{"id":182,"date":"2022-10-06T15:17:29","date_gmt":"2022-10-06T19:17:29","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/math025\/chapter\/adding-common-fractions\/"},"modified":"2025-06-28T00:02:31","modified_gmt":"2025-06-28T04:02:31","slug":"adding-common-fractions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/math025\/chapter\/adding-common-fractions\/","title":{"raw":"Topic A: Adding Common Fractions","rendered":"Topic A: Adding Common Fractions"},"content":{"raw":"Vocabulary Review:\r\n\r\n<img class=\"aligncenter wp-image-160 size-full\" title=\"Illustration of addend 4 plus 5 being equal to 9. With arrows to vocabulary examples of 9 as sum and 4,5 as addends. Also fraction three sevenths, illustrating the vocabular of 3 being the numerator and 7 being the denominator.\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-6-1.png\" alt=\"Illustration of addend 4 plus 5 being equal to 9. With arrows to vocabulary examples of 9 as sum and 4,5 as addends. Also fraction three sevenths, illustrating the vocabular of 3 being the numerator and 7 being the denominator.\" width=\"514\" height=\"438\" data-popupalt-original-title=\"null\" \/>\r\n\r\n<strong>Like Fractions:<\/strong> Fractions that have the same denominator\r\n\r\nExample: [latex]\\tfrac{1}{4}[\/latex],\u00a0 [latex]\\tfrac{2}{4}[\/latex],\u00a0 \u00a0[latex]\\tfrac{3}{4}[\/latex],\u00a0 \u00a0[latex]\\tfrac{4}{4}[\/latex],\u00a0 etc.\r\n\r\n<strong>Adding and subtracting fractions has some different rules from multiplying and dividing.<\/strong>\r\n\r\n<img class=\"aligncenter wp-image-161 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-1.png\" alt=\"Two circles. Circles are divided in three with shading of one of the three.\" width=\"124\" height=\"66\" \/>\r\n\r\nThere are two cakes that are left over. There is 1 piece of each cake left. If you were to put all the pieces left onto one plate, how much cake would you have?\r\n\r\n<img class=\"wp-image-162 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-4-A-3-1.jpg\" alt=\"Visual representation of equation about two cakes. Three circles. Circles are divided in three with shading of one of the three for the items to add together.\" width=\"265\" height=\"67\" \/>\r\n<p style=\"text-align: center\">Shade in your answer here<\/p>\r\nIf you made your plate like this: <img class=\"alignnone size-full wp-image-163\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-3.png\" alt=\"\" width=\"86\" height=\"80\" \/>\u00a0 then you are right!\r\n\r\n<strong>Try this example:<\/strong>\r\n\r\n<img class=\"aligncenter wp-image-164 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-4.png\" alt=\"Image of equation using three rectangle objects. One rectangle divided in four, shaded in one segment. Addition between. Second rectangle divided in four segments, shaded in first and third segments. Equals sign. And rectangle with three four segments. All unshaded.\" width=\"412\" height=\"110\" \/>\r\n\r\n<strong>The answer is:<\/strong>\r\n\r\n<img class=\"aligncenter wp-image-165 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-5.png\" alt=\"Image of equation using three rectangle objects. One rectangle divided in four, shaded in one segment. Addition between. Second rectangle divided in four segments, shaded in first and third segments. Equals sign. And rectangle with three shaded segments, one blank.\" width=\"416\" height=\"100\" \/>\r\n\r\n<strong>What you are doing is adding two <span style=\"text-decoration: underline\">like<\/span> fractions.<\/strong>\r\n<ul>\r\n \t<li>You are moving pieces of fractions that are the same size into one whole shape. The pieces do not change size, so the denominator must stay the same size.<\/li>\r\n \t<li>When adding two fractions, your answer is a fraction.<\/li>\r\n<\/ul>\r\n<strong>Look back at the two examples.<\/strong>\r\n\r\nWhen you add fractions, does the denominator or the numerator stay the same?\r\n<div class=\"textbox\">Common fractions must have the same denominator when you add them together.\u00a0<strong> Add the numerators<\/strong> and keep the denominators the same.<\/div>\r\nLook at the next two examples:\r\n\r\n<img class=\"wp-image-358 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-3.jpg\" alt=\"\" width=\"698\" height=\"91\" \/>\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{4} + \\dfrac{2}{4} = \\dfrac{3}{4}[\/latex]<\/p>\r\n<img class=\"wp-image-359 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-2.jpg\" alt=\"\" width=\"769\" height=\"58\" \/>\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{5}+ \\dfrac{2}{5}+ \\dfrac{1}{5} = \\dfrac{4}{5}[\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nTry a few for yourself\r\n<ol class=\"smallspace\" type=\"a\">\r\n \t<li style=\"text-align: center\">\r\n<p style=\"margin-top: 1em\"><img class=\"wp-image-168 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-13.png\" alt=\"Visual representation of the equation to solve. \" width=\"388\" height=\"90\" \/><\/p>\r\n<p style=\"text-align: center\">[latex]\\dfrac{2}{9} + \\dfrac{3}{9} = \\dfrac{ }{9}[\/latex]<\/p>\r\n<\/li>\r\n \t<li style=\"text-align: center\"><img class=\"wp-image-169 alignnone\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-14.png\" alt=\"Visual representation of the equation to solve. \" width=\"408\" height=\"40\" \/>\r\n<p style=\"text-align: center\">[latex]\\dfrac{2}{4} + \\dfrac{1}{4} = \\dfrac{ }{4}[\/latex]<\/p>\r\n<\/li>\r\n \t<li style=\"text-align: center\"><img class=\"wp-image-170 size-full alignnone\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-14b.png\" alt=\"Visual representation of the equation to solve. \" width=\"276\" height=\"70\" \/>\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{3} + \\dfrac{1}{3} = \\dfrac{ }{3}[\/latex]<\/p>\r\n<\/li>\r\n \t<li style=\"text-align: center\"><img class=\"wp-image-171 size-full alignnone\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-15.png\" alt=\"Visual representation of the equation to solve. \" width=\"380\" height=\"66\" \/>\r\n<p style=\"text-align: center\">[latex]\\dfrac{3}{6} + \\dfrac{2}{6} = \\dfrac{ }{6}[\/latex]<\/p>\r\n<\/li>\r\n \t<li style=\"text-align: center\"><img class=\"wp-image-172 size-full alignnone\" style=\"margin-top: 0.5em;margin-bottom: 0.5em;text-align: center\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-16.png\" alt=\"Visual representation of the equation to solve. \" width=\"412\" height=\"96\" \/>\r\n[latex]\\dfrac{3}{8} + \\dfrac{4}{8} = \\dfrac{ }{8}[\/latex]<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 1<\/strong>\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{5}{9}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{4}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{2}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{5}{6}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{7}{8}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Now find the answers to the additions without diagrams.<\/strong>\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{2}{4} + \\dfrac{1}{4} = \\dfrac{ }{4}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{3} + \\dfrac{1}{3} = \\dfrac{ }{3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{5} + \\dfrac{1}{5} = \\dfrac{ }{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{2}{11} + \\dfrac{7}{11} = \\dfrac{ }{11}[\/latex]<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise Two<\/strong>\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{3}{4}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{2}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{2}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{9}{11}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Add these common fractions.<\/strong>\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{1}{5} + \\dfrac{2}{5} = [\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{6} + \\dfrac{2}{6} = [\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{7} + \\dfrac{2}{7} = [\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{10} + \\dfrac{6}{10} = [\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{14}{20} + \\dfrac{3}{20} = [\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{7}{37} + \\dfrac{19}{37} = [\/latex]<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 3<\/strong>\r\n<ol class=\"threecolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{3}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{5}{6}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{5}{7}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{9}{10}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{17}{20}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{26}{37}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nSometimes the [pb_glossary id=\"269\"]sum[\/pb_glossary] of a fraction will need to be reduced (take a look at this example to remind yourself how to do this).\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p style=\"text-align: center\">[latex]\\dfrac{2}{8} + \\dfrac{2}{8} = \\dfrac{4}{8}\\rightarrow\\dfrac{\u00f7 4}{\u00f7 4} = \\dfrac{1}{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example B<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p style=\"text-align: center\">[latex]\\dfrac{3}{4} + \\dfrac{3}{4} = \\dfrac{6}{4}\\rightarrow\\dfrac{6}{4}[\/latex] [latex]\\dfrac{\u00f7 2}{\u00f7 2}[\/latex] = [latex]\\dfrac{3}{2}[\/latex] =\u00a0 [latex]1\\dfrac{1}{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the sums to the following additions. Make sure your answer is in the lowest terms.\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{1}{4} + \\dfrac{1}{4} = [\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{3} + \\dfrac{1}{3} = [\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{10} + \\dfrac{2}{10} = [\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{7}{25} + \\dfrac{8}{25} = [\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{5} + \\dfrac{1}{5} = [\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{9}{27} + \\dfrac{12}{27} = [\/latex]<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 4\r\n<\/strong>\r\n<ol class=\"threecolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{2}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{7}{9}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nSo far all your answers have been less than one (a [pb_glossary id=\"282\"]proper fraction[\/pb_glossary]). Sometimes adding fractions can result in more than one whole.\r\n\r\nLook at this example:\r\n\r\n<img class=\"aligncenter wp-image-173 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-4-A-14.jpg\" alt=\"\" width=\"499\" height=\"72\" \/>\r\n<p style=\"text-align: center\">[latex]\\dfrac{2}{4} + \\dfrac{3}{4}=\\dfrac{4}{4}\\text{and}\\dfrac{1}{4} [\/latex]\u00a0 \u00a0(or [latex]\\left(\\dfrac{5}{4}\\right)[\/latex])<\/p>\r\nThere are not enough parts in the first square to hold all your shaded parts, so you need to draw a second square to hold the extra shaded parts.\r\n\r\nYou would also have to convert this answer from an improper fraction to a mixed number:\r\n<p style=\"text-align: center\">[latex]\\dfrac{5}{4} = 1\\dfrac{1}{4}[\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Try these additions.<\/strong> Remember to always reduce!\r\n<ol class=\"smallspace\" type=\"a\">\r\n \t<li style=\"text-align: center\"><img class=\"aligncenter wp-image-174 size-full\" title=\"Visual represenation of the equation to solve.\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-21.png\" alt=\"Visual represenation of the equation to solve.\" width=\"480\" height=\"70\" data-popupalt-original-title=\"null\" \/>\r\n[latex]\\dfrac{4}{6}+\\dfrac{5}{6} = [\/latex]<\/li>\r\n \t<li style=\"text-align: center\"><img class=\"aligncenter wp-image-175 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-22.png\" alt=\"Visual representation of the equation to solve. \" width=\"452\" height=\"94\" \/>\r\n[latex]\\dfrac{6}{8}+\\dfrac{3}{8} = [\/latex]<\/li>\r\n \t<li style=\"text-align: center\"><img class=\"aligncenter wp-image-176 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-23.png\" alt=\"Visual representation of the equation to solve. \" width=\"508\" height=\"76\" \/>\r\n[latex]\\dfrac{3}{4}+\\dfrac{3}{4}=[\/latex]<\/li>\r\n \t<li style=\"text-align: center\"><img class=\"aligncenter wp-image-177 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-24.png\" alt=\"Visual representation of the equation to solve. \" width=\"444\" height=\"56\" \/>\r\n[latex]\\dfrac{8}{9}+\\dfrac{4}{9} = [\/latex]<\/li>\r\n \t<li style=\"text-align: center\"><img class=\"aligncenter wp-image-178 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-25.png\" alt=\"Visual representation of the equation to solve. \" width=\"482\" height=\"94\" \/>\r\n[latex]\\dfrac{3}{5}+\\dfrac{4}{5} = [\/latex]<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 5<\/strong>\r\n<ol class=\"threecolumn\" type=\"a\">\r\n \t<li>[latex]1\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]1\\dfrac{1}{8}[\/latex]<\/li>\r\n \t<li>[latex]1\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]1\\dfrac{1}{3}[\/latex]<\/li>\r\n \t<li>[latex]1\\dfrac{2}{5}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example C<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Sometimes you will have to add 3 or more fractions together.