frac{a+b}{a-b}\\) when<\/p><div data-type=\"newline\"><br><\/div>\\(a=-3,b=8\\)<\/div><div data-type=\"solution\" id=\"fs-id1167829752045\"><p id=\"fs-id1167829752047\">\\(-\\frac{5}{11}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829752062\"><div data-type=\"problem\" id=\"fs-id1167829752064\"><p id=\"fs-id1167829752066\">\\(\\frac{r-s}{r+s}\\) when<\/p><div data-type=\"newline\"><br><\/div>\\(r=10,s=-5\\)<\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167829752118\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167829752126\"><div data-type=\"problem\" id=\"fs-id1167829752128\"><p id=\"fs-id1167829752130\">Why do you need a common denominator to add or subtract fractions? Explain.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829752134\"><p id=\"fs-id1167829752136\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829752142\"><div data-type=\"problem\" id=\"fs-id1167829752144\"><p id=\"fs-id1167829752146\">How do you find the LCD of 2 fractions?<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829752158\"><div data-type=\"problem\" id=\"fs-id1167829752160\"><p id=\"fs-id1167829752162\">Explain how you find the reciprocal of a fraction.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829752166\"><p id=\"fs-id1167829752168\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829752174\"><div data-type=\"problem\" id=\"fs-id1167829752176\"><p id=\"fs-id1167829752178\">Explain how you find the reciprocal of a negative number.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829752191\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167829752196\"><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-id1167829752208\" data-alt=\"This table has 4 columns, 5 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: simplify fractions, multiply and divide fractions, add and subtract fractions, use the order of operations to simplify fractions, evaluate variable expressions with fractions. The remaining columns are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_201_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 5 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: simplify fractions, multiply and divide fractions, add and subtract fractions, use the order of operations to simplify fractions, evaluate variable expressions with fractions. The remaining columns are blank.\"><\/span><p id=\"fs-id1167829752220\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167829752234\"><dt>complex fraction<\/dt><dd id=\"fs-id1167829752239\">A fraction in which the numerator or the denominator is a fraction is called a complex fraction.<\/dd><\/dl><dl id=\"fs-id1167829752244\"><dt>denominator<\/dt><dd id=\"fs-id1167829752250\">In a fraction, written \\(\\frac{a}{b},\\) where \\(b\\ne 0,\\) the denominator <em data-effect=\"italics\">b<\/em> is the number of equal parts the whole has been divided into.<\/dd><\/dl><dl id=\"fs-id1167829752281\"><dt>equivalent fractions<\/dt><dd id=\"fs-id1167829752287\">Equivalent fractions are fractions that have the same value.<\/dd><\/dl><dl id=\"fs-id1167829752291\"><dt>fraction<\/dt><dd id=\"fs-id1167829752296\">A fraction is written \\(\\frac{a}{b},\\) where \\(b\\ne 0,\\) and <em data-effect=\"italics\">a<\/em> is the numerator and <em data-effect=\"italics\">b<\/em> is the denominator. A fraction represents parts of a whole.<\/dd><\/dl><dl id=\"fs-id1167829751654\"><dt>least common denominator<\/dt><dd id=\"fs-id1167829751659\">The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.<\/dd><\/dl><dl id=\"fs-id1167829751664\"><dt>numerator<\/dt><dd id=\"fs-id1167829751670\">In a fraction, written \\(\\frac{a}{b},\\) where \\(b\\ne 0,\\) the numerator <em data-effect=\"italics\">a<\/em> indicates how many parts are included.<\/dd><\/dl><dl id=\"fs-id1167829751702\"><dt>reciprocal<\/dt><dd id=\"fs-id1167829751707\">The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator.<\/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>Simplify fractions<\/li>\n<li>Multiply and divide fractions<\/li>\n<li>Add and subtract fractions<\/li>\n<li>Use the order of operations to simplify fractions<\/li>\n<li>Evaluate variable expressions with fractions<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836516264\" class=\"be-prepared\">\n<p id=\"fs-id1167836311643\">A more thorough introduction to the topics covered in this section can be found in the <em data-effect=\"italics\">Elementary Algebra<\/em> chapter, Foundations.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829716499\">\n<h3 data-type=\"title\">Simplify Fractions<\/h3>\n<p id=\"fs-id1167833058122\">A <span data-type=\"term\">fraction<\/span> is a way to represent parts of a whole. The fraction <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;\" \/> represents two of three equal parts. See <a href=\"#CNX_IntAlg_Figure_01_03_001\" class=\"autogenerated-content\">(Figure)<\/a>. In the fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71c765e3490b559d4d3bf38f46e24efa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"13\" style=\"vertical-align: -6px;\" \/> the 2 is called the <span data-type=\"term\">numerator<\/span> and the 3 is called the <span data-type=\"term\">denominator<\/span>. The line is called the fraction bar.<\/p>\n<div class=\"bc-figure figure\" id=\"CNX_IntAlg_Figure_01_03_001\">\n<div class=\"bc-figcaption figcaption\">In the circle, <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;\" \/> of the circle is shaded\u20142 of the 3 equal parts.<\/div>\n<p><span data-type=\"media\" id=\"fs-id1167836629535\" data-alt=\"Figure shows a circle divided in three equal parts. 2 of these are shaded.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_001_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Figure shows a circle divided in three equal parts. 2 of these are shaded.\" \/><\/span><\/div>\n<div data-type=\"note\" id=\"fs-id1167833412814\">\n<div data-type=\"title\">Fraction<\/div>\n<p id=\"fs-id1167836688384\">A <strong data-effect=\"bold\">fraction<\/strong> is written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7825cc57d4b3386857d6642115a67cd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"13\" style=\"vertical-align: -6px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1edd53778d94abb1dc1e19acff79e9da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"40\" style=\"vertical-align: -4px;\" \/> and<\/p>\n<p id=\"fs-id1167836408815\"><em data-effect=\"italics\">a<\/em> is the <strong data-effect=\"bold\">numerator<\/strong> and <em data-effect=\"italics\">b<\/em> is the <strong data-effect=\"bold\">denominator<\/strong>.<\/p>\n<p id=\"fs-id1167836688080\">A fraction represents parts of a whole. The denominator <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;\" \/> is the number of equal parts the whole has been divided into, and the numerator <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;\" \/> indicates how many parts are included.<\/p>\n<\/div>\n<p id=\"fs-id1167836521454\">Fractions that have the same value are <span data-type=\"term\">equivalent fractions<\/span>. The Equivalent Fractions<\/p>\n<p id=\"fs-id1167829755870\">Property allows us to find equivalent fractions and also simplify fractions.<\/p>\n<div data-type=\"note\" id=\"fs-id1167829596336\">\n<div data-type=\"title\">Equivalent Fractions Property<\/div>\n<p id=\"fs-id1167832940622\">If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db153ab0c4406f4cf2d57596a36fe4de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#99;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167836550529\">then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4826516972c8e413618db1ce751f6d44_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;&#97;&middot;&#99;&#125;&#123;&#98;&middot;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"49\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e9f76ab51c94dd9586c1559b694b610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&middot;&#99;&#125;&#123;&#98;&middot;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"54\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<p id=\"fs-id1167836327003\">A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.<\/p>\n<p id=\"fs-id1167836447306\">For example,<\/p>\n<p id=\"fs-id1167836388086\">\u2003\u2003<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;\" \/> is simplified because there are no common factors of 2 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a650b6362973a0817ee43187a6682e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1167829695451\">\u2003\u2003<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80f51ab0207944543454dc1900a35d61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/> is not simplified because 5 is a common factor of 10 and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4fdfe17e8406ced563dfc36040a9c920_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"21\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"fs-id1167836377781\">We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.<\/p>\n<p id=\"fs-id1167836493293\">Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the Equivalent Fractions Property.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836620030\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How To Simplify a Fraction<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836627469\">\n<div data-type=\"problem\" id=\"fs-id1167836340855\">\n<p id=\"fs-id1167836526280\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3463a15c51264a279b36a595755c3a87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#49;&#53;&#125;&#123;&#55;&#55;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"41\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836527773\"><span data-type=\"media\" id=\"fs-id1167836514898\" data-alt=\"Step 1 is to rewrite the numerator and denominator to show the common factors. If needed, use a factor tree. Here, we rewrite 315 and 770 as the product of the primes. Starting with minus 315 divided by 770, we get, minus 3 times 3 time 5 times 7 divided by 2 times 5 times 7 times 11.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_002a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to rewrite the numerator and denominator to show the common factors. If needed, use a factor tree. Here, we rewrite 315 and 770 as the product of the primes. Starting with minus 315 divided by 770, we get, minus 3 times 3 time 5 times 7 divided by 2 times 5 times 7 times 11.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836391701\" data-alt=\"Step 2 is to simplify using the Equivalent Fractions Property by dividing out common factors. We first mark out the common factors 5 and 7 and then divide them out. This leaves minus 3 times 3 divided by 2 times 11.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_002b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to simplify using the Equivalent Fractions Property by dividing out common factors. We first mark out the common factors 5 and 7 and then divide them out. This leaves minus 3 times 3 divided by 2 times 11.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836516667\" data-alt=\"Step 3 is to multiply the remaining factors, if necessary. We get minus 9 by 22.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_002c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to multiply the remaining factors, if necessary. We get minus 9 by 22.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836297017\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836318031\">\n<div data-type=\"problem\" id=\"fs-id1167836550421\">\n<p id=\"fs-id1167836321058\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-693d5f20fbfcc97054ef40b341196527_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#57;&#125;&#123;&#49;&#50;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"41\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836362368\">\n<p id=\"fs-id1167836356125\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05494b4de90093bb504cbbb040e78bf6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#51;&#125;&#123;&#52;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"29\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833047887\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836363380\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167836525114\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c8d23d352b80c676b42f577ae54bd12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#48;&#125;&#123;&#49;&#57;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"41\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836320202\">\n<p id=\"fs-id1167829717832\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f8db5feff6785204c5d807666d4af85_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836624000\">We now summarize the steps you should follow to simplify fractions.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836282262\" class=\"howto\">\n<div data-type=\"title\">Simplify a fraction.<\/div>\n<ol id=\"fs-id1167836693896\" type=\"1\" class=\"stepwise\">\n<li>Rewrite the numerator and denominator to show the common factors.\n<div data-type=\"newline\"><\/div>\n<p> If needed, factor the numerator and denominator into prime numbers first.<\/li>\n<li>Simplify using the Equivalent Fractions Property by dividing out common factors.<\/li>\n<li>Multiply any remaining factors.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836486885\">\n<h3 data-type=\"title\">Multiply and Divide Fractions<\/h3>\n<p id=\"fs-id1167836523502\">Many people find multiplying and dividing fractions easier than adding and subtracting fractions.<\/p>\n<p id=\"fs-id1167829750700\">To multiply fractions, we multiply the numerators and multiply the denominators.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836537188\">\n<div data-type=\"title\">Fraction Multiplication<\/div>\n<p id=\"fs-id1167836546369\">If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>, and <em data-effect=\"italics\">d<\/em> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6a0135701625ac5da01c134efeaebdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bbf7ff861c113467e719a3ea0798dd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> then<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836493747\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3cc0a8315ff009be9d3e043f0ff005ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#99;&#125;&#123;&#98;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"fs-id1167836612729\">To multiply fractions, multiply the numerators and multiply the denominators.<\/p>\n<\/div>\n<p id=\"fs-id1167836706036\">When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In <a href=\"#fs-id1167836390100\" class=\"autogenerated-content\">(Figure)<\/a>, we will multiply negative and a positive, so the product will be negative.<\/p>\n<p id=\"fs-id1167836511224\">When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, <em data-effect=\"italics\">a<\/em>, can be written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c802cfde6e83e36af67779da8e0c3bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"13\" style=\"vertical-align: -7px;\" \/> So, for example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4526bf29365f24a6d1ab03b9a4a1c5df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"47\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"example\" id=\"fs-id1167836390100\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829598267\">\n<div data-type=\"problem\" id=\"fs-id1167836701672\">\n<p id=\"fs-id1167836391386\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-568f47aadd3f7777497709337ef80dd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#48;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"96\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836688426\">\n<p id=\"fs-id1167836391689\">The first step is to find the sign of the product. Since the signs are the same, the product is positive.