<\/strong>\r\n\r\n<img class=\"aligncenter wp-image-179 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-26.png\" alt=\"Visual representation of the equation to solve. \" width=\"500\" height=\"80\" \/>\r\n<p style=\"text-align: center\">[latex]\\dfrac{2}{3} + \\dfrac{1}{3} + \\dfrac{2}{3} = \\dfrac{5}{3} = 1\\dfrac{2}{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example D<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<img class=\"aligncenter wp-image-180 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-27.png\" alt=\"Visual representation of the equation to solve. \" width=\"532\" height=\"88\" \/>\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{4} + \\dfrac{2}{4} + \\dfrac{1}{4} + \\dfrac{3}{4} = \\dfrac{7}{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Add these common fractions. Be sure your answers are in <span style=\"text-decoration: underline\">lowest terms<\/span>.<\/strong>\r\n<ol class=\"threecolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{2}{3} + \\dfrac{1}{3} = \\dfrac{3}{3} = 1[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{7}{10} + \\dfrac{3}{10} = [\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{5} + \\dfrac{2}{5} = [\/latex]<\/li>\r\n \t<li>\u00a0[latex]\\begin{array}{rr}&amp;\\dfrac{3}{4}\\\\+&amp;\\dfrac{1}{4}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;\\dfrac{5}{6}\\\\+&amp;\\dfrac{5}{6}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>\u00a0[latex]\\begin{array}{rr}&amp;\\dfrac{4}{8}\\\\+&amp;\\dfrac{3}{8}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>\u00a0[latex]\\begin{array}{rr}&amp;\\dfrac{1}{8}\\\\+&amp;\\dfrac{3}{8}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;\\dfrac{2}{5}\\\\&amp;\\dfrac{3}{5}\\\\+&amp;\\dfrac{3}{5}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;\\dfrac{3}{6}\\\\&amp;\\dfrac{1}{6}\\\\ +&amp;\\dfrac{1}{6}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 6<\/strong>\r\n<ol class=\"threecolumn\" start=\"2\" type=\"a\">\r\n \t<li>[latex]1[\/latex]<\/li>\r\n \t<li>[latex]1[\/latex]<\/li>\r\n \t<li>[latex]1[\/latex]<\/li>\r\n \t<li>[latex]1\\dfrac{2}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{7}{8}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]1\\dfrac{3}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{5}{6}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>Adding Mixed Numbers<\/h1>\r\n<strong>To add mixed numbers<\/strong>\r\n<div class=\"textbox\">\r\n<ul>\r\n \t<li>Be sure the denominators are the same.<\/li>\r\n \t<li>Add the common fractions.<\/li>\r\n \t<li>Add the whole numbers.Simplify the common fraction.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example E<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<p style=\"text-align: left\">[latex]\\begin{array}{rr}&amp;3\\dfrac{1}{8}\\\\+&amp;2\\dfrac{3}{8}\\\\ \\hline\\end{array}[\/latex]<\/p>\r\n\r\n<ul>\r\n \t<li style=\"text-align: left\">[latex]5\\dfrac{4}{8}[\/latex] = [latex]5\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li style=\"text-align: left\">[latex]\\dfrac{4}{8}[\/latex] =\u00a0[latex] \\dfrac{4}{8}\\left(\\dfrac{\u00f74}{\u00f74}\\right )[\/latex] = [latex]\\dfrac{1}{2}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example F<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[latex]\\begin{array}{rr}&amp;12\\dfrac{1}{3}\\\\+&amp;6\\dfrac{1}{3}\\\\\\hline&amp;18\\dfrac{2}{3}\\end{array}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAdd the following numbers. Reduce the answers to lowest terms.\r\n<ol class=\"twocolumn\" style=\"line-height: 7em;margin-bottom: 15px\" type=\"a\">\r\n \t<li>[latex]\\begin{array}{rr}&amp;6\\dfrac{1}{12}\\\\+&amp;8 \\dfrac{5}{12}\\\\ \\hline \\end{array}[\/latex]\r\n[latex] 14 \\dfrac{6}{12} = 14 \\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;22\\dfrac{1}{6}\\\\+&amp;14\\dfrac{1}{6}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;8\\dfrac{1}{4}\\\\+&amp;3\\dfrac{1}{4}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;18\\dfrac{1}{2}\\\\+&amp;10\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;4\\dfrac{1}{10}\\\\+&amp;\\dfrac{3}{10}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Answers to Exercise 7<\/strong>\r\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\r\n \t<li>[latex]36 \\dfrac{1}{3}[\/latex]<\/li>\r\n \t<li>[latex]11 \\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]28 \\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]4 \\dfrac{2}{5}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>A<\/strong><strong>d<\/strong><strong>d these numbers. Give your answers in lowest terms<\/strong>.\r\n<ol class=\"twocolumn\" style=\"line-height: 7em;margin-bottom: 15px\" type=\"a\">\r\n \t<li>[latex]\\begin{array}{ll}&amp;6\\dfrac{4}{5}\\\\+&amp;3\\dfrac{2}{5}\\\\ \\hline\\end{array}[\/latex]\r\n[latex]9\\dfrac{6}{5}=10\\dfrac{1}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}&amp;9\\dfrac{1}{3}\\\\+&amp;2\\dfrac{2}{3}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}&amp;3\\dfrac{3}{8}\\\\+&amp;12\\dfrac{7}{8}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}&amp;100\\dfrac{7}{10}\\\\+&amp;50\\dfrac{5}{10}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}&amp;3\\dfrac{4}{7}\\\\+&amp;6\\dfrac{5}{7}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}&amp;8\\dfrac{4}{5}\\\\+&amp;\\dfrac{4}{5}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Answers to Exercise 8<\/strong>\r\n<ol class=\"threecolumn\" start=\"2\" type=\"a\">\r\n \t<li>[latex]12[\/latex]<\/li>\r\n \t<li>[latex]16 \\dfrac{1}{4}[\/latex]<\/li>\r\n \t<li>[latex]151 \\dfrac{1}{5}[\/latex]<\/li>\r\n \t<li>[latex]10 \\dfrac{2}{7}[\/latex]<\/li>\r\n \t<li>[latex]9 \\dfrac{3}{5}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<strong>If you are not comfortable with this work so far, talk to your instructor and get some more practice before you go ahead.<\/strong>\r\n\r\nThe next question is:\r\n\r\nWhat happens when two fractions in an addition (the [pb_glossary id=\"278\"]addends[\/pb_glossary]) do not have the same [pb_glossary id=\"280\"]denominator[\/pb_glossary]? If the addends do not have a common denominator, you will need to find equivalent fractions to make the addends have a common denominator.\r\nRead on to find out how!\r\n<h1>Multiples and Least Common Multiples (LCM)<\/h1>\r\nWhen you learned the multiplication tables you learned the <strong>multiples<\/strong> of each number. Multiples are the answers when you multiply a whole number by 1, 2, 3, 4, 5, 6, 7, and so on.\r\n<table style=\"border-collapse: collapse;width: 100%;height: 234px\" border=\"0\">\r\n<tbody>\r\n<tr style=\"height: 18px\">\r\n<th style=\"width: 50%;text-align: center;height: 18px\" scope=\"col\">The <strong>multiples of 2<\/strong><\/th>\r\n<th style=\"width: 50%;text-align: center;height: 18px\" scope=\"col\">The<strong> multiples of 6<\/strong><\/th>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times1 = \\bf{2}[\/latex]<\/td>\r\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times1 = \\bf{6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times2 = \\bf{4}[\/latex]<\/td>\r\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times2 = \\bf{12}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times3 = \\bf{6}[\/latex]<\/td>\r\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times3 = \\bf{18}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times4 = \\bf{8}[\/latex]<\/td>\r\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times4 = \\bf{24}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times5 = \\bf{10}[\/latex]<\/td>\r\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times5 = \\bf{30}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times6 = \\bf{12}[\/latex]<\/td>\r\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times6 = \\bf{36}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times7 = \\bf{14}[\/latex]<\/td>\r\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times7 = \\bf{42}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times8 = \\bf{16}[\/latex]<\/td>\r\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times8 = \\bf{48}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times9 = \\bf{18}[\/latex]<\/td>\r\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times9 = \\bf{54}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times10 = \\bf{20}[\/latex]<\/td>\r\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times10 = \\bf{60}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times11 = \\bf{22}[\/latex]<\/td>\r\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times11 = \\bf{66}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times12 = \\bf{24}[\/latex]<\/td>\r\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times12 = \\bf{72}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nand you can keep going as high as you want.\r\n\r\nThe multiples of 2 are <strong>2, 4, 6, 8, 10, 12, 14,<\/strong> and so on. &amp; The multiples of 6 are <strong>6, 12, 18, 24, 30, 36,<\/strong> and so on.\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>List the first ten multiples of each number. This chart may be useful to you later.<\/strong>\r\n<ol type=\"a\">\r\n \t<li>2\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<strong>Multiples<\/strong> <em>2, 4, 6, 8, 10, 12, 14, 16, 18, 20<\/em><\/li>\r\n<\/ol>\r\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\r\n \t<li>3<\/li>\r\n \t<li>4<\/li>\r\n \t<li>5<\/li>\r\n \t<li>9<\/li>\r\n \t<li>10<\/li>\r\n \t<li>11<\/li>\r\n \t<li>12<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\n<strong>Answers to Exercise 9<\/strong>\r\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\r\n \t<li>3, 6, 9, 12, 15, 18, 21, 24, 27, 30<\/li>\r\n \t<li>4, 8, 12, 16, 20, 24, 28, 32, 36, 40<\/li>\r\n \t<li>5, 10, 15, 20, 25, 30, 35, 40, 45, 50<\/li>\r\n \t<li>9, 18, 27, 36, 45, 54, 63, 72, 81, 90<\/li>\r\n \t<li>10,20,30,40,50,60,70,80,90,100<\/li>\r\n \t<li>11, 22, 33, 44, 55, 66, 77, 88, 99, 110<\/li>\r\n \t<li>12,24,36,48,60,72,84,96,108,120<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div>This is a quick method to find the least common multiple (LCM).<\/div>\r\n<div style=\"padding-left: 40px\">least means smallest<\/div>\r\n<div style=\"padding-left: 40px\">common means shared<\/div>\r\n<div style=\"padding-left: 40px\">multiple means the answers when you multiply by 1, 2, 3, etc.<\/div>\r\n<div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example G<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the least common multiple (LCM) of 3 and 5?\r\n<ul>\r\n \t<li>Multiples:\r\n<ul>\r\n \t<li>Multiples of 3: 3, 6, 9, 12, <span style=\"text-decoration: underline\">15<\/span>, 18, 21, 24, 27, 30<\/li>\r\n \t<li>Multiples of 5: 5, 10, <span style=\"text-decoration: underline\">15<\/span>, 20, 25, 30\u2026<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nThe least common multiple of 3 and 5 is 15.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example H<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the LCM of 3 and 4?\r\n<ul>\r\n \t<li>Multiples:\r\n<ul>\r\n \t<li>Multiples of 3: 3, 6, 9, <span style=\"text-decoration: underline\">12<\/span>, 15, 18, 21, 24, 27, 30\u2026<\/li>\r\n \t<li>Multiples of 4: 4, 8, <span style=\"text-decoration: underline\">12<\/span>, 16, 20, 24, 28, 32 \u2026.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The least common multiple of 3 and 4 is 12.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example I<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the LCM of 4 and 8?\r\n<ul>\r\n \t<li>Multiples:\r\n<ul>\r\n \t<li>Multiples of 4: 4, <span style=\"text-decoration: underline\">8<\/span>, 12, 16, 20\u2026<\/li>\r\n \t<li>Multiples of 8:\u00a0 <span style=\"text-decoration: underline\">8<\/span>, 16, 24, 32, 40\u2026<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The least common multiple of 4 and 8 is 8.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div>Hint: Always check to see if the larger number is a multiple of the smaller number. If it is, then the larger number is the Least Common Multiple (LCM).<\/div>\r\n<ul>\r\n \t<li>LCM of 3 and 6 is 6<\/li>\r\n \t<li>LCM of 2 and 4 is 4<\/li>\r\n \t<li>LCM of 5 and 15 is 15<\/li>\r\n<\/ul>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the Least Common Multiple of these pairs of numbers. Use your chart from Exercise Nine to help you. You may want to add the multiples of other numbers to that chart.