<\/p>\n<table id=\"fs-id1167836521020\" class=\"unnumbered unstyled\" summary=\"The expression is minus 12 divided by 5 into minus 20 x. We first determine the sign of the product. The signs are the same, so the product is positive. Now we have 12 by 5 into 20 x. Now we write 20 x as a fraction. Hence, we get 12 by 5 open parentheses 20 x by 1 close parentheses. We multiply 12 times 20 x in the numerator and 5 times 1 in the denominator. We then rewrite 20 to show the common factor 5 and divide it out. We simplify to get 48 x.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836544266\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_003a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Determine the sign of the product. The signs\u2003\u2003\u2003\u2003\u2003\u2003<\/p>\n<div data-type=\"newline\"><\/div>\n<p> are the same, so the product is positive.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836318933\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_003b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write 20<em data-effect=\"italics\">x<\/em> as a fraction.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836547838\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_003c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836342022\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_003d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite 20 to show the common factor 5<\/p>\n<div data-type=\"newline\"><\/div>\n<p> and divide it out.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833055928\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_003e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836547919\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_003f_img-1.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-id1167833024061\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836352569\">\n<div data-type=\"problem\" id=\"fs-id1167836533364\">\n<p id=\"fs-id1167836518690\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f57332266ecbdf47e47dca256a8a6f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"71\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829693025\">\n<p id=\"fs-id1167836523426\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07517dccd8dab6bb9283a93771400624_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#51;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"40\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836285673\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836538840\">\n<div data-type=\"problem\" id=\"fs-id1167836627735\">\n<p id=\"fs-id1167836418131\">Multiply: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a5e8bed8525af1acfaf3b859be3492a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#55;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#52;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"78\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836613391\">\n<p id=\"fs-id1167833055111\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4cb18c12b2b979c7375bfcc0aeef3210_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#54;&#98;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836622994\">Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, we need some vocabulary. The <span data-type=\"term\">reciprocal<\/span> of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of <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;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d13295b2838b5da1e8fda3b798eabe7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"13\" style=\"vertical-align: -6px;\" \/> Since 4 is written in fraction form as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bdf4f14f1401e463cf67e5c481c3f06c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"13\" style=\"vertical-align: -7px;\" \/> the reciprocal of 4 is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-134d803fc1e11c50071777638b483ec0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"13\" style=\"vertical-align: -6px;\" \/><\/p>\n<p id=\"fs-id1167836335184\">To divide fractions, we multiply the first fraction by the reciprocal of the second.<\/p>\n<div data-type=\"note\" id=\"fs-id1167829693065\">\n<div data-type=\"title\">Fraction Division<\/div>\n<p id=\"fs-id1167836545781\">If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>, and <em data-effect=\"italics\">d<\/em> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db153ab0c4406f4cf2d57596a36fe4de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#99;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"92\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bbf7ff861c113467e719a3ea0798dd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> then<\/p>\n<div data-type=\"equation\" id=\"fs-id1167829695610\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-834cfd90259957f58511ed708e37fa79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#100;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"65\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"fs-id1167836706779\">To divide fractions, we multiply the first fraction by the <strong data-effect=\"bold\">reciprocal<\/strong> of the second.<\/p>\n<\/div>\n<p id=\"fs-id1167836544502\">We need to say <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6a0135701625ac5da01c134efeaebdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6fff7b8ea2aa79d8d5b4c39843a64a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bbf7ff861c113467e719a3ea0798dd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> to be sure we don\u2019t divide by zero!<\/p>\n<div data-type=\"example\" id=\"fs-id1167836356409\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836554684\">\n<div data-type=\"problem\" id=\"fs-id1167836349206\">\n<p id=\"fs-id1167836448362\">Find the quotient: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5103cf19d90c6d13c4d31a436f2e0b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#56;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#50;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836512548\">\n<table id=\"fs-id1167833021584\" class=\"unnumbered unstyled\" summary=\"The expression is minus 7 by 8 divided by minus 14 by 27. To divide, multiply the first fraction by the reciprocal of the second. We get minus 7 by 8 multiplied by minus 27 by 14. Determine the sign of the product, and then multiply. We get 7 times 27 divided by 18 times 14. We rewrite showing common factors to get 7 times 9 times 3 divided by 9 times 2 times 7 times 2. We remove the common factors between numerator and denominator. We get 3 in the numerator and 2 times 2 in the denominator. We simplify to get 3 by 4.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836386433\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_004a_img_Errata-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To divide, multiply the first fraction by the\u2003\u2003\u2003\u2003\u2003\u2003<\/p>\n<div data-type=\"newline\"><\/div>\n<p>reciprocal of the second.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836378177\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_004b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Determine the sign of the product, and<\/p>\n<div data-type=\"newline\"><\/div>\n<p>then multiply.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836295914\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_004c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Rewrite showing common factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836289041\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_004d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Remove common factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836622702\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_004e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836545813\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_004f_img-1.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-id1167836595955\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829690872\">\n<div data-type=\"problem\" id=\"fs-id1167836362778\">\n<p id=\"fs-id1167836398875\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e07d7d0a7f315dc0397ca8d7784ceac8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#50;&#55;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#53;&#125;&#123;&#51;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836362322\">\n<p id=\"fs-id1167836697174\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6061ea4d4da1a30eadbc5bfff4712dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836293918\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836378913\">\n<div data-type=\"problem\" id=\"fs-id1167829597190\">\n<p id=\"fs-id1167836417232\">Divide: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd5bf1059dc4cdae8d4239c1d2bc0d66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#52;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#50;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167829580094\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836700387\">The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a <span data-type=\"term\">complex fraction<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167833018456\">\n<div data-type=\"title\">Complex Fraction<\/div>\n<p id=\"fs-id1167836611505\">A <strong data-effect=\"bold\">complex fraction<\/strong> is a fraction in which the numerator or the denominator contains a fraction.<\/p>\n<\/div>\n<p id=\"fs-id1167829719773\">Some examples of complex fractions are:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836706809\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81872a20898257107d8a70859882408c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#55;&#125;&#125;&#123;&#51;&#125;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#125;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"216\" style=\"vertical-align: -13px;\" \/><\/div>\n<p id=\"fs-id1167829936808\">To simplify a complex fraction, remember that the fraction bar means division. For example, the complex fraction <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2684b90a3d8fb0811f29c1b1886d4a7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"10\" style=\"vertical-align: -13px;\" \/> means <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31d86092de9704fc422350b382eb6d07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"24\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"example\" id=\"fs-id1167829590640\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836596927\">\n<div data-type=\"problem\" id=\"fs-id1167836688056\">\n<p id=\"fs-id1167836652786\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b276c42be4ec515f9769d753986cabfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#121;&#125;&#123;&#54;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"23\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829594563\">\n<p id=\"fs-id1167836552744\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0b426acb2149ec6ca1da316c19ebee34_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;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#121;&#125;&#123;&#54;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#97;&#115;&#32;&#100;&#105;&#118;&#105;&#115;&#105;&#111;&#110;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#50;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#121;&#125;&#123;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#32;&#98;&#121;&#32;&#116;&#104;&#101;&#32;&#114;&#101;&#99;&#105;&#112;&#114;&#111;&#99;&#97;&#108;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#115;&#101;&#99;&#111;&#110;&#100;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#50;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#120;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&middot;&#54;&#125;&#123;&#50;&middot;&#120;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#111;&#111;&#107;&#32;&#102;&#111;&#114;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#120;&#125;&middot;&#51;&middot;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#50;&#125;&#125;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#50;&#125;&middot;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#120;&#125;&middot;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#118;&#105;&#100;&#101;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#32;&#97;&#110;&#100;&#32;&#115;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#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=\"160\" width=\"604\" style=\"vertical-align: -76px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836621356\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836689418\">\n<div data-type=\"problem\" id=\"fs-id1167836521629\">\n<p id=\"fs-id1167836521631\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa00b703d78416b401c38d88045247eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#56;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#98;&#125;&#123;&#54;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"22\" style=\"vertical-align: -14px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836522187\">\n<p id=\"fs-id1167829750716\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24cd9963d965f100ffa8a449f0b0aaf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"13\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829716940\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836613207\">\n<div data-type=\"problem\" id=\"fs-id1167836520162\">\n<p id=\"fs-id1167836520164\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d3df5209f4b420ffd1b964408d18ab1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#112;&#113;&#125;&#123;&#56;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"22\" style=\"vertical-align: -12px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833224683\">\n<p id=\"fs-id1167836551420\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4bb2b2f896bea9c920023fc4025013e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#113;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"7\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829720923\">\n<h3 data-type=\"title\">Add and Subtract Fractions<\/h3>\n<p id=\"fs-id1167836585263\">When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.<\/p>\n<div data-type=\"note\" id=\"fs-id1167829744486\">\n<div data-type=\"title\">Fraction Addition and Subtraction<\/div>\n<p id=\"fs-id1167836493307\">If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6fff7b8ea2aa79d8d5b4c39843a64a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> then<\/p>\n<div data-type=\"equation\" id=\"fs-id1171791493356\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-710898a028c666bd15ef38c80bab2dcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#43;&#98;&#125;&#123;&#99;&#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;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#45;&#98;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"231\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"fs-id1167833024310\">To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.<\/p>\n<\/div>\n<p id=\"fs-id1167829717535\">The <span data-type=\"term\">least common denominator<\/span> (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.<\/p>\n<div data-type=\"note\" id=\"fs-id1167829714220\">\n<div data-type=\"title\">Least Common Denominator<\/div>\n<p id=\"fs-id1167836546986\">The <strong data-effect=\"bold\">least common denominator<\/strong> (LCD) of two fractions is the least common multiple (LCM) of their denominators.<\/p>\n<\/div>\n<p id=\"fs-id1167829718952\">After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!<\/p>\n<div data-type=\"example\" id=\"fs-id1167836518722\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Add or Subtract Fractions<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836518880\">\n<div data-type=\"problem\" id=\"fs-id1167836550915\">\n<p id=\"fs-id1167836550917\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41d8215aa80e5a9edb60969e59a1919f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"59\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836522037\"><span data-type=\"media\" id=\"fs-id1167836522040\" data-alt=\"The expression is 7 by 12 plus 5 by 18. Step 1 is to check if the two numbers have a common denominator. Since they do not, rewrite each fraction with the LCD (least common denominator). For finding the LCD, we write the factors of 12 as 2 times 2 times 2 and the factors of 18 as 2 times 3 times 3. The LCD is 2 times 2 times 3 times 3, which is equal to 36.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_005a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The expression is 7 by 12 plus 5 by 18. Step 1 is to check if the two numbers have a common denominator. Since they do not, rewrite each fraction with the LCD (least common denominator). For finding the LCD, we write the factors of 12 as 2 times 2 times 2 and the factors of 18 as 2 times 3 times 3. The LCD is 2 times 2 times 3 times 3, which is equal to 36.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829721204\" data-alt=\"Step 2 is to add or subtract the fractions. We multiply the numerator and denominator of each fraction by the factor needed to get the denominator to be 36. Do not simplify the equivalent fractions. If you do, you\u2019ll get back to the original fractions and lose the common denominator. We multiply the numerator and denominator of 7 divided by 12, by 3 times. We multiply numerator and denominator of 5 divided by 18 by 2 times. We get the expression 21 by 36 plus 10 by 36.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_005b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to add or subtract the fractions. We multiply the numerator and denominator of each fraction by the factor needed to get the denominator to be 36. Do not simplify the equivalent fractions. If you do, you\u2019ll get back to the original fractions and lose the common denominator. We multiply the numerator and denominator of 7 divided by 12, by 3 times. We multiply numerator and denominator of 5 divided by 18 by 2 times. We get the expression 21 by 36 plus 10 by 36.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167836481443\" data-alt=\"Step 3 is to simplify is possible. Since 31 is prime, its only factors are 1and 31. Since 31 does not go into 36, the answer is simplified.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_005c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to simplify is possible. Since 31 is prime, its only factors are 1and 31. Since 31 does not go into 36, the answer is simplified.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836558604\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829692339\">\n<div data-type=\"problem\" id=\"fs-id1167829692341\">\n<p id=\"fs-id1167836537182\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5f52b3c453bb9cc42138f1786adeb17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#49;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"59\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829715998\">\n<p id=\"fs-id1167829716000\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-64e25e8ede9f97da22579b3e05475e39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#57;&#125;&#123;&#54;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836684166\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836684169\">\n<div data-type=\"problem\" id=\"fs-id1167829580090\">\n<p id=\"fs-id1167829580092\">Add: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7283b74368b84963b533a31d4612b6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#49;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#50;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"59\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829614382\">\n<p id=\"fs-id1167829691275\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0680ff1b2c79f14f201cbf5f6907bf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#51;&#125;&#123;&#54;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"21\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836610898\" class=\"howto\">\n<div data-type=\"title\">Add or subtract fractions.<\/div>\n<ol id=\"fs-id1167829715974\" type=\"1\" class=\"stepwise\">\n<li>Do they have a common denominator?\n<ul id=\"fs-id1167836546938\" data-bullet-style=\"bullet\">\n<li>Yes\u2014go to step 2.<\/li>\n<li>No\u2014rewrite each fraction with the LCD (least common denominator).\n<ul id=\"fs-id1167833053829\" data-bullet-style=\"bullet\">\n<li>Find the LCD.<\/li>\n<li>Change each fraction into an equivalent fraction with the LCD as its denominator.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Add or subtract the fractions.<\/li>\n<li>Simplify, if possible.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167829744496\">We now have all four operations for fractions. <a href=\"#fs-id1167829751987\" class=\"autogenerated-content\">(Figure)<\/a> summarizes fraction operations.<\/p>\n<table id=\"fs-id1167829751987\" summary=\"This table gives notes on fraction multiplication, division, addition and subtraction. For fraction multiplication, multiply the numerators and multiply the denominators. Hence, a by b times c by d is ac by bd. For fraction division, multiply the first fraction by the reciprocal of the second. Hence, a by b divided by c by d is ad by bc. For fraction addition, add the numerators and place the sum over the common denominator. Hence, a by c plus b by c is open parentheses a plus b close parentheses by c. For fraction subtraction, subtract the numerators and place the difference over the common denominator. Hence, a by c minus b by c is open parentheses a minus b close parentheses by c. To multiply or divide fractions, an LCD is NOT needed. To add or subtract fractions, an LCD is needed.\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Fraction Multiplication<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Fraction Division<\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3cc0a8315ff009be9d3e043f0ff005ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#99;&#125;&#123;&#98;&#100;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-834cfd90259957f58511ed708e37fa79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#100;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"65\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply the numerators and multiply the denominators<\/td>\n<td data-valign=\"top\" data-align=\"left\">Multiply the first fraction by the reciprocal of the second.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Fraction Addition<\/strong><\/td>\n<td data-valign=\"top\" data-align=\"left\"><strong data-effect=\"bold\">Fraction Subtraction<\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e19c83ca68864bbe71a6b53b7a1ad55c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#43;&#98;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"90\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-427fa88c44f4a55249b2a6a8d74e4a12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#45;&#98;&#125;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"90\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add the numerators and place the sum over the common denominator.<\/td>\n<td data-valign=\"top\" data-align=\"left\">Subtract the numerators and place the difference over the common denominator.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"top\" data-align=\"center\">To multiply or divide fractions, an LCD is NOT needed.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>To add or subtract fractions, an LCD is needed.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836367514\">When starting an exercise, always identify the operation and then recall the methods needed for that operation.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836553755\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836553757\">\n<div data-type=\"problem\" id=\"fs-id1167836553759\">\n<p id=\"fs-id1167833240062\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7358c319c115ea0b991ec4aa44ae325c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#125;&#123;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad677d7ba44b0622bd7464e5ddc64e5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#125;&#123;&#54;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"38\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836530461\">\n<p id=\"fs-id1167829754896\">First ask, \u201cWhat is the operation?\u201d Identifying the operation will determine whether or not we need a common denominator. Remember, we need a common denominator to add or subtract, but not to multiply or divide.<\/p>\n<p id=\"fs-id1167829754898\"><span class=\"token\">\u24d0<\/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-a76f9877afd74a12aa7f2eb407ae78d7_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;&#116;&#101;&#120;&#116;&#123;&#87;&#104;&#97;&#116;&#32;&#105;&#115;&#32;&#116;&#104;&#101;&#32;&#111;&#112;&#101;&#114;&#97;&#116;&#105;&#111;&#110;&#63;&#32;&#84;&#104;&#101;&#32;&#111;&#112;&#101;&#114;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#32;&#115;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#111;&#32;&#116;&#104;&#101;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#115;&#32;&#104;&#97;&#118;&#101;&#32;&#97;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#63;&#32;&#78;&#111;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&#125;&#123;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#76;&#67;&#68;&#32;&#111;&#102;&#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;&#54;&#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;&#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;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#76;&#67;&#68;&#32;&#105;&#115;&#32;&#51;&#48;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#54;&#61;&#50;&middot;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#123;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#48;&#61;&#50;&middot;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#68;&#125;&#61;&#50;&middot;&#51;&middot;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#68;&#125;&#61;&#51;&#48;&#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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#32;&#101;&#97;&#99;&#104;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#32;&#97;&#115;&#32;&#97;&#110;&#32;&#101;&#113;&#117;&#105;&#118;&#97;&#108;&#101;&#110;&#116;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#32;&#119;&#105;&#116;&#104;&#32;&#116;&#104;&#101;&#32;&#76;&#67;&#68;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#120;&middot;&#53;&#125;&#123;&#54;&middot;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&middot;&#51;&#125;&#123;&#49;&#48;&middot;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#120;&#125;&#123;&#51;&#48;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#51;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#116;&#114;&#97;&#99;&#116;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#115;&#32;&#97;&#110;&#100;&#32;&#112;&#108;&#97;&#99;&#101;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#111;&#118;&#101;&#114;&#32;&#116;&#104;&#101;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#115;&#125;&#46;&#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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#120;&#45;&#57;&#125;&#123;&#51;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#102;&#32;&#112;&#111;&#115;&#115;&#105;&#98;&#108;&#101;&#125;&#46;&#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;&#84;&#104;&#101;&#114;&#101;&#32;&#97;&#114;&#101;&#32;&#110;&#111;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#32;&#115;&#105;&#109;&#112;&#108;&#105;&#102;&#105;&#101;&#100;&#125;&#46;&#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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"524\" width=\"480\" style=\"vertical-align: -257px;\" \/><\/p>\n<p id=\"fs-id1167833340094\"><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-507fc8848fbf393ae4f92f26683a6587_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;&#87;&#104;&#97;&#116;&#32;&#105;&#115;&#32;&#116;&#104;&#101;&#32;&#111;&#112;&#101;&#114;&#97;&#116;&#105;&#111;&#110;&#63;&#32;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#105;&#99;&#97;&#116;&#105;&#111;&#110;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#120;&#125;&#123;&#54;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#102;&#114;&#97;&#99;&#116;&#105;&#111;&#110;&#115;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#32;&#109;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#115;&#125;&#46;&#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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#120;&middot;&#51;&#125;&#123;&#54;&middot;&#49;&#48;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#119;&#114;&#105;&#116;&#101;&#44;&#32;&#115;&#104;&#111;&#119;&#105;&#110;&#103;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#101;&#109;&#111;&#118;&#101;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#53;&#125;&#120;&middot;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#51;&#125;&#125;&#123;&#50;&middot;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#51;&#125;&middot;&#50;&middot;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#53;&#125;&#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;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#55;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#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=\"132\" width=\"591\" style=\"vertical-align: -61px;\" \/><\/p>\n<p id=\"fs-id1167836692915\">Notice, we needed an LCD to add <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac901f41d2f49ab9e5759fa2c533220a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#120;&#125;&#123;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"67\" style=\"vertical-align: -7px;\" \/> but not to multiply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7167af49fb7261567aee60252f1dd40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#120;&#125;&#123;&#54;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"45\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836698608\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836698612\">\n<div data-type=\"problem\" id=\"fs-id1167836684139\">\n<p id=\"fs-id1167836684141\">Simplify: <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-411d53ecd301a9144b5b1a52146bf41c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#97;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"47\" 