\r\n<ol class=\"threecolumn\" type=\"a\">\r\n \t<li>3,6<\/li>\r\n \t<li>2,5<\/li>\r\n \t<li>12, 3<\/li>\r\n \t<li>6, 12<\/li>\r\n \t<li>5, 4<\/li>\r\n \t<li>4, 8<\/li>\r\n \t<li>8, 16<\/li>\r\n \t<li>4, 7<\/li>\r\n \t<li>25, 5<\/li>\r\n \t<li>2, 9<\/li>\r\n \t<li>6, 10<\/li>\r\n \t<li>8, 12<\/li>\r\n<\/ol>\r\n<strong>\r\nAnswers Exercise 10<\/strong>\r\n<ol class=\"threecolumn\" type=\"a\">\r\n \t<li>6<\/li>\r\n \t<li>10<\/li>\r\n \t<li>12<\/li>\r\n \t<li>12<\/li>\r\n \t<li>20<\/li>\r\n \t<li>8<\/li>\r\n \t<li>16<\/li>\r\n \t<li>28<\/li>\r\n \t<li>25<\/li>\r\n \t<li>18<\/li>\r\n \t<li>30<\/li>\r\n \t<li>24<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nNow that you know how to find an LCM, you can apply this knowledge to adding and subtracting fractions.\r\n<h1>Least Common Denominator (LCD)<\/h1>\r\nTo find the Least Common Denominator of common fractions: <strong>find the least common multiple of the denominators<\/strong>.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example J<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the least common denominator of [latex]\\dfrac{1}{2}[\/latex] and [latex]\\dfrac{3}{4}[\/latex]?\r\n\r\nThe denominators are 2 and 4.\r\n\r\nThe <strong>least common multiple<\/strong> of 2 and 4 is <strong>4<\/strong>.\r\n\r\nSo the <strong>least common denominator (LCD)<\/strong> for [latex]\\dfrac{1}{2}[\/latex] and [latex]\\dfrac{3}{4}[\/latex] is <strong>4<\/strong>.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example K<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the <strong>LCD<\/strong> of [latex]\\dfrac{3}{4}[\/latex] and [latex]\\dfrac{2}{3}[\/latex]?\r\n\r\nThe denominators are 4 and 3.\r\n\r\nThe <strong>least common multiple<\/strong> of 4 and 3 is <strong>12<\/strong>.\r\n\r\nSo the<strong> least common denominator<\/strong> for [latex]\\dfrac{3}{4}[\/latex] and [latex]\\dfrac{2}{3}[\/latex] is <strong>12<\/strong>.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the Least Common Denominator (LCD) for these pairs of fractions.\r\n<table style=\"border-collapse: collapse;width: 100%\" border=\"0\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 7.91317%\" scope=\"col\"><\/th>\r\n<th style=\"width: 42.0868%\" scope=\"col\">Fractions<\/th>\r\n<th style=\"width: 25%\" scope=\"col\">Denominators<\/th>\r\n<th style=\"width: 25%\" scope=\"col\">Least Common Denominators<\/th>\r\n<\/tr>\r\n<tr>\r\n<th style=\"width: 7.91317%\" scope=\"row\">a)<\/th>\r\n<td style=\"width: 42.0868%\">[latex]\\dfrac{5}{8}[\/latex], [latex]\\dfrac{2}{3}[\/latex]<\/td>\r\n<td style=\"width: 25%\">8, 3<\/td>\r\n<td style=\"width: 25%\">24<\/td>\r\n<\/tr>\r\n<tr>\r\n<th style=\"width: 7.91317%\" scope=\"row\">b)<\/th>\r\n<td style=\"width: 42.0868%\">[latex]\\dfrac{1}{5}[\/latex], [latex]\\dfrac{1}{10}[\/latex]<\/td>\r\n<td style=\"width: 25%\"><\/td>\r\n<td style=\"width: 25%\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<th style=\"width: 7.91317%\" scope=\"row\">c)<\/th>\r\n<td style=\"width: 42.0868%\">[latex]\\dfrac{1}{3}[\/latex], [latex]\\dfrac{3}{4}[\/latex]<\/td>\r\n<td style=\"width: 25%\"><\/td>\r\n<td style=\"width: 25%\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<th style=\"width: 7.91317%\" scope=\"row\">d)<\/th>\r\n<td style=\"width: 42.0868%\">[latex]\\dfrac{2}{3}[\/latex], [latex]\\dfrac{1}{5}[\/latex]<\/td>\r\n<td style=\"width: 25%\"><\/td>\r\n<td style=\"width: 25%\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<th style=\"width: 7.91317%\" scope=\"row\">e)<\/th>\r\n<td style=\"width: 42.0868%\">[latex]\\dfrac{5}{8}[\/latex], [latex]\\dfrac{1}{16}[\/latex]<\/td>\r\n<td style=\"width: 25%\"><\/td>\r\n<td style=\"width: 25%\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Answers to Exercise 11 (only least common denominator is given)<\/strong>\r\n<ol class=\"threecolumn\" start=\"2\" type=\"a\">\r\n \t<li>10<\/li>\r\n \t<li>12<\/li>\r\n \t<li>15<\/li>\r\n \t<li>16<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nYou know how to find the least common denominator <strong>(LCD).<\/strong> The next step is to make <strong>equivalent fractions<\/strong> using the <strong>LCD<\/strong>.\r\n\r\n<strong>Step 1:<\/strong> Find the least common denominator.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{rr}&amp;\\dfrac{3}{4}\\\\+&amp;\\dfrac{1}{3}\\\\ \\hline\\end{array}[\/latex]<\/p>\r\nLCD of 4 and 3 is 12.\r\n\r\n<strong> Step 2:<\/strong>\u00a0Write an = sign after each fraction, followed by the common denominator.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{rrrr}&amp;\\dfrac{3}{4}=\\dfrac{ }{12}\\\\+&amp;\\dfrac{1}{3} = \\dfrac{ }{12}\\\\ \\hline\\end{array}[\/latex]<\/p>\r\n<strong>Step 3:<\/strong> Rename the fractions as equivalent fractions with the <strong>LCD<\/strong>.\r\n<p style=\"text-align: center\">[latex]\\dfrac{3}{4}[\/latex] = [latex]\\dfrac{ }{12}[\/latex]<\/p>\r\n<p style=\"text-align: center\">4 times what = 12?<\/p>\r\n<p style=\"text-align: center\">4 \u00d7 3 = 12<\/p>\r\nIf the denominator was multiplied by 3, the numerator must be multiplied by 3.\r\n<p style=\"text-align: center\">[latex]\\dfrac{3}{4}[\/latex] [latex]\\dfrac{\u00d73}{\u00d73}[\/latex] = [latex]\\dfrac{9}{12}[\/latex]<\/p>\r\nNow rename the other fraction.\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{3}[\/latex] = [latex]\\dfrac{ }{12}[\/latex]<\/p>\r\n3 times what = 12?\r\n<p style=\"text-align: center\">[latex]3\\times 4 = 12[\/latex]<\/p>\r\nIf this denominator was multiplied by 4, this numerator must be multiplied by 4.\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{3}[\/latex]\u00a0 [latex]\\dfrac{\u00d74}{\u00d74}[\/latex] = [latex]\\dfrac{4}{12}[\/latex]<\/p>\r\nNow rename the other fraction.\r\n\r\n<strong>Step 4:<\/strong> The question now looks like this and can be added.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{rrrr}&amp;\\dfrac{3}{4}&amp;=&amp;\\dfrac{9}{12}\\\\ +&amp;\\dfrac{1}{3} &amp;= &amp;\\dfrac{4}{12}\\\\ \\hline \\\\ &amp; \\dfrac{13}{12}&amp;=&amp; 1\\dfrac{1}{12}\\end{array}[\/latex]<\/p>\r\n&nbsp;\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example L<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[latex]\\dfrac{1}{4} + \\dfrac{3}{8}[\/latex] =\r\n\r\n<strong>Step 1 and 2:<\/strong> Find the least common denominator\r\n<p style=\"text-align: center\">[latex]\\begin{array}{ll}&amp;\\dfrac{1}{4} = \\dfrac{ }{8}\\\\+&amp;\\dfrac{3}{8}= \\dfrac{ }{8}\\\\ \\hline\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\"><strong>Step 3:<\/strong> Rename as equivalent fractions<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{ll}&amp;\\dfrac{1}{4}\\left(\\dfrac{\\times2}{\\times2}\\right) = \\dfrac{2}{8}\\\\+&amp;\\dfrac{3}{8}\\left(\\dfrac{\\times1}{\\times1}\\right)= \\dfrac{3}{8}\\\\ \\hline\\end{array}[\/latex]<\/p>\r\n<strong>Step 4:<\/strong> Add and simplify the answer.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{lll}&amp;\\dfrac{1}{4}\\left(\\dfrac{\u00d72}{\u00d72}\\right) = &amp;\\dfrac{2}{8}\\\\+&amp;\\dfrac{3}{8}\\left(\\dfrac{\u00d71}{\u00d71}\\right)= &amp;\\dfrac{3}{8}\\\\ \\hline&amp;&amp;\\dfrac{5}{8}\\end{array}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 12<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>A<\/strong><strong>d<\/strong><strong>d these common fractions.<\/strong> Express the answer in lowest <span style=\"text-decoration: underline\">terms<\/span>.\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>[latex]\\begin{array}{rr}&amp;\\dfrac{1}{2}\\left(\\dfrac{\u00d74}{\u00d74}\\right) =&amp;\\dfrac{4}{8}\\\\+&amp;\\dfrac{3}{8}\\left(\\dfrac{\u00d71}{\u00d71}\\right)= &amp;\\dfrac{3}{8}\\\\ \\hline&amp;&amp;\\dfrac{7}{8}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;\\dfrac{1}{4}\\left(\\dfrac{\u00d72}{\u00d72}\\right) =&amp;\\dfrac{2}{8}\\\\+&amp;\\dfrac{3}{8}\\left(\\dfrac{\u00d71}{\u00d71}\\right)= &amp;\\dfrac{3}{8}\\\\ \\hline&amp;&amp;\\dfrac{5}{8}\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;\\dfrac{1}{5}\\\\+&amp;\\dfrac{1}{10}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;\\dfrac{5}{16}\\\\+&amp;\\dfrac{1}{4}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;\\dfrac{1}{3}\\\\+&amp;\\dfrac{7}{12}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;\\dfrac{2}{3}\\\\+&amp;\\dfrac{1}{6}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;\\dfrac{3}{10}\\\\ +&amp;\\dfrac{2}{5}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;\\dfrac{1}{12}\\\\ +&amp;\\dfrac{1}{4}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 12<\/strong>\r\n<ol class=\"twocolumn\" start=\"3\" type=\"a\">\r\n \t<li>[latex]\\dfrac{3}{10}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{9}{16}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{11}{12}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{5}{6}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{7}{10}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{3}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<strong>\u00a0How did you do? If you are struggling with this process, speak to your instructor for help.<\/strong>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 13<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>More practice.<\/strong> Do only as many as you think you need.\r\n<ol type=\"a\">\r\n \t<li style=\"margin-bottom: 2em\">[latex]\\begin{array}{rrr}&amp;\\dfrac{2}{3}\\left(\\dfrac{\u00d74}{\u00d74}\\right) &amp;=\\dfrac{8}{12}\\\\&amp;\\dfrac{1}{2}\\left(\\dfrac{\u00d76}{\u00d76}\\right)&amp;=\\dfrac{6}{12}\\\\+&amp;\\dfrac{3}{4}\\left(\\dfrac{\u00d73}{\u00d73}\\right)&amp; = \\dfrac{9}{12}\\\\ \\hline&amp;&amp;\\dfrac{23}{12}&amp;=1\\dfrac{11}{12}\\end{array}[\/latex]<\/li>\r\n \t<li style=\"margin-bottom: 2em\">[latex]\\begin{array}{rrr}&amp;\\dfrac{5}{24}\\left(\\dfrac{\u00d71}{\u00d71}\\right)&amp; =\\dfrac{5}{24}\\\\&amp;\\dfrac{1}{3}\\left(\\dfrac{\u00d78}{\u00d78}\\right)&amp;= \\dfrac{8}{24}\\\\+&amp;\\dfrac{3}{8}\\left(\\dfrac{\u00d73}{\u00d73}\\right)&amp;= \\dfrac{9}{24}\\\\ \\hline&amp;&amp;\\dfrac{22}{24}&amp;=\\dfrac{11}{12}\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<ol type=\"a\">\r\n \t<li style=\"list-style-type: none\">\r\n<ol class=\"threecolumn\" start=\"3\" type=\"a\">\r\n \t<li style=\"margin-bottom: 3em\">[latex]\\begin{array}{rr}&amp;\\dfrac{5}{12}\\\\&amp;\\dfrac{5}{6}\\\\+&amp;\\dfrac{3}{4}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n \t<li style=\"margin-bottom: 3em\">[latex]\\begin{array}{rr}&amp;\\dfrac{3}{10}\\\\&amp;\\dfrac{3}{4}\\\\+&amp;\\dfrac{4}{5}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n \t<li style=\"margin-bottom: 3em\">[latex]\\begin{array}{rr}&amp;\\dfrac{1}{2}\\\\&amp;\\dfrac{2}{5}\\\\+&amp;\\dfrac{7}{10}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n \t<li style=\"margin-bottom: 3em\">[latex]\\begin{array}{rr}&amp;\\dfrac{5}{6}\\\\&amp;\\dfrac{3}{4}\\\\+&amp;\\dfrac{1}{3}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n \t<li style=\"margin-bottom: 3em\">[latex]\\begin{array}{rr}&amp;\\dfrac{7}{16}\\\\+&amp;\\dfrac{3}{4}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n \t<li style=\"margin-bottom: 3em\">[latex]\\begin{array}{rr}&amp;\\dfrac{4}{5}\\\\+&amp;\\dfrac{1}{3}\\\\ \\hline\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 13<\/strong>\r\n<ol class=\"twocolumn\" start=\"3\" type=\"a\">\r\n \t<li>[latex]2[\/latex]<\/li>\r\n \t<li>[latex]1\\dfrac{17}{20}[\/latex]<\/li>\r\n \t<li>[latex]1\\dfrac{3}{5}[\/latex]<\/li>\r\n \t<li>[latex]1\\dfrac{11}{12}[\/latex]<\/li>\r\n \t<li>[latex]1\\dfrac{3}{16}[\/latex]<\/li>\r\n \t<li>[latex]1\\dfrac{2}{15}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nAddition questions are often written with the fractions side by side instead of one fraction above the other. For example:\r\n<p style=\"text-align: center\">[latex]\\dfrac{2}{3}[\/latex] + [latex]\\dfrac{5}{8}[\/latex] =<\/p>\r\nYou may solve as shown in this example or rewrite the question with the fractions one above the other.\r\n<p style=\"text-align: center\">[latex]\\dfrac{2}{3} + \\dfrac{5}{8} = \\dfrac{2}{3}\\dfrac{\u00d78}{\u00d78} + \\dfrac{5}{8}\\dfrac{\u00d73}{\u00d73}= \\dfrac{16}{24}+ \\dfrac{15}{24}=\\dfrac{31}{24}= 1 \\dfrac{7}{24}[\/latex]<\/p>\r\n<p style=\"text-align: center\">or<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{rrr}&amp;\\dfrac{2}{3}\\left(\\dfrac{\u00d78}{\u00d78}\\right) &amp;=\\dfrac{16}{24}\\\\&amp;\\dfrac{5}{8}\\left(\\dfrac{\u00d73}{\u00d73}\\right)&amp;=\\dfrac{15}{24}\\\\ \\hline&amp;&amp;\\dfrac{31}{24}&amp;=1\\dfrac{7}{24}\\end{array}[\/latex]<\/p>\r\n&nbsp;\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 14<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Find the sum<\/strong>. Do enough questions to be confident in your skill.