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-990940c895a07d2279d9b5e235e46c54_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#97;&#125;&#123;&#52;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#57;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"31\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836684124\">\n<p id=\"fs-id1167836684126\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-771cead83cbad1a98aabcbdaf077da21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#55;&#97;&#45;&#51;&#50;&#125;&#123;&#51;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" 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-be69a6d34d3868c943e0fbe2397aef0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#97;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"15\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829744140\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829744143\">\n<div data-type=\"problem\" id=\"fs-id1167836595910\">\n<p id=\"fs-id1167836595912\">Simplify: <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce24463d4dbcc01440449650233412f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#107;&#125;&#123;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"47\" 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-b872a930ac22bc5e6a4922d12313805f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#107;&#125;&#123;&#53;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"31\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829589696\">\n<p id=\"fs-id1167829589698\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed1735b812a75e3b1bd7d90c1e859cdb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#52;&#107;&#45;&#53;&#125;&#123;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" 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-b0ba5bedb9faff3a56f9805330bc79fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#107;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"15\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829720090\">\n<h3 data-type=\"title\">Use the Order of Operations to Simplify Fractions<\/h3>\n<p id=\"fs-id1167836688781\">The fraction bar in a fraction acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836688785\" class=\"howto\">\n<div data-type=\"title\">Simplify an expression with a fraction bar.<\/div>\n<ol id=\"fs-id1167833412506\" type=\"1\" class=\"stepwise\">\n<li>Simplify the expression in the numerator. Simplify the expression in the denominator.<\/li>\n<li>Simplify the fraction.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167829744159\">Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167829754611\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56d4be07adb02bc790499f3ec4e4c523_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#125;&#123;&#51;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"262\" style=\"vertical-align: -9px;\" \/><\/div>\n<div data-type=\"equation\" id=\"fs-id1167833050784\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-98c282022af9537bb4567f4c512f1f03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#45;&#51;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#112;&#111;&#115;&#105;&#116;&#105;&#118;&#101;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#101;&#103;&#97;&#116;&#105;&#118;&#101;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"262\" style=\"vertical-align: -9px;\" \/><\/div>\n<div data-type=\"note\" id=\"fs-id1167829742311\">\n<div data-type=\"title\">Placement of Negative Sign in a Fraction<\/div>\n<p id=\"fs-id1167829741605\">For any positive numbers <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>,<\/p>\n<div data-type=\"equation\" id=\"fs-id1167829754672\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91ec66c282c8274333b51d850e50e5ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#125;&#123;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167829755877\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829755879\">\n<div data-type=\"problem\" id=\"fs-id1167829755881\">\n<p id=\"fs-id1167836484393\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47994d3a199c787c16bc2e519e577f49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"87\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750490\">\n<p id=\"fs-id1167829750492\">The fraction bar acts like a grouping symbol. So completely simplify the numerator and the denominator separately.<\/p>\n<p id=\"fs-id1167829750496\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fe4a6ff2017fff366b34944ab6965fd_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;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#77;&#117;&#108;&#116;&#105;&#112;&#108;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#54;&#45;&#50;&#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;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#52;&#125;&#123;&#45;&#56;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#118;&#105;&#100;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"93\" width=\"274\" style=\"vertical-align: -38px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833412533\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829627923\">\n<div data-type=\"problem\" id=\"fs-id1167829627925\">\n<p id=\"fs-id1167829627927\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e846cb91f81e9190bf2a57cd71cd63e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"87\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829753437\">\n<p id=\"fs-id1167829753439\">4<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829743958\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829743962\">\n<div data-type=\"problem\" id=\"fs-id1167829743964\">\n<p id=\"fs-id1167829691090\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd6955b6890ae11d90742f811a30c282_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"87\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829754202\">\n<p id=\"fs-id1167829754205\">2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829754211\">Now we\u2019ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator as the fraction bar means division.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829627815\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Simplify Complex Fractions<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829627820\">\n<div data-type=\"problem\" id=\"fs-id1167829754213\">\n<p id=\"fs-id1167829752824\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-911e3fbb97b90d8822735cdaf5116f62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#43;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"37\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829754739\"><span data-type=\"media\" id=\"fs-id1167829754742\" data-alt=\"The expression is 1 by 2 the whole squared divided by 4 plus 3 squared. Step 1 is to simplify the numerator, which becomes 1 by 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_006a_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"The expression is 1 by 2 the whole squared divided by 4 plus 3 squared. Step 1 is to simplify the numerator, which becomes 1 by 4.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829754924\" data-alt=\"Step 2 is to simplify the denominator, which becomes 4 plus 9 equals 13.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_006b_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to simplify the denominator, which becomes 4 plus 9 equals 13.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167829720638\" data-alt=\"Step 3 is to divide the numerator by the denominator and simplify if possible. Now the expression becomes 1 by 4 divided by 13 by 1, which equals 1 by 4 multiplied by 1 by 13, which equals 1 by 52\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_006c_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to divide the numerator by the denominator and simplify if possible. Now the expression becomes 1 by 4 divided by 13 by 1, which equals 1 by 4 multiplied by 1 by 13, which equals 1 by 52\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829753146\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829753149\">\n<div data-type=\"problem\" id=\"fs-id1167829753151\">\n<p id=\"fs-id1167829753154\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5554acaa0e11b8c41ab92c9886aae2b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#50;&#125;&#94;&#123;&#51;&#125;&#43;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"37\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833025479\">\n<p id=\"fs-id1167833025481\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4ec3cb5149ae42207f9367831b4719a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836477915\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836477918\">\n<div data-type=\"problem\" id=\"fs-id1167836477920\">\n<p id=\"fs-id1167836580186\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d03c1af046fa6a163f8d42f084f44477_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#43;&#123;&#52;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"37\" style=\"vertical-align: -16px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829751585\">\n<p id=\"fs-id1167829751588\">272<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829753394\" class=\"howto\">\n<div data-type=\"title\">Simplify complex fractions.<\/div>\n<ol id=\"fs-id1167829753400\" type=\"1\" class=\"stepwise\">\n<li>Simplify the numerator.<\/li>\n<li>Simplify the denominator.<\/li>\n<li>Divide the numerator by the denominator. Simplify if possible.<\/li>\n<\/ol>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167829750481\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829750483\">\n<div data-type=\"problem\" id=\"fs-id1167829750485\">\n<p id=\"fs-id1167829755820\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1ab860e98f6be4abefa8f4106ecf570_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"36\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750436\">\n<p id=\"fs-id1167829750438\">It may help to put parentheses around the numerator and the denominator.<\/p>\n<p id=\"fs-id1167829750442\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16954605ecb09599b57ce4b27f4568f2_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;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#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;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#68;&#125;&#61;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#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;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#67;&#68;&#125;&#61;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#49;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#118;&#105;&#100;&#101;&#32;&#116;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#32;&#98;&#121;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#54;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#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;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#54;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#55;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#68;&#105;&#118;&#105;&#100;&#101;&#32;&#111;&#117;&#116;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#55;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#54;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#125;&#123;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#54;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#111;&#118;&#101;&#114;&#108;&#105;&#110;&#101;&#123;&#41;&#55;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#49;&#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;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"217\" width=\"509\" style=\"vertical-align: -102px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836407228\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836407232\">\n<div data-type=\"problem\" id=\"fs-id1167836407234\">\n<p id=\"fs-id1167836407236\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48018c639453a7b655cb665cdd40a072_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"36\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829754361\">\n<p id=\"fs-id1167829754363\">2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836585276\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836585279\">\n<div data-type=\"problem\" id=\"fs-id1167836585281\">\n<p id=\"fs-id1167836585283\">Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-52a9ebac116313fc9b5ca5821b663e0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"36\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750741\">\n<p id=\"fs-id1167829750743\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-014e796b7998038837413334971b044f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829754323\">\n<h3 data-type=\"title\">Evaluate Variable Expressions with Fractions<\/h3>\n<p id=\"fs-id1167829754328\">We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829754333\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829754335\">\n<div data-type=\"problem\" id=\"fs-id1167829754337\">\n<p id=\"fs-id1167836477223\">Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4de408ec4623b772a39d43de95651bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"36\" style=\"vertical-align: -4px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8359433f92a78fa959edae8995b107d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"43\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cb18e4863ea2dd61bc520040db28271_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829741825\">\n<p id=\"fs-id1167829741827\">Substitute the values into the expression.<\/p>\n<table id=\"fs-id1167829741831\" class=\"unnumbered unstyled\" summary=\"The expression is 2 x squared y. Substitute 1 by 4 for x and minus 2 by 3 for y. We now have 2 into open parentheses 1 by 4 close parentheses squared open parentheses minus 2 by 3 close parentheses. Simplifying the exponents first, we get, 2 into open parentheses 1 by 16 close parentheses open parentheses minus 2 by 3 close parentheses. We multiply to get minus 2 times 1 times 2 divided by 2 times 2 times 4 times 3. We write 16 as 2 times 2 times 4 to make it easy to remove common factors. Now divide out the common factors. We are left with minus 1 in the numerator and 4 times 3 in the denominator. We simplify to get minus 1 by 12.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829756310\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_007b_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829756324\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_007a_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829756334\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_007c_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify exponents first.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836409091\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_007d_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Multiply; divide out the common factors.<\/p>\n<div data-type=\"newline\"><\/div>\n<p>Notice we write 16 as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fbb11417f8edb877f2231b68fd8528c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&middot;&#50;&middot;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"27\" style=\"vertical-align: -1px;\" \/> to make it easy to<\/p>\n<div data-type=\"newline\"><\/div>\n<p>remove common factors.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829749843\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_007e_img-1.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829756113\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_007f_img-1.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-id1167829756127\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829756130\">\n<div data-type=\"problem\" id=\"fs-id1167829756133\">\n<p id=\"fs-id1167829756135\">Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9cab041def1e31eae0acc93bc9b9f12f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#97;&#123;&#98;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"33\" style=\"vertical-align: 0px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a466e522113eaea0f7b5b8f0d943d267_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d12c2a9117006477f75ab1aae47ce003_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829743935\">\n<p id=\"fs-id1167829743938\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-413f537f0ad6eef7c0df18690b364ca0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829741554\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829741557\">\n<div data-type=\"problem\" id=\"fs-id1167829741560\">\n<p id=\"fs-id1167829741562\">Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0f516837ca3f402aca672a0b6af6fb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#99;&#125;&#94;&#123;&#51;&#125;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"34\" style=\"vertical-align: -1px;\" \/> when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b725f8ce0609963cd45880ad77bc101d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"54\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65f708d588e0e6a6417f6607d6928628_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829742275\">\n<p id=\"fs-id1167829742277\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829742290\" class=\"media-2\">\n<p id=\"fs-id1167829742293\">Access this online resource for additional instruction and practice with fractions.<\/p>\n<ul id=\"fs-id1167829742297\" data-bullet-style=\"bullet\">\n<li><a href=\"https:\/\/openstax.org\/l\/25addfractions\">Adding Fractions with Unlike Denominators<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836700134\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167835361771\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Equivalent Fractions Property<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db153ab0c4406f4cf2d57596a36fe4de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#99;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"92\" style=\"vertical-align: -4px;\" \/> then<\/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-733078c459f16336ffe7d91bd2b37256_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;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&middot;&#99;&#125;&#123;&#98;&middot;&#99;&#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;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&middot;&#99;&#125;&#123;&#98;&middot;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><strong data-effect=\"bold\">How to simplify a fraction.<\/strong>\n<ol id=\"fs-id1167829690649\" type=\"1\" class=\"stepwise\">\n<li>Rewrite the numerator and denominator to show the common factors.\n<div data-type=\"newline\"><\/div>\n<p> If needed, factor the numerator and denominator into prime numbers first.<\/li>\n<li>Simplify using the Equivalent Fractions Property by dividing out common factors.<\/li>\n<li>Multiply any remaining factors.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Fraction Multiplication<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>, and <em data-effect=\"italics\">d<\/em> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6a0135701625ac5da01c134efeaebdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bbf7ff861c113467e719a3ea0798dd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> then<\/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-ce7b05e123b199881b4dca14b5473626_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;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#99;&#125;&#123;&#98;&#100;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> To multiply fractions, multiply the numerators and multiply the denominators.<\/li>\n<li><strong data-effect=\"bold\">Fraction Division<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, <em data-effect=\"italics\">c<\/em>, and <em data-effect=\"italics\">d<\/em> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db153ab0c4406f4cf2d57596a36fe4de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;&#99;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"92\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0bbf7ff861c113467e719a3ea0798dd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"46\" style=\"vertical-align: -4px;\" \/> then<\/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-9f53b3588ad8f1d839f1859ee9bd4081_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;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#99;&#125;&#123;&#100;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#100;&#125;&#123;&#99;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"70\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> To divide fractions, we multiply the first fraction by the reciprocal of the second.<\/li>\n<li><strong data-effect=\"bold\">Fraction Addition and Subtraction<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> If <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are numbers where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6fff7b8ea2aa79d8d5b4c39843a64a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> then<\/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-413ee91e5a206f92818d80d694b6c507_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;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#43;&#98;&#125;&#123;&#99;&#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;&#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;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#99;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#99;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#45;&#98;&#125;&#123;&#99;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"237\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.<\/li>\n<li><strong data-effect=\"bold\">How to add or subtract fractions.<\/strong>\n<div data-type=\"newline\"><\/div>\n<ol id=\"fs-id1167829751330\" type=\"1\" class=\"stepwise\">\n<li>Do they have a common denominator?\n<ul id=\"fs-id1167829751341\" data-bullet-style=\"open-circle\">\n<li>Yes\u2014go to step 2.<\/li>\n<li>No\u2014rewrite each fraction with the LCD (least common denominator).\n<ul id=\"fs-id1167829751354\" data-bullet-style=\"bullet\">\n<li>Find the LCD.<\/li>\n<li>Change each fraction into an equivalent fraction with the LCD as its denominator.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li>Add or subtract the fractions.<\/li>\n<li>Simplify, if possible.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">How to simplify an expression with a fraction bar.<\/strong>\n<ol id=\"fs-id1167829751381\" type=\"1\" class=\"stepwise\">\n<li>Simplify the expression in the numerator. Simplify the expression in the denominator.<\/li>\n<li>Simplify the fraction.<\/li>\n<\/ol>\n<\/li>\n<li><strong data-effect=\"bold\">Placement of Negative Sign in a Fraction<\/strong>\n<div data-type=\"newline\"><\/div>\n<p> For any positive numbers <em data-effect=\"italics\">a<\/em> and <em data-effect=\"italics\">b<\/em>,<\/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-fb95806e610e8f3d587a76161bb381c9_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;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#125;&#123;&#98;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -6px;\" \/><\/li>\n<li><strong data-effect=\"bold\">How to simplify complex fractions.<\/strong>\n<div data-type=\"newline\"><\/div>\n<ol id=\"fs-id1167829754290\" type=\"1\" class=\"stepwise\">\n<li>Simplify the numerator.<\/li>\n<li>Simplify the denominator.<\/li>\n<li>Divide the numerator by the denominator. Simplify if possible.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836597065\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836597069\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167836597076\"><strong data-effect=\"bold\">Simplify Fractions<\/strong><\/p>\n<p id=\"fs-id1167836597082\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836597086\">\n<div data-type=\"problem\" id=\"fs-id1167836597088\">\n<p id=\"fs-id1167836597090\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44a2bd977bdf95da77259e6016612c20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#56;&#125;&#123;&#54;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"36\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836597105\">\n<p id=\"fs-id1167836597107\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-378cf07c6565fe6391dbfb8500e08c53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"29\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836597122\">\n<div data-type=\"problem\" id=\"fs-id1167836597124\">\n<p id=\"fs-id1167836597126\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-616229f3cb951c20c20d910f662f2cb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#52;&#125;&#123;&#52;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"36\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829624446\">\n<div data-type=\"problem\" id=\"fs-id1167829624448\">\n<p id=\"fs-id1167829624450\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ee9238e528605b5556823d46c9ace98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#48;&#125;&#123;&#50;&#53;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"21\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829624463\">\n<p id=\"fs-id1167829624465\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5dd5aeceb6caade5b6ce45cb93b3c000_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829624479\">\n<div data-type=\"problem\" id=\"fs-id1167829624481\">\n<p id=\"fs-id1167829624483\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae0d0f21ced7705df2bcd5d1a4d7cf7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#50;&#125;&#123;&#50;&#57;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"21\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829624512\">\n<div data-type=\"problem\" id=\"fs-id1167829624514\">\n<p id=\"fs-id1167829624516\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c62f1df1f12e1a65108a3bb8bd03514_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#49;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"29\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829717620\">\n<p id=\"fs-id1167829717622\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e36cb9195f6da25b62a152db746fcdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"22\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829717644\">\n<div data-type=\"problem\" id=\"fs-id1167829717646\">\n<p id=\"fs-id1167829717648\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c20d1231bdb86a218db79e10fbb87ca6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#52;&#97;&#125;&#123;&#51;&#50;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"27\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829614578\">\n<div data-type=\"problem\" id=\"fs-id1167829614580\">\n<p id=\"fs-id1167829614582\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-01a5b25fef58c081b6497efa33f49cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#48;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#49;&#48;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"50\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829614608\">\n<p id=\"fs-id1167829614610\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9cbb56908262582605d581d960245137_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#49;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"43\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829614637\">\n<div data-type=\"problem\" id=\"fs-id1167829614639\">\n<p id=\"fs-id1167829614641\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c0c8678ef7d9a9ec6357adbfc2ba7fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#48;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"50\" style=\"vertical-align: -10px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836698533\"><strong data-effect=\"bold\">Multiply and Divide Fractions<\/strong><\/p>\n<p id=\"fs-id1167836698539\">In the following exercises, perform the indicated operation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836698542\">\n<div data-type=\"problem\" id=\"fs-id1167836698544\">\n<p id=\"fs-id1167836698547\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-190d0ab3062663940c05357a99bf7b5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"65\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836698573\">\n<p id=\"fs-id1167836698575\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34f0e870957984f6c69249b8cf4f5813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829755551\">\n<div data-type=\"problem\" id=\"fs-id1167829755553\">\n<p id=\"fs-id1167829755556\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77d14e926c808d95ded981be5e224d64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"39\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829755594\">\n<div data-type=\"problem\" id=\"fs-id1167829755596\">\n<p id=\"fs-id1167829755598\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cff2f5302dd9c424f606a08ef4f310f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#49;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"80\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829755632\">\n<p id=\"fs-id1167829755634\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8883c4e33b5e5dc0a3738852c662f146_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#49;&#125;&#123;&#53;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"29\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829755734\">\n<div data-type=\"problem\" id=\"fs-id1167829755737\">\n<p id=\"fs-id1167829755739\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a47f119266d79f5c19de6027e1e1279f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#125;&#123;&#51;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"80\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829755791\">\n<div data-type=\"problem\" id=\"fs-id1167829755793\">\n<p id=\"fs-id1167829755795\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4f0e9fc97e53807b90413cc0663ceb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#51;&#125;&#123;&#56;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#52;&#125;&#123;&#57;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"94\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836541862\">\n<p id=\"fs-id1167836541864\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4dc0d5b1b056d8ad35b3581c25971d0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836541878\">\n<div data-type=\"problem\" id=\"fs-id1167836541880\">\n<p id=\"fs-id1167836541882\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-737736ea7b372d743a42658ce84e9013_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#51;&#125;&#123;&#54;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#48;&#125;&#123;&#56;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"94\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836541934\">\n<div data-type=\"problem\" id=\"fs-id1167836541936\">\n<p