\r\n<ol type=\"a\">\r\n \t<li>[latex]\\begin{array}{rr} \\\\ \\dfrac{1}{2} + \\dfrac{1}{6} = &amp;\\\\ \\dfrac{1}{2}\\left( \\dfrac{\\times 3}{\\times 3}\\right) + \\dfrac{1}{6} = &amp;\\\\ \\dfrac{3}{6} + \\dfrac{1}{6} = &amp;\\dfrac{4}{6} = \\dfrac{2}{3} \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\r\n \t<li>[latex]\\dfrac{1}{4} + \\dfrac{7}{8}[\/latex] =<\/li>\r\n \t<li>[latex]\\dfrac{1}{5} + \\dfrac{3}{5}[\/latex] =<\/li>\r\n \t<li>[latex]\\dfrac{1}{12}+\\dfrac{2}{3}[\/latex] =<\/li>\r\n \t<li>[latex]\\dfrac{1}{3} + \\dfrac{2}{3}[\/latex] =<\/li>\r\n \t<li>[latex]\\dfrac{1}{6} + \\dfrac{3}{8}[\/latex] =<\/li>\r\n \t<li>[latex]\\dfrac{3}{4} + \\dfrac{1}{2}[\/latex] =<\/li>\r\n \t<li>[latex]\\dfrac{1}{3} + \\dfrac{5}{8}[\/latex] =<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 14<\/strong>\r\n<ol class=\"threecolumn\" start=\"2\" type=\"a\">\r\n \t<li>[latex]1 \\dfrac{1}{8}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{4}[\/latex]<\/li>\r\n \t<li>[latex]1[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{13}{24}[\/latex]<\/li>\r\n \t<li>1 [latex]\\dfrac{1}{4}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{23}{24}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nYou already know how to add mixed numbers which have the same (like) denominators.\r\n\r\n<strong>To add mixed numbers with different denominators, you must:<\/strong>\r\n<div class=\"textbox\">\r\n<ul>\r\n \t<li>Find the least common denominator (LCD) for the fractions.<\/li>\r\n \t<li>Rename the fractions as equivalent fractions using the LCD<\/li>\r\n \t<li>Be sure to bring the whole number across the equal sign when you rename.<\/li>\r\n \t<li>Add the fractions.<\/li>\r\n \t<li>Add the whole numbers.<\/li>\r\n \t<li>Simplify the answer.<\/li>\r\n \t<li>Remember that if the sum of the fractions is an improper fraction, you must rename it as a mixed number that is added to the whole number in your answer.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example M<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[latex]\\begin{array}{rr}&amp;3\\dfrac{3}{4}\\left(\\dfrac{\\times5}{\\times5}\\right)=3\\dfrac{15}{20}\r\n\\\\ +&amp;6\\dfrac{1}{5}\\left(\\dfrac{\\times 4}{\\times 4}\\right)=6\\dfrac{4}{20} \\\\ \\hline \\\\ &amp; =9\\dfrac{19}{20}\\end{array}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example N<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[latex]\\begin{array}{rr}&amp;3\\dfrac{3}{4}\\left(\\dfrac{\\times3}{\\times3}\\right)=3\\dfrac{9}{12}\r\n\\\\ &amp;8\\dfrac{2}{3}\\left(\\dfrac{\\times 4}{\\times 4}\\right)=8\\dfrac{8}{12} \\\\ +&amp; 2 \\dfrac{1}{2} \\left(\\dfrac{\\times 6}{\\times 6}\\right) = 2 \\dfrac{6}{12} \\\\ \\hline \\\\ &amp; =14\\dfrac{11}{12}\\end{array}[\/latex]\r\n[latex]\\dfrac{23}{12}[\/latex] is an improper fraction so we simplify it: [latex]\\dfrac{23}{12} = 1 \\dfrac{11}{12}[\/latex]\r\n\r\nTherefore, the answer becomes:\r\n\r\n[latex]13 \\dfrac{23}{12} = 13 + 1 \\dfrac{11}{12} = 14 \\dfrac{11}{12}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 15<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAdd. Express the sums in lowest terms.\r\n<ol type=\"a\">\r\n \t<li>[latex]\\begin{array}{rrrrr}&amp;1\\dfrac{3}{8}\\left(\\dfrac{\\times1}{\\times1}\\right)&amp;=&amp;1\\dfrac{3}{8}&amp;\\\\+&amp;1\\dfrac{1}{4}\\left(\\dfrac{\\times2}{\\times2}\\right)&amp;=&amp;1\\dfrac{2}{8}&amp;\\\\\\hline&amp;&amp;&amp;2\\dfrac{5}{8}\r\n\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\r\n \t<li>[latex]\\begin{array}{rr}&amp;3\\dfrac{1}{5}\\\\+&amp;2\\dfrac{3}{10}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;6\\dfrac{2}{15}\\\\+&amp;1\\dfrac{3}{10}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>\u00a0[latex]\\begin{array}{rr}&amp;8\\dfrac{1}{4}\\\\+&amp;4\\dfrac{1}{3}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;5\\dfrac{2}{3}\\\\+&amp;6\\dfrac{1}{4}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;116\\dfrac{5}{8}\\\\+&amp;9\\dfrac{1}{24}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 15<\/strong>\r\n<ol class=\"threecolumn\" start=\"2\" type=\"a\">\r\n \t<li>[latex]5\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]7\\dfrac{13}{30}[\/latex]<\/li>\r\n \t<li>[latex]12\\dfrac{7}{12}[\/latex]<\/li>\r\n \t<li>[latex]11\\dfrac{11}{12}[\/latex]<\/li>\r\n \t<li>[latex]125\\dfrac{2}{3}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 16<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAdd. Express the sums in lowest terms.\r\n<ol type=\"a\">\r\n \t<li>[latex]\\begin{array}{rrrrr}&amp;4\\dfrac{1}{2}\\left(\\dfrac{\\times6}{\\times6}\\right)&amp;=&amp;4\\dfrac{6}{12}&amp;\\\\+&amp;2\\dfrac{1}{3}\\left(\\dfrac{\\times4}{\\times4}\\right)&amp;=&amp;2\\dfrac{4}{12}&amp;\\\\\\hline&amp;&amp;&amp;6\\dfrac{10}{12}&amp;=6\\dfrac{5}{6}\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\r\n \t<li>[latex]\\begin{array}{rr}&amp;3\\dfrac{2}{3}\\\\+&amp;1\\dfrac{1}{2}\\\\\r\n\\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;6\\dfrac{1}{2}\\\\+&amp;4\\dfrac{1}{4}\\\\\r\n\\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;2\\dfrac{1}{8}\\\\+&amp;4\\dfrac{3}{16}\\\\\r\n\\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;2\\dfrac{1}{5}\\\\+&amp;3\\dfrac{2}{3}\\\\\r\n\\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;3\\dfrac{3}{8}\\\\&amp;2\\dfrac{3}{4}\\\\+&amp;1\\dfrac{1}{2}\\\\\\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;4\\dfrac{3}{4}\\\\&amp;2\\dfrac{1}{5}\\\\+&amp;4\\dfrac{1}{2}\\\\\r\n\\hline&amp;\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 16<\/strong>\r\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\r\n \t<li>[latex]5\\dfrac{1}{6}[\/latex]<\/li>\r\n \t<li>[latex]10\\dfrac{3}{4}[\/latex]<\/li>\r\n \t<li>[latex]6\\dfrac{5}{16}[\/latex]<\/li>\r\n \t<li>[latex]5\\dfrac{13}{15}[\/latex]<\/li>\r\n \t<li>[latex]7\\dfrac{5}{8}[\/latex]<\/li>\r\n \t<li>[latex]11\\dfrac{9}{20}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>Problems Using Addition of Common Fractions<\/h1>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 17<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Solve these problems.<\/strong>\r\n<ol type=\"a\">\r\n \t<li>The bathroom shelf is crowded with hand lotion bottles, each with a little lotion left inside. Everyone always likes to try the new bottle before the old one is emptied! One bottle is [latex]\\tfrac{1}{3}[\/latex] full, another is [latex]\\tfrac{1}{4}[\/latex] full, one is only [latex]\\tfrac{1}{8}[\/latex] full and one is still [latex]\\tfrac{1}{2}[\/latex] full. How much lotion is in the bottles altogether?<\/li>\r\n \t<li>Sometimes Joan thinks she will go crazy when she packs the lunches for her family. Little Sarah has decided she only wants [latex]\\tfrac{3}{4}[\/latex] of a sandwich, Megan wants [latex]\\tfrac{1}{4}[\/latex] of a sandwich, Joan's husband takes [latex]1\\tfrac{1}{2}[\/latex] sandwiches, and their son, who does heavy work, takes 3 sandwiches! How many sandwiches does Joan make?<\/li>\r\n \t<li>Dave paid the babysitter for the week. The sitter worked [latex]3\\tfrac{3}{4}[\/latex] hours on Monday, [latex]4\\tfrac{1}{4}[\/latex] hours on Tuesday and [latex]6\\tfrac{1}{2}[\/latex] hours on Friday. How many hours did the babysitter work looking after Dave\u2019s children that week?<\/li>\r\n \t<li>Quite a lot of watermelon was left after the watermelon-eating contest: [latex]1\\tfrac{1}{2}[\/latex] watermelons on one table, [latex]2\\tfrac{3}{4}[\/latex] of a watermelon on another table and [latex]\\tfrac{5}{8}[\/latex] of a watermelon on the third table. The organizers want to know exactly how much was left over so they will not buy so much next year. Calculate the amount of watermelon left over.<\/li>\r\n \t<li>Jeanette has a novel to read for English. She read [latex]\\tfrac{1}{2}[\/latex] of the book on the weekend, only had time to read [latex]\\tfrac{1}{8}[\/latex] of the book on Monday and another [latex]\\tfrac{1}{4}[\/latex] on Wednesday. How much of the book has she read?<\/li>\r\n \t<li>Dion walks around this route each day for exercise. How far does he walk each day? Is this a perimeter or area question?<img class=\"aligncenter wp-image-181\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/4A-ex-17.f-1.png\" alt=\"Rectangle. 1 half kilometres by 1 and 2 thirds kilometres.\" width=\"342\" height=\"211\" \/><\/li>\r\n \t<li>How many metres of baseboard are needed for a rectangular room [latex]4\\tfrac{1}{2}[\/latex] m by [latex]3\\tfrac{1}{5}[\/latex]m? Deduct 1 m for the doorway. (TIP: Draw a picture)<\/li>\r\n \t<li>Sana is going to frame a large piece of art with a wooden frame. The art piece is [latex]1\\tfrac{1}{10}[\/latex] m by [latex]\\tfrac{3}{5}[\/latex]\u00a0 m. How much framing material should she buy?<\/li>\r\n \t<li>Find the perimeter of the following rectangle.<img class=\"size-medium wp-image-373 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_i-300x183.jpg\" alt=\"\" width=\"300\" height=\"183\" \/><\/li>\r\n \t<li>Find the perimeter of a picture frame if one side is [latex]12 \\dfrac{1}{10}[\/latex] cm and the other side measures [latex]14 \\dfrac{1}{5} [\/latex]cm.<\/li>\r\n \t<li>Find the perimeter of this triangle.\r\n<div><img class=\"wp-image-370 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_k.jpg\" alt=\"a triangle\" width=\"233\" height=\"319\" \/><\/div><\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 17<\/strong>\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>[latex]1 \\dfrac{5}{24}[\/latex] bottles total<\/li>\r\n \t<li>[latex]5 \\dfrac{1}{2}[\/latex] sandwiches<\/li>\r\n \t<li>[latex]14 \\dfrac{1}{2}[\/latex] hours<\/li>\r\n \t<li>[latex]4 \\dfrac{7}{8}[\/latex] watermelons<\/li>\r\n \t<li>[latex]\\dfrac{7}{8}[\/latex] of the book<\/li>\r\n \t<li>He walks [latex]4 \\dfrac{1}{3}[\/latex] km each day, perimeter<\/li>\r\n \t<li>[latex]15 \\dfrac{2}{5}[\/latex] m of material<\/li>\r\n \t<li>[latex]3 \\dfrac{2}{5}[\/latex] m of material<\/li>\r\n \t<li>[latex]15 \\dfrac{2}{3}[\/latex] cm<\/li>\r\n \t<li>[latex]52 \\dfrac{3}{5}[\/latex] cm<\/li>\r\n \t<li>[latex]17 \\dfrac{11}{24}[\/latex] cm<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>Topic A: Self-Test<\/h1>\r\n<strong>Mark\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \/14\u00a0 \u00a0Aim 11\/14<\/strong>\r\n<ol type=\"A\">\r\n \t<li>Add and express the answers in lowest terms (6 marks).\r\n<ol class=\"twocolumn space\" type=\"a\">\r\n \t<li>\u00a0[latex]\\begin{array}{rr}&amp;\\dfrac{1}{4}\\\\+&amp;\\dfrac{3}{4}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>\u00a0[latex]\\begin{array}{rr}&amp;1\\dfrac{3}{5}\\\\+&amp;3\\dfrac{4}{5}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;\\dfrac{3}{8}\\\\+&amp;\\dfrac{3}{4}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;2\\dfrac{1}{6}\\\\+&amp;3\\dfrac{5}{12}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;6\\dfrac{3}{4}\\\\+&amp;2\\dfrac{1}{2}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{rr}&amp;6\\dfrac{7}{8}\\\\+&amp;9\\dfrac{1}{3}\\\\ \\hline&amp;\\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Word Problems (8 marks).\r\n<ol type=\"a\">\r\n \t<li>The flight from Vancouver to Sandspit took [latex]1\\dfrac{1}{4}[\/latex] hours. The wait in Sandspit was [latex]1\\dfrac{1}{2}[\/latex] hours and the flight from there to Ketchican, Alaska was [latex]\\dfrac{3}{4}[\/latex] of an hour. How long did it take to make the trip from Vancouver, BC to Ketchican, Alaska?<\/li>\r\n \t<li>Dave built\u00a0[latex]\\dfrac{1}{8}[\/latex] of the fence around his house on Monday, [latex]\\dfrac{1}{4}[\/latex] of it on Tuesday and another [latex]\\dfrac{1}{4}[\/latex] on Wednesday. How much of the fence has he built?<\/li>\r\n \t<li>John bought snacks in bulk for the class party. His items weighed [latex]\\dfrac{2}{5}[\/latex] kg of chips, [latex]\\dfrac{3}{5}[\/latex] kg of peanuts, [latex]\\dfrac{1}{2}[\/latex] kg of cheese and [latex]1\\dfrac{1}{4}[\/latex] kg of fresh veggies. How much did all his snacks weigh?<\/li>\r\n \t<li>Clarence is making a frame for his favourite photo. The frame needs to be [latex]\\dfrac{1}{8}[\/latex] m by [latex]\\dfrac{5}{6}[\/latex] m. How much material should he buy?<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<strong>Answers to Topic A Self-Test<\/strong>\r\n<ol type=\"A\">\r\n \t<li>\r\n<ol class=\"threecolumn\" type=\"a\">\r\n \t<li>1<\/li>\r\n \t<li>[latex]5\\dfrac{2}{5}[\/latex]<\/li>\r\n \t<li>[latex]1\\dfrac{1}{8}[\/latex]<\/li>\r\n \t<li>[latex]5\\dfrac{7}{12}[\/latex]<\/li>\r\n \t<li>[latex]9\\dfrac{1}{4}[\/latex]<\/li>\r\n \t<li>[latex]16\\dfrac{5}{24}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>[latex]3 \\dfrac{1}{2}[\/latex] hr<\/li>\r\n \t<li>[latex]\\dfrac{5}{8}[\/latex] of the fence<\/li>\r\n \t<li>[latex]2 \\dfrac{3}{4}[\/latex] kg of food<\/li>\r\n \t<li>[latex]1 \\dfrac{11}{12}[\/latex] m of material<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>","rendered":"<p>Vocabulary Review:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-160 size-full\" title=\"Illustration of addend 4 plus 5 being equal to 9. With arrows to vocabulary examples of 9 as sum and 4,5 as addends. Also fraction three sevenths, illustrating the vocabular of 3 being the numerator and 7 being the denominator.