id=\"fs-id1167836541938\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91323eda9b9bff5002f6176700d3c921_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#55;&#125;&middot;&#50;&#49;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"38\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829720481\">\n<p id=\"fs-id1167829720483\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a122c0441d8c81283d7de8708678e7e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"20\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829720494\">\n<div data-type=\"problem\" id=\"fs-id1167829720496\">\n<p id=\"fs-id1167829720498\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0c8b50cbdbd1371ea5ab70077a4a4c5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&middot;&#51;&#48;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"42\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829720528\">\n<div data-type=\"problem\" id=\"fs-id1167829720530\">\n<p id=\"fs-id1167829720532\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a30ab4e87da62c4c43fa0d840aba32a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"25\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829720551\">\n<p id=\"fs-id1167829720553\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-282742f04238f02cbdbd823c12e1946a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#51;&#125;&#123;&#52;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"15\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750069\">\n<div data-type=\"problem\" id=\"fs-id1167829750071\">\n<p id=\"fs-id1167829750073\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9d617f617d4e5213bd0d46b72982cd16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"19\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750109\">\n<div data-type=\"problem\" id=\"fs-id1167829750111\">\n<p id=\"fs-id1167829750114\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e57c2e72bb674bb089a46b80d3e15fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#56;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#50;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"64\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750143\">\n<p id=\"fs-id1167829750146\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1ed596631867bfaf25f7f4ead1e7c84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750159\">\n<div data-type=\"problem\" id=\"fs-id1167829750162\">\n<p id=\"fs-id1167829750164\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7efef0649b34ba38b35b15f4643678c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#56;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#50;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"64\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829741742\">\n<div data-type=\"problem\" id=\"fs-id1167829741745\">\n<p id=\"fs-id1167829741747\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18bbd41acf288ef2b6717618081aa850_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#117;&#125;&#123;&#49;&#53;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#118;&#125;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829741773\">\n<p id=\"fs-id1167829741776\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1ee5bee25f61a30fa7519873db9938e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#117;&#125;&#123;&#57;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829741794\">\n<div data-type=\"problem\" id=\"fs-id1167829754089\">\n<p id=\"fs-id1167829754091\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38ed2aeb5b3347d0dfd4b7802f315a6a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#114;&#125;&#123;&#50;&#53;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#115;&#125;&#123;&#51;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"45\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829754138\">\n<div data-type=\"problem\" id=\"fs-id1167829754140\">\n<p id=\"fs-id1167829754142\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0306d191816172344f0c2989f25ef30a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829754163\">\n<p id=\"fs-id1167829754165\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48b09578cdff2d298e4c2b3d9990e04e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"29\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829754180\">\n<div data-type=\"problem\" id=\"fs-id1167829754182\">\n<p id=\"fs-id1167829754185\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2c8a9dff418aa0fa65672b0b933c20c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"72\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833350831\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833350834\">\n<div data-type=\"problem\" id=\"fs-id1167833350836\">\n<p id=\"fs-id1167833350838\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc8d8dbbc0d201af4cb463930e651d34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#50;&#49;&#125;&#125;&#123;&#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;&#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;&#102;&#114;&#97;&#99;&#123;&#49;&#50;&#125;&#123;&#51;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"27\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833350868\">\n<p id=\"fs-id1167833350870\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-750dbc1dd4c16f9f53a493376bf17b80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"29\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833350885\">\n<div data-type=\"problem\" id=\"fs-id1167833350888\">\n<p id=\"fs-id1167833350890\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f6223e61ace83d2b0339aeaaad0dc5f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#54;&#125;&#125;&#123;&#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;&#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;&#102;&#114;&#97;&#99;&#123;&#51;&#51;&#125;&#123;&#52;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"27\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829743148\">\n<div data-type=\"problem\" id=\"fs-id1167829743150\">\n<p id=\"fs-id1167829743152\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33b6373af83358aff284a243c3c98fca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#125;&#123;&#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;&#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;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"21\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829743178\">\n<p id=\"fs-id1167829743180\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e47fdbcee43f2762a6da60b82f8ba7b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829743194\">\n<div data-type=\"problem\" id=\"fs-id1167829743196\">\n<p id=\"fs-id1167829743198\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b00447e275e1e278d0612ea61bcef76d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#125;&#125;&#123;&#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;&#49;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"18\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836409504\">\n<div data-type=\"problem\" id=\"fs-id1167836409506\">\n<p id=\"fs-id1167836409508\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19c021c5d733dd5b43a4f886d9492b35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#51;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"31\" width=\"15\" style=\"vertical-align: -11px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836409527\">\n<p id=\"fs-id1167836409529\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-180c1565231d608028cdd2154a4e8b26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#109;&#125;&#123;&#51;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"20\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836409548\">\n<div data-type=\"problem\" id=\"fs-id1167836409550\">\n<p id=\"fs-id1167836409552\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f017303864f5b2c722161e0babe101f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#125;&#123;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#49;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"27\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836409593\"><strong data-effect=\"bold\">Add and Subtract Fractions<\/strong><\/p>\n<p id=\"fs-id1167836409599\">In the following exercises, add or subtract.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836409602\">\n<div data-type=\"problem\" id=\"fs-id1167833025150\">\n<p id=\"fs-id1167833025152\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45141b729bd349147e7f675d38d6428a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"46\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833025171\">\n<p id=\"fs-id1167833025173\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51b5de452d11711f395ef24be1c2a813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#57;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833025187\">\n<div data-type=\"problem\" id=\"fs-id1167833025189\">\n<p id=\"fs-id1167833025191\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-742270a78fdcd6cb3d6dd075f521a2cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"46\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833025226\">\n<div data-type=\"problem\" id=\"fs-id1167833025229\">\n<p id=\"fs-id1167833025231\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8973586423995284cf345229c1f8042a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"53\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833025251\">\n<p id=\"fs-id1167833025253\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b66a5ee1ae0ac052a0c18be3ae508b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829753938\">\n<div data-type=\"problem\" id=\"fs-id1167829753940\">\n<p id=\"fs-id1167829753943\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2606865a8e3e3938bb2403289f2e999_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"53\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829753978\">\n<div data-type=\"problem\" id=\"fs-id1167829753980\">\n<p id=\"fs-id1167829753982\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fce0d7822499b4e318ccf88470dca76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#51;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#125;&#123;&#52;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"68\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829754007\">\n<p id=\"fs-id1167829754009\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-06778df6698326d2490a23fb2ad5ac61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#55;&#125;&#123;&#49;&#48;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"21\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829754023\">\n<div data-type=\"problem\" id=\"fs-id1167829754025\">\n<p id=\"fs-id1167829754027\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c46da80479388fd6e3d3532e4156604_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#51;&#125;&#123;&#51;&#48;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"68\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829756176\">\n<div data-type=\"problem\" id=\"fs-id1167829756178\">\n<p id=\"fs-id1167829756180\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44a9cee3ae77e61ac4747355538cbbed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#57;&#125;&#123;&#53;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#50;&#125;&#123;&#51;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"68\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829756204\">\n<p id=\"fs-id1167829756206\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19343b62fcc2d3d6bcbd30e1d5a4fe5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#51;&#125;&#123;&#52;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"29\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829756222\">\n<div data-type=\"problem\" id=\"fs-id1167829756225\">\n<p id=\"fs-id1167829756227\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7de9f80ab9eda342a17362a11d33d86a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#51;&#125;&#123;&#52;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#56;&#125;&#123;&#51;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"68\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829752841\">\n<div data-type=\"problem\" id=\"fs-id1167829752843\">\n<p id=\"fs-id1167829752845\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f25bb3044411aa1bbe4a3411eb9b225_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"84\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829752874\">\n<p id=\"fs-id1167829752876\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829752889\">\n<div data-type=\"problem\" id=\"fs-id1167829752891\">\n<p id=\"fs-id1167829752893\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-789fc16ff408a47ca1f7d2bf1f06574e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"84\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829752937\">\n<div data-type=\"problem\" id=\"fs-id1167829752939\">\n<p id=\"fs-id1167829752941\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-75b4a4b8b50d0f84a237570dafc182d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"40\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829742577\">\n<p id=\"fs-id1167829742579\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0df9abc80f6908961a66fe5194e96013_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#43;&#51;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"33\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829742599\">\n<div data-type=\"problem\" id=\"fs-id1167829742601\">\n<p id=\"fs-id1167829742604\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-781f6ee5d2a222a84a4c78e918778f5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"40\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829742644\">\n<div data-type=\"problem\" id=\"fs-id1167829742646\">\n<p id=\"fs-id1167829742648\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20a502a64addaa180d63478263a0305a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9336d591d38f00e848dbc4241251c243_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"18\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167836595709\">\n<p id=\"fs-id1167836595711\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b589e486fc233b12d903a213a61a086_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" 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-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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836595737\">\n<div data-type=\"problem\" id=\"fs-id1167836595740\">\n<p id=\"fs-id1167836595742\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbf12d99b54b7d794c577b28e9ad89e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"54\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef9788cbe23f877f530300982c217aef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"32\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829743231\">\n<div data-type=\"problem\" id=\"fs-id1167829743233\">\n<p id=\"fs-id1167829743235\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1361b60ce6dd9d1cf1820def91da653d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#110;&#125;&#123;&#54;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"33\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad2c8c449cac3fd52ed9968ccb764be6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#110;&#125;&#123;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"55\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167829743288\">\n<p id=\"fs-id1167829743290\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ff7ae7980557d49872c513d462b6b4e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#110;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"23\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f19624ef513906660946a60d57a9d38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#53;&#110;&#45;&#49;&#54;&#125;&#123;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"47\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829743334\">\n<div data-type=\"problem\" id=\"fs-id1167829743336\">\n<p id=\"fs-id1167829743338\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a71f9b9d20c982afcdd49a7f808a0c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#97;&#125;&#123;&#56;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"32\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d120774f97d2a1a1586f978960c91c2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#97;&#125;&#123;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"54\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829744261\">\n<div data-type=\"problem\" id=\"fs-id1167829744263\">\n<p id=\"fs-id1167829744266\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b92f27777cd81cb4aeed6a0cf1c12130_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#120;&#125;&#123;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77fbacad972776e2029fd18e3bbd8083_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#107;&#125;&#123;&#57;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"40\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167829744812\">\n<p id=\"fs-id1167829744814\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9300dbc350c4afdfd17c2f99ff12c097_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#56;&#120;&#45;&#49;&#53;&#125;&#123;&#49;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"51\" style=\"vertical-align: -7px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5d344b40abc1342a3566249b04ab5385_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#107;&#125;&#123;&#50;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"37\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829744860\">\n<div data-type=\"problem\" id=\"fs-id1167829744862\">\n<p id=\"fs-id1167829744864\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-048b8528b2d78000cb545a15cc84cf00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#121;&#125;&#123;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"61\" style=\"vertical-align: -6px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a03eb8fd190fe181aa473ed80d2f2804_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#121;&#125;&#123;&#56;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829744383\">\n<div data-type=\"problem\" id=\"fs-id1167829744385\">\n<p id=\"fs-id1167829744387\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34f0e9bd5c564758a1a2c42fae416bb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#97;&#125;&#123;&#51;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"98\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7c7aaa71ab3a610a1dade9d8c7fd05a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#97;&#125;&#123;&#51;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"79\" style=\"vertical-align: -7px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167829744462\">\n<p id=\"fs-id1167829744464\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-790f0a13ee01471eecd6d587662233e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"54\" 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-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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829753473\">\n<div data-type=\"problem\" id=\"fs-id1167829753475\">\n<p id=\"fs-id1167829753477\">\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-835a5a41c3080cd021b38ec080de953a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#98;&#125;&#123;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"52\" style=\"vertical-align: -7px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58afd36e699aeb4864f250f2f8574873_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#98;&#125;&#123;&#53;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"31\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167829753575\"><strong data-effect=\"bold\">Use the Order of Operations to Simplify Fractions<\/strong><\/p>\n<p id=\"fs-id1167829741857\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829741860\">\n<div data-type=\"problem\" id=\"fs-id1167829741862\">\n<p id=\"fs-id1167829741864\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-886d5945692bb468e6b30f5e7ba71ad6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&middot;&#54;&#45;&#51;&middot;&#52;&#125;&#123;&#52;&middot;&#53;&#45;&#50;&middot;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829741903\">\n<p id=\"fs-id1167829741905\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1968ddfbabcddff8b28c71b75a144ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#55;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829741916\">\n<div data-type=\"problem\" id=\"fs-id1167829741919\">\n<p id=\"fs-id1167829741921\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d95848b5e6e7016d53f8424ba231d578_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&middot;&#57;&#45;&#55;&middot;&#54;&#125;&#123;&#53;&middot;&#54;&#45;&#57;&middot;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"39\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829741973\">\n<div data-type=\"problem\" id=\"fs-id1167829741975\">\n<p id=\"fs-id1167829741977\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5941371979474f963757bac3384812a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#53;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#45;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"38\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829755927\">\n<p id=\"fs-id1167829755929\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829755937\">\n<div data-type=\"problem\" id=\"fs-id1167829755939\">\n<p id=\"fs-id1167829755942\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6df711cce06c2a7943cd73c9fc62a456_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#54;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#52;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#45;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"38\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829755980\">\n<div data-type=\"problem\" id=\"fs-id1167829755982\">\n<p id=\"fs-id1167829755984\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7c27c286fd0e322a2e27fdd7fbd3f1a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&middot;&#52;&#45;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#57;&middot;&#51;&#45;&#51;&middot;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"67\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829756031\">\n<p id=\"fs-id1167829756033\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f60306c12e0b1c2107d7595d78b20a5e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829756046\">\n<div data-type=\"problem\" id=\"fs-id1167829756048\">\n<p id=\"fs-id1167829756050\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67056ab18ecdc7ed7aa8b46ea14e05b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&middot;&#55;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#56;&middot;&#55;&#45;&#54;&middot;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"74\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829752395\">\n<div data-type=\"problem\" id=\"fs-id1167829752397\">\n<p id=\"fs-id1167829752399\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-468532a0852ffe469ec0a17cfcedc527_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#45;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#54;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#55;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"102\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829752472\">\n<p id=\"fs-id1167829752474\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c8a34714cd9e14f96438eaca16625df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829752486\">\n<div data-type=\"problem\" id=\"fs-id1167829752488\">\n<p id=\"fs-id1167829752490\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d988df0be5cf18655d6a38b0da4265d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#57;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#55;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#54;&#45;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"102\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829744633\">\n<div data-type=\"problem\" id=\"fs-id1167829744635\">\n<p id=\"fs-id1167829744637\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d5b985e9cb87c770bb5e4856b70ad405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#50;&#125;&#94;&#123;&#51;&#125;&#43;&#123;&#52;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"38\" style=\"vertical-align: -16px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829744674\">\n<p id=\"fs-id1167829744676\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66ceeedc341151a1adecf7dd22c17553_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"18\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829744685\">\n<div data-type=\"problem\" id=\"fs-id1167829744687\">\n<p id=\"fs-id1167829744689\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a1b33198a44ffb9b418064e57a5427e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#51;&#125;&#94;&#123;&#51;&#125;&#45;&#123;&#51;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"38\" style=\"vertical-align: -16px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750519\">\n<div data-type=\"problem\" id=\"fs-id1167829750521\">\n<p id=\"fs-id1167829750523\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b151b8a502ef6fe85908b804124b29dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"31\" style=\"vertical-align: -16px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750564\">\n<p id=\"fs-id1167829750566\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad3eef5a6b9153f5169702e6f3b6482e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#57;&#125;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750580\">\n<div data-type=\"problem\" id=\"fs-id1167829750582\">\n<p id=\"fs-id1167829750584\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b81545e6b19e4f0873135f695a6790e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"31\" style=\"vertical-align: -16px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750640\">\n<div data-type=\"problem\" id=\"fs-id1167829750643\">\n<p id=\"fs-id1167829750645\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9d7e6a70f6fadd1069804f71df0cedc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750667\">\n<p id=\"fs-id1167829750669\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51269b0555b78cd51fc1dea9fa3b0890_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750682\">\n<div data-type=\"problem\" id=\"fs-id1167833049496\">\n<p id=\"fs-id1167833049498\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9afa45fa4b2cb97f4e7a6acf11c228d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833049535\">\n<div data-type=\"problem\" id=\"fs-id1167833049537\">\n<p id=\"fs-id1167833049540\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7dce0d0f4e7b168d0944616123ece495_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833049574\">\n<p id=\"fs-id1167833049576\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6941d24b8eac415464621657779b5b7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"14\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833049589\">\n<div data-type=\"problem\" id=\"fs-id1167833049591\">\n<p id=\"fs-id1167833049593\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-50764f2ba66c327e47920bb68d8814bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833049642\"><strong data-effect=\"bold\">Mixed Practice<\/strong><\/p>\n<p id=\"fs-id1167833049648\">In the following exercises, simplify.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167833049651\">\n<div data-type=\"problem\" id=\"fs-id1167833049654\">\n<p id=\"fs-id1167833049656\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35c36896823219ea6c659b664c4b471e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"72\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829742352\">\n<p id=\"fs-id1167829742354\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27f26f48f049d16d7bf76d51e1f91cef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829742366\">\n<div data-type=\"problem\" id=\"fs-id1167829742368\">\n<p id=\"fs-id1167829742370\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58a90b37d3ebd0d56bb754a563cfe44f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#50;&#125;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" 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-id1167829742414\">\n<div data-type=\"problem\" id=\"fs-id1167829742416\">\n<p id=\"fs-id1167829742418\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c77d253a473fa82738a3c7bdd2d9f438_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"61\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829742439\">\n<p id=\"fs-id1167829742442\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0a1bbf5668139f1530fc35acfc9cae9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829742454\">\n<div data-type=\"problem\" id=\"fs-id1167829742456\">\n<p id=\"fs-id1167829742459\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38e95a25da33ea5fdfccb1ba63f11a49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"61\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829742496\">\n<div data-type=\"problem\" id=\"fs-id1167829742498\">\n<p id=\"fs-id1167829742500\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf66854c0c0a2cd9a184d8d5f8c27e4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#121;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"62\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829742522\">\n<p id=\"fs-id1167829742524\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e61dee592736a465c9af6cb2526da40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#50;&#56;&#45;&#49;&#53;&#121;&#125;&#123;&#54;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836540350\">\n<div data-type=\"problem\" id=\"fs-id1167836540353\">\n<p id=\"fs-id1167836540355\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bd7e78cf443ce64b7d2a834294c227f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"61\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836540398\">\n<div data-type=\"problem\" id=\"fs-id1167836540401\">\n<p id=\"fs-id1167836540403\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ebdc27d728c974c5fdfb4b5d81cb22a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#49;&#50;&#97;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#97;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"40\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836540429\">\n<p id=\"fs-id1167836540432\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-abe99fd913ea65fb2f9fc0f49262ca3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#51;&#125;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836540445\">\n<div data-type=\"problem\" id=\"fs-id1167836540448\">\n<p id=\"fs-id1167836540450\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b995924ef713380034efb96d73f461d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#48;&#121;&#125;&#123;&#49;&#51;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#49;&#53;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"47\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836540491\">\n<div