\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-6-1.png\" alt=\"Illustration of addend 4 plus 5 being equal to 9. With arrows to vocabulary examples of 9 as sum and 4,5 as addends. Also fraction three sevenths, illustrating the vocabular of 3 being the numerator and 7 being the denominator.\" width=\"514\" height=\"438\" data-popupalt-original-title=\"null\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-6-1.png 514w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-6-1-300x256.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-6-1-65x55.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-6-1-225x192.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-6-1-350x298.png 350w\" sizes=\"auto, (max-width: 514px) 100vw, 514px\" \/><\/p>\n<p><strong>Like Fractions:<\/strong> Fractions that have the same denominator<\/p>\n<p>Example: [latex]\\tfrac{1}{4}[\/latex],\u00a0 [latex]\\tfrac{2}{4}[\/latex],\u00a0 \u00a0[latex]\\tfrac{3}{4}[\/latex],\u00a0 \u00a0[latex]\\tfrac{4}{4}[\/latex],\u00a0 etc.<\/p>\n<p><strong>Adding and subtracting fractions has some different rules from multiplying and dividing.<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-161 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-1.png\" alt=\"Two circles. Circles are divided in three with shading of one of the three.\" width=\"124\" height=\"66\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-1.png 124w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-1-65x35.png 65w\" sizes=\"auto, (max-width: 124px) 100vw, 124px\" \/><\/p>\n<p>There are two cakes that are left over. There is 1 piece of each cake left. If you were to put all the pieces left onto one plate, how much cake would you have?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-162 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-4-A-3-1.jpg\" alt=\"Visual representation of equation about two cakes. Three circles. Circles are divided in three with shading of one of the three for the items to add together.\" width=\"265\" height=\"67\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-4-A-3-1.jpg 241w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-4-A-3-1-65x16.jpg 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-4-A-3-1-225x57.jpg 225w\" sizes=\"auto, (max-width: 265px) 100vw, 265px\" \/><\/p>\n<p style=\"text-align: center\">Shade in your answer here<\/p>\n<p>If you made your plate like this: <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-163\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-3.png\" alt=\"\" width=\"86\" height=\"80\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-3.png 86w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-3-65x60.png 65w\" sizes=\"auto, (max-width: 86px) 100vw, 86px\" \/>\u00a0 then you are right!<\/p>\n<p><strong>Try this example:<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-164 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-4.png\" alt=\"Image of equation using three rectangle objects. One rectangle divided in four, shaded in one segment. Addition between. Second rectangle divided in four segments, shaded in first and third segments. Equals sign. And rectangle with three four segments. All unshaded.\" width=\"412\" height=\"110\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-4.png 412w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-4-300x80.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-4-65x17.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-4-225x60.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-4-350x93.png 350w\" sizes=\"auto, (max-width: 412px) 100vw, 412px\" \/><\/p>\n<p><strong>The answer is:<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-165 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-5.png\" alt=\"Image of equation using three rectangle objects. One rectangle divided in four, shaded in one segment. Addition between. Second rectangle divided in four segments, shaded in first and third segments. Equals sign. And rectangle with three shaded segments, one blank.\" width=\"416\" height=\"100\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-5.png 416w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-5-300x72.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-5-65x16.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-5-225x54.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-5-350x84.png 350w\" sizes=\"auto, (max-width: 416px) 100vw, 416px\" \/><\/p>\n<p><strong>What you are doing is adding two <span style=\"text-decoration: underline\">like<\/span> fractions.<\/strong><\/p>\n<ul>\n<li>You are moving pieces of fractions that are the same size into one whole shape. The pieces do not change size, so the denominator must stay the same size.<\/li>\n<li>When adding two fractions, your answer is a fraction.<\/li>\n<\/ul>\n<p><strong>Look back at the two examples.<\/strong><\/p>\n<p>When you add fractions, does the denominator or the numerator stay the same?<\/p>\n<div class=\"textbox\">Common fractions must have the same denominator when you add them together.\u00a0<strong> Add the numerators<\/strong> and keep the denominators the same.<\/div>\n<p>Look at the next two examples:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-358 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-3.jpg\" alt=\"\" width=\"698\" height=\"91\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-3.jpg 698w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-3-300x39.jpg 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-3-65x8.jpg 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-3-225x29.jpg 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-3-350x46.jpg 350w\" sizes=\"auto, (max-width: 698px) 100vw, 698px\" \/><\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{4} + \\dfrac{2}{4} = \\dfrac{3}{4}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-359 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-2.jpg\" alt=\"\" width=\"769\" height=\"58\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-2.jpg 769w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-2-300x23.jpg 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-2-65x5.jpg 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-2-225x17.jpg 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Picture-4-A-2-350x26.jpg 350w\" sizes=\"auto, (max-width: 769px) 100vw, 769px\" \/><\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{5}+ \\dfrac{2}{5}+ \\dfrac{1}{5} = \\dfrac{4}{5}[\/latex]<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Try a few for yourself<\/p>\n<ol class=\"smallspace\" type=\"a\">\n<li style=\"text-align: center\">\n<p style=\"margin-top: 1em\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-168 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-13.png\" alt=\"Visual representation of the equation to solve.\" width=\"388\" height=\"90\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-13.png 388w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-13-300x70.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-13-65x15.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-13-225x52.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-13-350x81.png 350w\" sizes=\"auto, (max-width: 388px) 100vw, 388px\" \/><\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{2}{9} + \\dfrac{3}{9} = \\dfrac{ }{9}[\/latex]<\/p>\n<\/li>\n<li style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-169 alignnone\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-14.png\" alt=\"Visual representation of the equation to solve.\" width=\"408\" height=\"40\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-14.png 408w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-14-300x29.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-14-65x6.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-14-225x22.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-14-350x34.png 350w\" sizes=\"auto, (max-width: 408px) 100vw, 408px\" \/>\n<p style=\"text-align: center\">[latex]\\dfrac{2}{4} + \\dfrac{1}{4} = \\dfrac{ }{4}[\/latex]<\/p>\n<\/li>\n<li style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-170 size-full alignnone\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-14b.png\" alt=\"Visual representation of the equation to solve.\" width=\"276\" height=\"70\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-14b.png 276w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-14b-65x16.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-14b-225x57.png 225w\" sizes=\"auto, (max-width: 276px) 100vw, 276px\" \/>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{3} + \\dfrac{1}{3} = \\dfrac{ }{3}[\/latex]<\/p>\n<\/li>\n<li style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-171 size-full alignnone\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-15.png\" alt=\"Visual representation of the equation to solve.\" width=\"380\" height=\"66\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-15.png 380w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-15-300x52.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-15-65x11.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-15-225x39.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-15-350x61.png 350w\" sizes=\"auto, (max-width: 380px) 100vw, 380px\" \/>\n<p style=\"text-align: center\">[latex]\\dfrac{3}{6} + \\dfrac{2}{6} = \\dfrac{ }{6}[\/latex]<\/p>\n<\/li>\n<li style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-172 size-full alignnone\" style=\"margin-top: 0.5em;margin-bottom: 0.5em;text-align: center\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-16.png\" alt=\"Visual representation of the equation to solve.\" width=\"412\" height=\"96\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-16.png 412w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-16-300x70.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-16-65x15.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-16-225x52.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-16-350x82.png 350w\" sizes=\"auto, (max-width: 412px) 100vw, 412px\" \/><br \/>\n[latex]\\dfrac{3}{8} + \\dfrac{4}{8} = \\dfrac{ }{8}[\/latex]<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 1<\/strong><\/p>\n<ol class=\"twocolumn\" type=\"a\">\n<li>[latex]\\dfrac{5}{9}[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{4}[\/latex]<\/li>\n<li>[latex]\\dfrac{2}{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{5}{6}[\/latex]<\/li>\n<li>[latex]\\dfrac{7}{8}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Now find the answers to the additions without diagrams.<\/strong><\/p>\n<ol class=\"twocolumn\" type=\"a\">\n<li>[latex]\\dfrac{2}{4} + \\dfrac{1}{4} = \\dfrac{ }{4}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{3} + \\dfrac{1}{3} = \\dfrac{ }{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{5} + \\dfrac{1}{5} = \\dfrac{ }{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{2}{11} + \\dfrac{7}{11} = \\dfrac{ }{11}[\/latex]<\/li>\n<\/ol>\n<p><strong>Answers to Exercise Two<\/strong><\/p>\n<ol class=\"twocolumn\" type=\"a\">\n<li>[latex]\\dfrac{3}{4}[\/latex]<\/li>\n<li>[latex]\\dfrac{2}{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{2}{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{9}{11}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Add these common fractions.<\/strong><\/p>\n<ol class=\"twocolumn\" type=\"a\">\n<li>[latex]\\dfrac{1}{5} + \\dfrac{2}{5} =[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{6} + \\dfrac{2}{6} =[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{7} + \\dfrac{2}{7} =[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{10} + \\dfrac{6}{10} =[\/latex]<\/li>\n<li>[latex]\\dfrac{14}{20} + \\dfrac{3}{20} =[\/latex]<\/li>\n<li>[latex]\\dfrac{7}{37} + \\dfrac{19}{37} =[\/latex]<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 3<\/strong><\/p>\n<ol class=\"threecolumn\" type=\"a\">\n<li>[latex]\\dfrac{3}{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{5}{6}[\/latex]<\/li>\n<li>[latex]\\dfrac{5}{7}[\/latex]<\/li>\n<li>[latex]\\dfrac{9}{10}[\/latex]<\/li>\n<li>[latex]\\dfrac{17}{20}[\/latex]<\/li>\n<li>[latex]\\dfrac{26}{37}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>Sometimes the <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_182_269\">sum<\/a> of a fraction will need to be reduced (take a look at this example to remind yourself how to do this).<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p style=\"text-align: center\">[latex]\\dfrac{2}{8} + \\dfrac{2}{8} = \\dfrac{4}{8}\\rightarrow\\dfrac{\u00f7 4}{\u00f7 4} = \\dfrac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example B<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p style=\"text-align: center\">[latex]\\dfrac{3}{4} + \\dfrac{3}{4} = \\dfrac{6}{4}\\rightarrow\\dfrac{6}{4}[\/latex] [latex]\\dfrac{\u00f7 2}{\u00f7 2}[\/latex] = [latex]\\dfrac{3}{2}[\/latex] =\u00a0 [latex]1\\dfrac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the sums to the following additions. Make sure your answer is in the lowest terms.<\/p>\n<ol class=\"twocolumn\" type=\"a\">\n<li>[latex]\\dfrac{1}{4} + \\dfrac{1}{4} =[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{3} + \\dfrac{1}{3} =[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{10} + \\dfrac{2}{10} =[\/latex]<\/li>\n<li>[latex]\\dfrac{7}{25} + \\dfrac{8}{25} =[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{5} + \\dfrac{1}{5} =[\/latex]<\/li>\n<li>[latex]\\dfrac{9}{27} + \\dfrac{12}{27} =[\/latex]<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 4<br \/>\n<\/strong><\/p>\n<ol class=\"threecolumn\" type=\"a\">\n<li>[latex]\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]\\dfrac{2}{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{7}{9}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>So far all your answers have been less than one (a <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_182_282\">proper fraction<\/a>). Sometimes adding fractions can result in more than one whole.<\/p>\n<p>Look at this example:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-173 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-4-A-14.jpg\" alt=\"\" width=\"499\" height=\"72\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-4-A-14.jpg 499w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-4-A-14-300x43.jpg 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-4-A-14-65x9.jpg 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-4-A-14-225x32.jpg 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture-4-A-14-350x51.jpg 350w\" sizes=\"auto, (max-width: 499px) 100vw, 499px\" \/><\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{2}{4} + \\dfrac{3}{4}=\\dfrac{4}{4}\\text{and}\\dfrac{1}{4}[\/latex]\u00a0 \u00a0(or [latex]\\left(\\dfrac{5}{4}\\right)[\/latex])<\/p>\n<p>There are not enough parts in the first square to hold all your shaded parts, so you need to draw a second square to hold the extra shaded parts.<\/p>\n<p>You would also have to convert this answer from an improper fraction to a mixed number:<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{5}{4} = 1\\dfrac{1}{4}[\/latex]<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Try these additions.<\/strong> Remember to always reduce!<\/p>\n<ol class=\"smallspace\" type=\"a\">\n<li style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-174 size-full\" title=\"Visual represenation of the equation to solve.\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-21.png\" alt=\"Visual represenation of the equation to solve.\" width=\"480\" height=\"70\" data-popupalt-original-title=\"null\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-21.png 480w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-21-300x44.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-21-65x9.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-21-225x33.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-21-350x51.png 350w\" sizes=\"auto, (max-width: 480px) 100vw, 480px\" \/><br \/>\n[latex]\\dfrac{4}{6}+\\dfrac{5}{6} =[\/latex]<\/li>\n<li style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-175 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-22.png\" alt=\"Visual representation of the equation to solve.\" width=\"452\" height=\"94\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-22.png 452w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-22-300x62.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-22-65x14.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-22-225x47.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-22-350x73.png 350w\" sizes=\"auto, (max-width: 452px) 100vw, 452px\" \/><br \/>\n[latex]\\dfrac{6}{8}+\\dfrac{3}{8} =[\/latex]<\/li>\n<li style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-176 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-23.png\" alt=\"Visual representation of the equation to solve.\" width=\"508\" height=\"76\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-23.png 508w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-23-300x45.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-23-65x10.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-23-225x34.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-23-350x52.png 350w\" sizes=\"auto, (max-width: 508px) 100vw, 508px\" \/><br \/>\n[latex]\\dfrac{3}{4}+\\dfrac{3}{4}=[\/latex]<\/li>\n<li style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-177 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-24.png\" alt=\"Visual representation of the equation to solve.\" width=\"444\" height=\"56\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-24.png 444w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-24-300x38.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-24-65x8.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-24-225x28.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-24-350x44.png 350w\" sizes=\"auto, (max-width: 444px) 100vw, 444px\" \/><br \/>\n[latex]\\dfrac{8}{9}+\\dfrac{4}{9} =[\/latex]<\/li>\n<li style=\"text-align: center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-178 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-25.png\" alt=\"Visual representation of the equation to solve.\" width=\"482\" height=\"94\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-25.png 482w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-25-300x59.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-25-65x13.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-25-225x44.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-25-350x68.png 350w\" sizes=\"auto, (max-width: 482px) 100vw, 482px\" \/><br \/>\n[latex]\\dfrac{3}{5}+\\dfrac{4}{5} =[\/latex]<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 5<\/strong><\/p>\n<ol class=\"threecolumn\" type=\"a\">\n<li>[latex]1\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]1\\dfrac{1}{8}[\/latex]<\/li>\n<li>[latex]1\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]1\\dfrac{1}{3}[\/latex]<\/li>\n<li>[latex]1\\dfrac{2}{5}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example C<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Sometimes you will have to add 3 or more fractions together.<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-179 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-26.png\" alt=\"Visual representation of the equation to solve.\" width=\"500\" height=\"80\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-26.png 500w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-26-300x48.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-26-65x10.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-26-225x36.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-26-350x56.png 350w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{2}{3} + \\dfrac{1}{3} + \\dfrac{2}{3} = \\dfrac{5}{3} = 1\\dfrac{2}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example D<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-180 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-27.png\" alt=\"Visual representation of the equation to solve.\" width=\"532\" height=\"88\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-27.png 532w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-27-300x50.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-27-65x11.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-27-225x37.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Picture4-27-350x58.png 350w\" sizes=\"auto, (max-width: 532px) 100vw, 532px\" \/><\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{4} + \\dfrac{2}{4} + \\dfrac{1}{4} + \\dfrac{3}{4} = \\dfrac{7}{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Add these common fractions. Be sure your answers are in <span style=\"text-decoration: underline\">lowest terms<\/span>.<\/strong><\/p>\n<ol class=\"threecolumn\" type=\"a\">\n<li>[latex]\\dfrac{2}{3} + \\dfrac{1}{3} = \\dfrac{3}{3} = 1[\/latex]<\/li>\n<li>[latex]\\dfrac{7}{10} + \\dfrac{3}{10} =[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{5} + \\dfrac{2}{5} =[\/latex]<\/li>\n<li>\u00a0[latex]\\begin{array}{rr}&\\dfrac{3}{4}\\\\+&\\dfrac{1}{4}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&\\dfrac{5}{6}\\\\+&\\dfrac{5}{6}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>\u00a0[latex]\\begin{array}{rr}&\\dfrac{4}{8}\\\\+&\\dfrac{3}{8}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>\u00a0[latex]\\begin{array}{rr}&\\dfrac{1}{8}\\\\+&\\dfrac{3}{8}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&\\dfrac{2}{5}\\\\&\\dfrac{3}{5}\\\\+&\\dfrac{3}{5}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&\\dfrac{3}{6}\\\\&\\dfrac{1}{6}\\\\ +&\\dfrac{1}{6}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 6<\/strong><\/p>\n<ol class=\"threecolumn\" start=\"2\" type=\"a\">\n<li>[latex]1[\/latex]<\/li>\n<li>[latex]1[\/latex]<\/li>\n<li>[latex]1[\/latex]<\/li>\n<li>[latex]1\\dfrac{2}{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{7}{8}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]1\\dfrac{3}{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{5}{6}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>Adding Mixed Numbers<\/h1>\n<p><strong>To add mixed numbers<\/strong><\/p>\n<div class=\"textbox\">\n<ul>\n<li>Be sure the denominators are the same.<\/li>\n<li>Add the common fractions.<\/li>\n<li>Add the whole numbers.Simplify the common fraction.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example E<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p style=\"text-align: left\">[latex]\\begin{array}{rr}&3\\dfrac{1}{8}\\\\+&2\\dfrac{3}{8}\\\\ \\hline\\end{array}[\/latex]<\/p>\n<ul>\n<li style=\"text-align: left\">[latex]5\\dfrac{4}{8}[\/latex] = [latex]5\\dfrac{1}{2}[\/latex]<\/li>\n<li style=\"text-align: left\">[latex]\\dfrac{4}{8}[\/latex] =\u00a0[latex]\\dfrac{4}{8}\\left(\\dfrac{\u00f74}{\u00f74}\\right )[\/latex] = [latex]\\dfrac{1}{2}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example F<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>[latex]\\begin{array}{rr}&12\\dfrac{1}{3}\\\\+&6\\dfrac{1}{3}\\\\\\hline&18\\dfrac{2}{3}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Add the following numbers. Reduce the answers to lowest terms.<\/p>\n<ol class=\"twocolumn\" style=\"line-height: 7em;margin-bottom: 15px\" type=\"a\">\n<li>[latex]\\begin{array}{rr}&6\\dfrac{1}{12}\\\\+&8 \\dfrac{5}{12}\\\\ \\hline \\end{array}[\/latex]<br \/>\n[latex]14 \\dfrac{6}{12} = 14 \\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&22\\dfrac{1}{6}\\\\+&14\\dfrac{1}{6}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&8\\dfrac{1}{4}\\\\+&3\\dfrac{1}{4}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&18\\dfrac{1}{2}\\\\+&10\\\\ \\hline\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&4\\dfrac{1}{10}\\\\+&\\dfrac{3}{10}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox__content\">\n<p><strong>Answers to Exercise 7<\/strong><\/p>\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\n<li>[latex]36 \\dfrac{1}{3}[\/latex]<\/li>\n<li>[latex]11 \\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]28 \\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]4 \\dfrac{2}{5}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>A<\/strong><strong>d<\/strong><strong>d these numbers. Give your answers in lowest terms<\/strong>.<\/p>\n<ol class=\"twocolumn\" style=\"line-height: 7em;margin-bottom: 15px\" type=\"a\">\n<li>[latex]\\begin{array}{ll}&6\\dfrac{4}{5}\\\\+&3\\dfrac{2}{5}\\\\ \\hline\\end{array}[\/latex]<br \/>\n[latex]9\\dfrac{6}{5}=10\\dfrac{1}{5}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}&9\\dfrac{1}{3}\\\\+&2\\dfrac{2}{3}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}&3\\dfrac{3}{8}\\\\+&12\\dfrac{7}{8}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}&100\\dfrac{7}{10}\\\\+&50\\dfrac{5}{10}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}&3\\dfrac{4}{7}\\\\+&6\\dfrac{5}{7}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}&8\\dfrac{4}{5}\\\\+&\\dfrac{4}{5}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox__content\">\n<p><strong>Answers to Exercise 8<\/strong><\/p>\n<ol class=\"threecolumn\" start=\"2\" type=\"a\">\n<li>[latex]12[\/latex]<\/li>\n<li>[latex]16 \\dfrac{1}{4}[\/latex]<\/li>\n<li>[latex]151 \\dfrac{1}{5}[\/latex]<\/li>\n<li>[latex]10 \\dfrac{2}{7}[\/latex]<\/li>\n<li>[latex]9 \\dfrac{3}{5}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><strong>If you are not comfortable with this work so far, talk to your instructor and get some more practice before you go ahead.