data-type=\"problem\" id=\"fs-id1167836540493\">\n<p id=\"fs-id1167836540496\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b73c02e027bbb96a0caefb476c12175_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"57\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836540522\">\n<p id=\"fs-id1167836540524\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e8ce2f90e409bd9f90b29b7e9dacb23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836540536\">\n<div data-type=\"problem\" id=\"fs-id1167836540538\">\n<p id=\"fs-id1167836540540\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2640c3b8464952e2181231bbafe125ba_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;&#50;&#125;&#123;&#53;&#125;&middot;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750204\">\n<div data-type=\"problem\" id=\"fs-id1167829750206\">\n<p id=\"fs-id1167829750208\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b89e088edecf2f71b8f03fc89bf6893_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"56\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750232\">\n<p id=\"fs-id1167829750234\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b5b9d9f382b11767d19f257afca0019_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750242\">\n<div data-type=\"problem\" id=\"fs-id1167829750244\">\n<p id=\"fs-id1167829750246\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b409124d97896808726fe32355e03d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&divide;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"56\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750281\">\n<div data-type=\"problem\" id=\"fs-id1167829750283\">\n<p id=\"fs-id1167829750285\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47fc7226a7b99e4c617e75939709874d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750310\">\n<p id=\"fs-id1167829750313\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-458b8ef89dedd7cf464d6aa24d1f60c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#51;&#125;&#123;&#50;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750326\">\n<div data-type=\"problem\" id=\"fs-id1167829750329\">\n<p id=\"fs-id1167829750331\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7f294b9fbd99d93d9e82c9098bc64329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750372\">\n<div data-type=\"problem\" id=\"fs-id1167829750374\">\n<p id=\"fs-id1167829750377\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a52e0591454ee05629df1a98c79c6b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#50;&#48;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#53;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"91\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829750405\">\n<p id=\"fs-id1167829750408\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1239857d5f8f2f1f3625981723d94b50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829750420\">\n<div data-type=\"problem\" id=\"fs-id1167829750422\">\n<p id=\"fs-id1167829750425\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77aeb619cedca750f79e38fa4ee70500_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"76\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829738275\">\n<div data-type=\"problem\" id=\"fs-id1167829738277\">\n<p id=\"fs-id1167829738279\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e501375f1dd409ba7b728c0939b66c2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#57;&#125;&#123;&#50;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"31\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829738308\">\n<p id=\"fs-id1167829738310\"><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;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829738318\">\n<div data-type=\"problem\" id=\"fs-id1167829738320\">\n<p id=\"fs-id1167829738322\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1f431eedc3777d95fb81abcb7b06aae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#49;&#48;&#125;&#125;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#125;&#123;&#51;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"37\" style=\"vertical-align: -13px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829738361\">\n<div data-type=\"problem\" id=\"fs-id1167829738364\">\n<p id=\"fs-id1167829738366\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a9c52deb9dcdbdea5cfc854a9796556_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"121\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829738412\">\n<p id=\"fs-id1167829738414\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a8dc9e92eaa09a95e5c650903ec1cf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#51;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829738426\">\n<div data-type=\"problem\" id=\"fs-id1167829738429\">\n<p id=\"fs-id1167829738431\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c7d8cd51eb207df28c4864c0d961002_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&divide;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"121\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829738492\"><strong data-effect=\"bold\">Evaluate Variable Expressions with Fractions<\/strong><\/p>\n<p id=\"fs-id1167829738497\">In the following exercises, evaluate.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829738500\">\n<div data-type=\"problem\" id=\"fs-id1167829738502\">\n<p id=\"fs-id1167829738504\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-31bf02c5ed3b48fb5faae33d6083d22b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#49;&#48;&#125;&#45;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"50\" style=\"vertical-align: -7px;\" \/> when<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d99af0e768e77a0e7bfe6c7fa2e2a868_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" 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-82ef685750aca2af136f60da8e8f9c86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167829753672\">\n<p id=\"fs-id1167829753674\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e174a73678f6bd6f70d7aaec8f911349_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" 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-05094d976698f8323b4ee8760d5e2dfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"7\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829753705\">\n<div data-type=\"problem\" id=\"fs-id1167829753707\">\n<p id=\"fs-id1167829753709\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51105d37cecfbaaa3c9ba6d51bdbfb93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#50;&#125;&#45;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"50\" style=\"vertical-align: -7px;\" \/> when<\/p>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bb6f3d8761b18f21695b81f749b94a05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"46\" 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-aa8a92418e950109dbfdb14af810a06b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#119;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829753796\">\n<div data-type=\"problem\" id=\"fs-id1167829753798\">\n<p id=\"fs-id1167829753801\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95cdd65cb472a9114184d59d48d871fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#123;&#121;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"43\" style=\"vertical-align: -4px;\" \/> when<\/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-9af98521b89333b949a5950d447fdf84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d73f92d367c6005c529d143de2bb9ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167829753849\">\n<p id=\"fs-id1167829753851\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-579bbd4687145ea7c88f861b7d47e73e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"22\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829753865\">\n<div data-type=\"problem\" id=\"fs-id1167829753867\">\n<p id=\"fs-id1167829753869\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eca35b24268f54105a5d857a329c3e35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#123;&#117;&#125;&#94;&#123;&#50;&#125;&#123;&#118;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"43\" style=\"vertical-align: 0px;\" \/> when<\/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-0345297275f6da1f430319b3e3e061ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"57\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5925d1393d05c290510093a67ba6d04b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#118;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"56\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829751999\">\n<div data-type=\"problem\" id=\"fs-id1167829752001\">\n<p id=\"fs-id1167829752003\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59236f3cbcf26efdb0275f4d3a65929e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#43;&#98;&#125;&#123;&#97;&#45;&#98;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"24\" style=\"vertical-align: -6px;\" \/> when<\/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-04c321d1a555ca2cbaf2d58dbfe5bfc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#45;&#51;&#44;&#98;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"104\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167829752045\">\n<p id=\"fs-id1167829752047\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1d927a8f46d1af4ce6e1e2dc2c0fb4b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"29\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829752062\">\n<div data-type=\"problem\" id=\"fs-id1167829752064\">\n<p id=\"fs-id1167829752066\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56f0da7c47af05537de38f4f5537d21e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#45;&#115;&#125;&#123;&#114;&#43;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"24\" style=\"vertical-align: -8px;\" \/> when<\/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-246265d7ddd6c812aa815f1fb1ea4cb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#61;&#49;&#48;&#44;&#115;&#61;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167829752118\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167829752126\">\n<div data-type=\"problem\" id=\"fs-id1167829752128\">\n<p id=\"fs-id1167829752130\">Why do you need a common denominator to add or subtract fractions? Explain.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829752134\">\n<p id=\"fs-id1167829752136\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829752142\">\n<div data-type=\"problem\" id=\"fs-id1167829752144\">\n<p id=\"fs-id1167829752146\">How do you find the LCD of 2 fractions?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829752158\">\n<div data-type=\"problem\" id=\"fs-id1167829752160\">\n<p id=\"fs-id1167829752162\">Explain how you find the reciprocal of a fraction.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829752166\">\n<p id=\"fs-id1167829752168\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829752174\">\n<div data-type=\"problem\" id=\"fs-id1167829752176\">\n<p id=\"fs-id1167829752178\">Explain how you find the reciprocal of a negative number.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829752191\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167829752196\"><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-id1167829752208\" data-alt=\"This table has 4 columns, 5 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: simplify fractions, multiply and divide fractions, add and subtract fractions, use the order of operations to simplify fractions, evaluate variable expressions with fractions. The remaining columns are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_01_03_201_img_new-1.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 5 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: simplify fractions, multiply and divide fractions, add and subtract fractions, use the order of operations to simplify fractions, evaluate variable expressions with fractions. The remaining columns are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1167829752220\"><span class=\"token\">\u24d1<\/span> What does this checklist tell you about your mastery of this section? What steps will you take to improve?<\/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-id1167829752234\">\n<dt>complex fraction<\/dt>\n<dd id=\"fs-id1167829752239\">A fraction in which the numerator or the denominator is a fraction is called a complex fraction.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167829752244\">\n<dt>denominator<\/dt>\n<dd id=\"fs-id1167829752250\">In a fraction, written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7825cc57d4b3386857d6642115a67cd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"13\" style=\"vertical-align: -6px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6a0135701625ac5da01c134efeaebdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> the denominator <em data-effect=\"italics\">b<\/em> is the number of equal parts the whole has been divided into.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167829752281\">\n<dt>equivalent fractions<\/dt>\n<dd id=\"fs-id1167829752287\">Equivalent fractions are fractions that have the same value.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167829752291\">\n<dt>fraction<\/dt>\n<dd id=\"fs-id1167829752296\">A fraction is written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7825cc57d4b3386857d6642115a67cd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"13\" style=\"vertical-align: -6px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6a0135701625ac5da01c134efeaebdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> and <em data-effect=\"italics\">a<\/em> is the numerator and <em data-effect=\"italics\">b<\/em> is the denominator. A fraction represents parts of a whole.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167829751654\">\n<dt>least common denominator<\/dt>\n<dd id=\"fs-id1167829751659\">The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167829751664\">\n<dt>numerator<\/dt>\n<dd id=\"fs-id1167829751670\">In a fraction, written <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7825cc57d4b3386857d6642115a67cd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"13\" style=\"vertical-align: -6px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6a0135701625ac5da01c134efeaebdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#92;&#110;&#101;&#32;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\" \/> the numerator <em data-effect=\"italics\">a<\/em> indicates how many parts are included.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167829751702\">\n<dt>reciprocal<\/dt>\n<dd id=\"fs-id1167829751707\">The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-911","chapter","type-chapter","status-publish","hentry"],"part":762,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/911","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\/911\/revisions"}],"predecessor-version":[{"id":14621,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/911\/revisions\/14621"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/762"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/911\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=911"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=911"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=911"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=911"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}