<\/strong><\/p>\n<p>The next question is:<\/p>\n<p>What happens when two fractions in an addition (the <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_182_278\">addends<\/a>) do not have the same <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_182_280\">denominator<\/a>? If the addends do not have a common denominator, you will need to find equivalent fractions to make the addends have a common denominator.<br \/>\nRead on to find out how!<\/p>\n<h1>Multiples and Least Common Multiples (LCM)<\/h1>\n<p>When you learned the multiplication tables you learned the <strong>multiples<\/strong> of each number. Multiples are the answers when you multiply a whole number by 1, 2, 3, 4, 5, 6, 7, and so on.<\/p>\n<table style=\"border-collapse: collapse;width: 100%;height: 234px\">\n<tbody>\n<tr style=\"height: 18px\">\n<th style=\"width: 50%;text-align: center;height: 18px\" scope=\"col\">The <strong>multiples of 2<\/strong><\/th>\n<th style=\"width: 50%;text-align: center;height: 18px\" scope=\"col\">The<strong> multiples of 6<\/strong><\/th>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times1 = \\bf{2}[\/latex]<\/td>\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times1 = \\bf{6}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times2 = \\bf{4}[\/latex]<\/td>\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times2 = \\bf{12}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times3 = \\bf{6}[\/latex]<\/td>\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times3 = \\bf{18}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times4 = \\bf{8}[\/latex]<\/td>\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times4 = \\bf{24}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times5 = \\bf{10}[\/latex]<\/td>\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times5 = \\bf{30}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times6 = \\bf{12}[\/latex]<\/td>\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times6 = \\bf{36}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times7 = \\bf{14}[\/latex]<\/td>\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times7 = \\bf{42}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times8 = \\bf{16}[\/latex]<\/td>\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times8 = \\bf{48}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times9 = \\bf{18}[\/latex]<\/td>\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times9 = \\bf{54}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times10 = \\bf{20}[\/latex]<\/td>\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times10 = \\bf{60}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times11 = \\bf{22}[\/latex]<\/td>\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times11 = \\bf{66}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 50%;text-align: center;height: 18px\">[latex]2\\times12 = \\bf{24}[\/latex]<\/td>\n<td style=\"width: 50%;height: 18px;text-align: center\">[latex]6\\times12 = \\bf{72}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>and you can keep going as high as you want.<\/p>\n<p>The multiples of 2 are <strong>2, 4, 6, 8, 10, 12, 14,<\/strong> and so on. &amp; The multiples of 6 are <strong>6, 12, 18, 24, 30, 36,<\/strong> and so on.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>List the first ten multiples of each number. This chart may be useful to you later.<\/strong><\/p>\n<ol type=\"a\">\n<li>2\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<strong>Multiples<\/strong> <em>2, 4, 6, 8, 10, 12, 14, 16, 18, 20<\/em><\/li>\n<\/ol>\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\n<li>3<\/li>\n<li>4<\/li>\n<li>5<\/li>\n<li>9<\/li>\n<li>10<\/li>\n<li>11<\/li>\n<li>12<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<p><strong>Answers to Exercise 9<\/strong><\/p>\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\n<li>3, 6, 9, 12, 15, 18, 21, 24, 27, 30<\/li>\n<li>4, 8, 12, 16, 20, 24, 28, 32, 36, 40<\/li>\n<li>5, 10, 15, 20, 25, 30, 35, 40, 45, 50<\/li>\n<li>9, 18, 27, 36, 45, 54, 63, 72, 81, 90<\/li>\n<li>10,20,30,40,50,60,70,80,90,100<\/li>\n<li>11, 22, 33, 44, 55, 66, 77, 88, 99, 110<\/li>\n<li>12,24,36,48,60,72,84,96,108,120<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div>This is a quick method to find the least common multiple (LCM).<\/div>\n<div style=\"padding-left: 40px\">least means smallest<\/div>\n<div style=\"padding-left: 40px\">common means shared<\/div>\n<div style=\"padding-left: 40px\">multiple means the answers when you multiply by 1, 2, 3, etc.<\/div>\n<div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example G<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the least common multiple (LCM) of 3 and 5?<\/p>\n<ul>\n<li>Multiples:\n<ul>\n<li>Multiples of 3: 3, 6, 9, 12, <span style=\"text-decoration: underline\">15<\/span>, 18, 21, 24, 27, 30<\/li>\n<li>Multiples of 5: 5, 10, <span style=\"text-decoration: underline\">15<\/span>, 20, 25, 30\u2026<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>The least common multiple of 3 and 5 is 15.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example H<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the LCM of 3 and 4?<\/p>\n<ul>\n<li>Multiples:\n<ul>\n<li>Multiples of 3: 3, 6, 9, <span style=\"text-decoration: underline\">12<\/span>, 15, 18, 21, 24, 27, 30\u2026<\/li>\n<li>Multiples of 4: 4, 8, <span style=\"text-decoration: underline\">12<\/span>, 16, 20, 24, 28, 32 \u2026.<\/li>\n<\/ul>\n<\/li>\n<li>The least common multiple of 3 and 4 is 12.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example I<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the LCM of 4 and 8?<\/p>\n<ul>\n<li>Multiples:\n<ul>\n<li>Multiples of 4: 4, <span style=\"text-decoration: underline\">8<\/span>, 12, 16, 20\u2026<\/li>\n<li>Multiples of 8:\u00a0 <span style=\"text-decoration: underline\">8<\/span>, 16, 24, 32, 40\u2026<\/li>\n<\/ul>\n<\/li>\n<li>The least common multiple of 4 and 8 is 8.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div>Hint: Always check to see if the larger number is a multiple of the smaller number. If it is, then the larger number is the Least Common Multiple (LCM).<\/div>\n<ul>\n<li>LCM of 3 and 6 is 6<\/li>\n<li>LCM of 2 and 4 is 4<\/li>\n<li>LCM of 5 and 15 is 15<\/li>\n<\/ul>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the Least Common Multiple of these pairs of numbers. Use your chart from Exercise Nine to help you. You may want to add the multiples of other numbers to that chart.<\/p>\n<ol class=\"threecolumn\" type=\"a\">\n<li>3,6<\/li>\n<li>2,5<\/li>\n<li>12, 3<\/li>\n<li>6, 12<\/li>\n<li>5, 4<\/li>\n<li>4, 8<\/li>\n<li>8, 16<\/li>\n<li>4, 7<\/li>\n<li>25, 5<\/li>\n<li>2, 9<\/li>\n<li>6, 10<\/li>\n<li>8, 12<\/li>\n<\/ol>\n<p><strong><br \/>\nAnswers Exercise 10<\/strong><\/p>\n<ol class=\"threecolumn\" type=\"a\">\n<li>6<\/li>\n<li>10<\/li>\n<li>12<\/li>\n<li>12<\/li>\n<li>20<\/li>\n<li>8<\/li>\n<li>16<\/li>\n<li>28<\/li>\n<li>25<\/li>\n<li>18<\/li>\n<li>30<\/li>\n<li>24<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>Now that you know how to find an LCM, you can apply this knowledge to adding and subtracting fractions.<\/p>\n<h1>Least Common Denominator (LCD)<\/h1>\n<p>To find the Least Common Denominator of common fractions: <strong>find the least common multiple of the denominators<\/strong>.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example J<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the least common denominator of [latex]\\dfrac{1}{2}[\/latex] and [latex]\\dfrac{3}{4}[\/latex]?<\/p>\n<p>The denominators are 2 and 4.<\/p>\n<p>The <strong>least common multiple<\/strong> of 2 and 4 is <strong>4<\/strong>.<\/p>\n<p>So the <strong>least common denominator (LCD)<\/strong> for [latex]\\dfrac{1}{2}[\/latex] and [latex]\\dfrac{3}{4}[\/latex] is <strong>4<\/strong>.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example K<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the <strong>LCD<\/strong> of [latex]\\dfrac{3}{4}[\/latex] and [latex]\\dfrac{2}{3}[\/latex]?<\/p>\n<p>The denominators are 4 and 3.<\/p>\n<p>The <strong>least common multiple<\/strong> of 4 and 3 is <strong>12<\/strong>.<\/p>\n<p>So the<strong> least common denominator<\/strong> for [latex]\\dfrac{3}{4}[\/latex] and [latex]\\dfrac{2}{3}[\/latex] is <strong>12<\/strong>.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the Least Common Denominator (LCD) for these pairs of fractions.<\/p>\n<table style=\"border-collapse: collapse;width: 100%\">\n<tbody>\n<tr>\n<th style=\"width: 7.91317%\" scope=\"col\"><\/th>\n<th style=\"width: 42.0868%\" scope=\"col\">Fractions<\/th>\n<th style=\"width: 25%\" scope=\"col\">Denominators<\/th>\n<th style=\"width: 25%\" scope=\"col\">Least Common Denominators<\/th>\n<\/tr>\n<tr>\n<th style=\"width: 7.91317%\" scope=\"row\">a)<\/th>\n<td style=\"width: 42.0868%\">[latex]\\dfrac{5}{8}[\/latex], [latex]\\dfrac{2}{3}[\/latex]<\/td>\n<td style=\"width: 25%\">8, 3<\/td>\n<td style=\"width: 25%\">24<\/td>\n<\/tr>\n<tr>\n<th style=\"width: 7.91317%\" scope=\"row\">b)<\/th>\n<td style=\"width: 42.0868%\">[latex]\\dfrac{1}{5}[\/latex], [latex]\\dfrac{1}{10}[\/latex]<\/td>\n<td style=\"width: 25%\"><\/td>\n<td style=\"width: 25%\"><\/td>\n<\/tr>\n<tr>\n<th style=\"width: 7.91317%\" scope=\"row\">c)<\/th>\n<td style=\"width: 42.0868%\">[latex]\\dfrac{1}{3}[\/latex], [latex]\\dfrac{3}{4}[\/latex]<\/td>\n<td style=\"width: 25%\"><\/td>\n<td style=\"width: 25%\"><\/td>\n<\/tr>\n<tr>\n<th style=\"width: 7.91317%\" scope=\"row\">d)<\/th>\n<td style=\"width: 42.0868%\">[latex]\\dfrac{2}{3}[\/latex], [latex]\\dfrac{1}{5}[\/latex]<\/td>\n<td style=\"width: 25%\"><\/td>\n<td style=\"width: 25%\"><\/td>\n<\/tr>\n<tr>\n<th style=\"width: 7.91317%\" scope=\"row\">e)<\/th>\n<td style=\"width: 42.0868%\">[latex]\\dfrac{5}{8}[\/latex], [latex]\\dfrac{1}{16}[\/latex]<\/td>\n<td style=\"width: 25%\"><\/td>\n<td style=\"width: 25%\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Answers to Exercise 11 (only least common denominator is given)<\/strong><\/p>\n<ol class=\"threecolumn\" start=\"2\" type=\"a\">\n<li>10<\/li>\n<li>12<\/li>\n<li>15<\/li>\n<li>16<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>You know how to find the least common denominator <strong>(LCD).<\/strong> The next step is to make <strong>equivalent fractions<\/strong> using the <strong>LCD<\/strong>.<\/p>\n<p><strong>Step 1:<\/strong> Find the least common denominator.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{rr}&\\dfrac{3}{4}\\\\+&\\dfrac{1}{3}\\\\ \\hline\\end{array}[\/latex]<\/p>\n<p>LCD of 4 and 3 is 12.<\/p>\n<p><strong> Step 2:<\/strong>\u00a0Write an = sign after each fraction, followed by the common denominator.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{rrrr}&\\dfrac{3}{4}=\\dfrac{ }{12}\\\\+&\\dfrac{1}{3} = \\dfrac{ }{12}\\\\ \\hline\\end{array}[\/latex]<\/p>\n<p><strong>Step 3:<\/strong> Rename the fractions as equivalent fractions with the <strong>LCD<\/strong>.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{3}{4}[\/latex] = [latex]\\dfrac{ }{12}[\/latex]<\/p>\n<p style=\"text-align: center\">4 times what = 12?<\/p>\n<p style=\"text-align: center\">4 \u00d7 3 = 12<\/p>\n<p>If the denominator was multiplied by 3, the numerator must be multiplied by 3.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{3}{4}[\/latex] [latex]\\dfrac{\u00d73}{\u00d73}[\/latex] = [latex]\\dfrac{9}{12}[\/latex]<\/p>\n<p>Now rename the other fraction.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{3}[\/latex] = [latex]\\dfrac{ }{12}[\/latex]<\/p>\n<p>3 times what = 12?<\/p>\n<p style=\"text-align: center\">[latex]3\\times 4 = 12[\/latex]<\/p>\n<p>If this denominator was multiplied by 4, this numerator must be multiplied by 4.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{3}[\/latex]\u00a0 [latex]\\dfrac{\u00d74}{\u00d74}[\/latex] = [latex]\\dfrac{4}{12}[\/latex]<\/p>\n<p>Now rename the other fraction.<\/p>\n<p><strong>Step 4:<\/strong> The question now looks like this and can be added.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{rrrr}&\\dfrac{3}{4}&=&\\dfrac{9}{12}\\\\ +&\\dfrac{1}{3} &= &\\dfrac{4}{12}\\\\ \\hline \\\\ & \\dfrac{13}{12}&=& 1\\dfrac{1}{12}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example L<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>[latex]\\dfrac{1}{4} + \\dfrac{3}{8}[\/latex] =<\/p>\n<p><strong>Step 1 and 2:<\/strong> Find the least common denominator<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ll}&\\dfrac{1}{4} = \\dfrac{ }{8}\\\\+&\\dfrac{3}{8}= \\dfrac{ }{8}\\\\ \\hline\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\"><strong>Step 3:<\/strong> Rename as equivalent fractions<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ll}&\\dfrac{1}{4}\\left(\\dfrac{\\times2}{\\times2}\\right) = \\dfrac{2}{8}\\\\+&\\dfrac{3}{8}\\left(\\dfrac{\\times1}{\\times1}\\right)= \\dfrac{3}{8}\\\\ \\hline\\end{array}[\/latex]<\/p>\n<p><strong>Step 4:<\/strong> Add and simplify the answer.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{lll}&\\dfrac{1}{4}\\left(\\dfrac{\u00d72}{\u00d72}\\right) = &\\dfrac{2}{8}\\\\+&\\dfrac{3}{8}\\left(\\dfrac{\u00d71}{\u00d71}\\right)= &\\dfrac{3}{8}\\\\ \\hline&&\\dfrac{5}{8}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 12<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>A<\/strong><strong>d<\/strong><strong>d these common fractions.<\/strong> Express the answer in lowest <span style=\"text-decoration: underline\">terms<\/span>.<\/p>\n<ol class=\"twocolumn\" type=\"a\">\n<li>[latex]\\begin{array}{rr}&\\dfrac{1}{2}\\left(\\dfrac{\u00d74}{\u00d74}\\right) =&\\dfrac{4}{8}\\\\+&\\dfrac{3}{8}\\left(\\dfrac{\u00d71}{\u00d71}\\right)= &\\dfrac{3}{8}\\\\ \\hline&&\\dfrac{7}{8}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&\\dfrac{1}{4}\\left(\\dfrac{\u00d72}{\u00d72}\\right) =&\\dfrac{2}{8}\\\\+&\\dfrac{3}{8}\\left(\\dfrac{\u00d71}{\u00d71}\\right)= &\\dfrac{3}{8}\\\\ \\hline&&\\dfrac{5}{8}\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&\\dfrac{1}{5}\\\\+&\\dfrac{1}{10}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&\\dfrac{5}{16}\\\\+&\\dfrac{1}{4}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&\\dfrac{1}{3}\\\\+&\\dfrac{7}{12}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&\\dfrac{2}{3}\\\\+&\\dfrac{1}{6}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&\\dfrac{3}{10}\\\\ +&\\dfrac{2}{5}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&\\dfrac{1}{12}\\\\ +&\\dfrac{1}{4}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 12<\/strong><\/p>\n<ol class=\"twocolumn\" start=\"3\" type=\"a\">\n<li>[latex]\\dfrac{3}{10}[\/latex]<\/li>\n<li>[latex]\\dfrac{9}{16}[\/latex]<\/li>\n<li>[latex]\\dfrac{11}{12}[\/latex]<\/li>\n<li>[latex]\\dfrac{5}{6}[\/latex]<\/li>\n<li>[latex]\\dfrac{7}{10}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{3}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p><strong>\u00a0How did you do? If you are struggling with this process, speak to your instructor for help.<\/strong><\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 13<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>More practice.<\/strong> Do only as many as you think you need.<\/p>\n<ol type=\"a\">\n<li style=\"margin-bottom: 2em\">[latex]\\begin{array}{rrr}&\\dfrac{2}{3}\\left(\\dfrac{\u00d74}{\u00d74}\\right) &=\\dfrac{8}{12}\\\\&\\dfrac{1}{2}\\left(\\dfrac{\u00d76}{\u00d76}\\right)&=\\dfrac{6}{12}\\\\+&\\dfrac{3}{4}\\left(\\dfrac{\u00d73}{\u00d73}\\right)& = \\dfrac{9}{12}\\\\ \\hline&&\\dfrac{23}{12}&=1\\dfrac{11}{12}\\end{array}[\/latex]<\/li>\n<li style=\"margin-bottom: 2em\">[latex]\\begin{array}{rrr}&\\dfrac{5}{24}\\left(\\dfrac{\u00d71}{\u00d71}\\right)& =\\dfrac{5}{24}\\\\&\\dfrac{1}{3}\\left(\\dfrac{\u00d78}{\u00d78}\\right)&= \\dfrac{8}{24}\\\\+&\\dfrac{3}{8}\\left(\\dfrac{\u00d73}{\u00d73}\\right)&= \\dfrac{9}{24}\\\\ \\hline&&\\dfrac{22}{24}&=\\dfrac{11}{12}\\end{array}[\/latex]<\/li>\n<\/ol>\n<ol type=\"a\">\n<li style=\"list-style-type: none\">\n<ol class=\"threecolumn\" start=\"3\" type=\"a\">\n<li style=\"margin-bottom: 3em\">[latex]\\begin{array}{rr}&\\dfrac{5}{12}\\\\&\\dfrac{5}{6}\\\\+&\\dfrac{3}{4}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<li style=\"margin-bottom: 3em\">[latex]\\begin{array}{rr}&\\dfrac{3}{10}\\\\&\\dfrac{3}{4}\\\\+&\\dfrac{4}{5}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<li style=\"margin-bottom: 3em\">[latex]\\begin{array}{rr}&\\dfrac{1}{2}\\\\&\\dfrac{2}{5}\\\\+&\\dfrac{7}{10}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<li style=\"margin-bottom: 3em\">[latex]\\begin{array}{rr}&\\dfrac{5}{6}\\\\&\\dfrac{3}{4}\\\\+&\\dfrac{1}{3}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<li style=\"margin-bottom: 3em\">[latex]\\begin{array}{rr}&\\dfrac{7}{16}\\\\+&\\dfrac{3}{4}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<li style=\"margin-bottom: 3em\">[latex]\\begin{array}{rr}&\\dfrac{4}{5}\\\\+&\\dfrac{1}{3}\\\\ \\hline\\end{array}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 13<\/strong><\/p>\n<ol class=\"twocolumn\" start=\"3\" type=\"a\">\n<li>[latex]2[\/latex]<\/li>\n<li>[latex]1\\dfrac{17}{20}[\/latex]<\/li>\n<li>[latex]1\\dfrac{3}{5}[\/latex]<\/li>\n<li>[latex]1\\dfrac{11}{12}[\/latex]<\/li>\n<li>[latex]1\\dfrac{3}{16}[\/latex]<\/li>\n<li>[latex]1\\dfrac{2}{15}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>Addition questions are often written with the fractions side by side instead of one fraction above the other. For example:<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{2}{3}[\/latex] + [latex]\\dfrac{5}{8}[\/latex] =<\/p>\n<p>You may solve as shown in this example or rewrite the question with the fractions one above the other.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{2}{3} + \\dfrac{5}{8} = \\dfrac{2}{3}\\dfrac{\u00d78}{\u00d78} + \\dfrac{5}{8}\\dfrac{\u00d73}{\u00d73}= \\dfrac{16}{24}+ \\dfrac{15}{24}=\\dfrac{31}{24}= 1 \\dfrac{7}{24}[\/latex]<\/p>\n<p style=\"text-align: center\">or<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{rrr}&\\dfrac{2}{3}\\left(\\dfrac{\u00d78}{\u00d78}\\right) &=\\dfrac{16}{24}\\\\&\\dfrac{5}{8}\\left(\\dfrac{\u00d73}{\u00d73}\\right)&=\\dfrac{15}{24}\\\\ \\hline&&\\dfrac{31}{24}&=1\\dfrac{7}{24}\\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 14<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Find the sum<\/strong>. Do enough questions to be confident in your skill.<\/p>\n<ol type=\"a\">\n<li>[latex]\\begin{array}{rr} \\\\ \\dfrac{1}{2} + \\dfrac{1}{6} = &\\\\ \\dfrac{1}{2}\\left( \\dfrac{\\times 3}{\\times 3}\\right) + \\dfrac{1}{6} = &\\\\ \\dfrac{3}{6} + \\dfrac{1}{6} = &\\dfrac{4}{6} = \\dfrac{2}{3} \\end{array}[\/latex]<\/li>\n<\/ol>\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\n<li>[latex]\\dfrac{1}{4} + \\dfrac{7}{8}[\/latex] =<\/li>\n<li>[latex]\\dfrac{1}{5} + \\dfrac{3}{5}[\/latex] =<\/li>\n<li>[latex]\\dfrac{1}{12}+\\dfrac{2}{3}[\/latex] =<\/li>\n<li>[latex]\\dfrac{1}{3} + \\dfrac{2}{3}[\/latex] =<\/li>\n<li>[latex]\\dfrac{1}{6} + \\dfrac{3}{8}[\/latex] =<\/li>\n<li>[latex]\\dfrac{3}{4} + \\dfrac{1}{2}[\/latex] =<\/li>\n<li>[latex]\\dfrac{1}{3} + \\dfrac{5}{8}[\/latex] =<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 14<\/strong><\/p>\n<ol class=\"threecolumn\" start=\"2\" type=\"a\">\n<li>[latex]1 \\dfrac{1}{8}[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{4}[\/latex]<\/li>\n<li>[latex]1[\/latex]<\/li>\n<li>[latex]\\dfrac{13}{24}[\/latex]<\/li>\n<li>1 [latex]\\dfrac{1}{4}[\/latex]<\/li>\n<li>[latex]\\dfrac{23}{24}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>You already know how to add mixed numbers which have the same (like) denominators.<\/p>\n<p><strong>To add mixed numbers with different denominators, you must:<\/strong><\/p>\n<div class=\"textbox\">\n<ul>\n<li>Find the least common denominator (LCD) for the fractions.<\/li>\n<li>Rename the fractions as equivalent fractions using the LCD<\/li>\n<li>Be sure to bring the whole number across the equal sign when you rename.<\/li>\n<li>Add the fractions.<\/li>\n<li>Add the whole numbers.<\/li>\n<li>Simplify the answer.<\/li>\n<li>Remember that if the sum of the fractions is an improper fraction, you must rename it as a mixed number that is added to the whole number in your answer.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example M<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>[latex]\\begin{array}{rr}&3\\dfrac{3}{4}\\left(\\dfrac{\\times5}{\\times5}\\right)=3\\dfrac{15}{20}  \\\\ +&6\\dfrac{1}{5}\\left(\\dfrac{\\times 4}{\\times 4}\\right)=6\\dfrac{4}{20} \\\\ \\hline \\\\ & =9\\dfrac{19}{20}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example N<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>[latex]\\begin{array}{rr}&3\\dfrac{3}{4}\\left(\\dfrac{\\times3}{\\times3}\\right)=3\\dfrac{9}{12}  \\\\ &8\\dfrac{2}{3}\\left(\\dfrac{\\times 4}{\\times 4}\\right)=8\\dfrac{8}{12} \\\\ +& 2 \\dfrac{1}{2} \\left(\\dfrac{\\times 6}{\\times 6}\\right) = 2 \\dfrac{6}{12} \\\\ \\hline \\\\ & =14\\dfrac{11}{12}\\end{array}[\/latex]<br \/>\n[latex]\\dfrac{23}{12}[\/latex] is an improper fraction so we simplify it: [latex]\\dfrac{23}{12} = 1 \\dfrac{11}{12}[\/latex]<\/p>\n<p>Therefore, the answer becomes:<\/p>\n<p>[latex]13 \\dfrac{23}{12} = 13 + 1 \\dfrac{11}{12} = 14 \\dfrac{11}{12}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 15<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Add. Express the sums in lowest terms.<\/p>\n<ol type=\"a\">\n<li>[latex]\\begin{array}{rrrrr}&1\\dfrac{3}{8}\\left(\\dfrac{\\times1}{\\times1}\\right)&=&1\\dfrac{3}{8}&\\\\+&1\\dfrac{1}{4}\\left(\\dfrac{\\times2}{\\times2}\\right)&=&1\\dfrac{2}{8}&\\\\\\hline&&&2\\dfrac{5}{8}  \\end{array}[\/latex]<\/li>\n<\/ol>\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\n<li>[latex]\\begin{array}{rr}&3\\dfrac{1}{5}\\\\+&2\\dfrac{3}{10}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&6\\dfrac{2}{15}\\\\+&1\\dfrac{3}{10}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>\u00a0[latex]\\begin{array}{rr}&8\\dfrac{1}{4}\\\\+&4\\dfrac{1}{3}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&5\\dfrac{2}{3}\\\\+&6\\dfrac{1}{4}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&116\\dfrac{5}{8}\\\\+&9\\dfrac{1}{24}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 15<\/strong><\/p>\n<ol class=\"threecolumn\" start=\"2\" type=\"a\">\n<li>[latex]5\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]7\\dfrac{13}{30}[\/latex]<\/li>\n<li>[latex]12\\dfrac{7}{12}[\/latex]<\/li>\n<li>[latex]11\\dfrac{11}{12}[\/latex]<\/li>\n<li>[latex]125\\dfrac{2}{3}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 16<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Add. Express the sums in lowest terms.<\/p>\n<ol type=\"a\">\n<li>[latex]\\begin{array}{rrrrr}&4\\dfrac{1}{2}\\left(\\dfrac{\\times6}{\\times6}\\right)&=&4\\dfrac{6}{12}&\\\\+&2\\dfrac{1}{3}\\left(\\dfrac{\\times4}{\\times4}\\right)&=&2\\dfrac{4}{12}&\\\\\\hline&&&6\\dfrac{10}{12}&=6\\dfrac{5}{6}\\end{array}[\/latex]<\/li>\n<\/ol>\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\n<li>[latex]\\begin{array}{rr}&3\\dfrac{2}{3}\\\\+&1\\dfrac{1}{2}\\\\  \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&6\\dfrac{1}{2}\\\\+&4\\dfrac{1}{4}\\\\  \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&2\\dfrac{1}{8}\\\\+&4\\dfrac{3}{16}\\\\  \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&2\\dfrac{1}{5}\\\\+&3\\dfrac{2}{3}\\\\  \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&3\\dfrac{3}{8}\\\\&2\\dfrac{3}{4}\\\\+&1\\dfrac{1}{2}\\\\\\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&4\\dfrac{3}{4}\\\\&2\\dfrac{1}{5}\\\\+&4\\dfrac{1}{2}\\\\  \\hline&\\end{array}[\/latex]<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 16<\/strong><\/p>\n<ol class=\"twocolumn\" start=\"2\" type=\"a\">\n<li>[latex]5\\dfrac{1}{6}[\/latex]<\/li>\n<li>[latex]10\\dfrac{3}{4}[\/latex]<\/li>\n<li>[latex]6\\dfrac{5}{16}[\/latex]<\/li>\n<li>[latex]5\\dfrac{13}{15}[\/latex]<\/li>\n<li>[latex]7\\dfrac{5}{8}[\/latex]<\/li>\n<li>[latex]11\\dfrac{9}{20}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>Problems Using Addition of Common Fractions<\/h1>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 17<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Solve these problems.<\/strong><\/p>\n<ol type=\"a\">\n<li>The bathroom shelf is crowded with hand lotion bottles, each with a little lotion left inside. Everyone always likes to try the new bottle before the old one is emptied! One bottle is [latex]\\tfrac{1}{3}[\/latex] full, another is [latex]\\tfrac{1}{4}[\/latex] full, one is only [latex]\\tfrac{1}{8}[\/latex] full and one is still [latex]\\tfrac{1}{2}[\/latex] full. How much lotion is in the bottles altogether?<\/li>\n<li>Sometimes Joan thinks she will go crazy when she packs the lunches for her family. Little Sarah has decided she only wants [latex]\\tfrac{3}{4}[\/latex] of a sandwich, Megan wants [latex]\\tfrac{1}{4}[\/latex] of a sandwich, Joan&#8217;s husband takes [latex]1\\tfrac{1}{2}[\/latex] sandwiches, and their son, who does heavy work, takes 3 sandwiches! How many sandwiches does Joan make?<\/li>\n<li>Dave paid the babysitter for the week. The sitter worked [latex]3\\tfrac{3}{4}[\/latex] hours on Monday, [latex]4\\tfrac{1}{4}[\/latex] hours on Tuesday and [latex]6\\tfrac{1}{2}[\/latex] hours on Friday. How many hours did the babysitter work looking after Dave\u2019s children that week?<\/li>\n<li>Quite a lot of watermelon was left after the watermelon-eating contest: [latex]1\\tfrac{1}{2}[\/latex] watermelons on one table, [latex]2\\tfrac{3}{4}[\/latex] of a watermelon on another table and [latex]\\tfrac{5}{8}[\/latex] of a watermelon on the third table. The organizers want to know exactly how much was left over so they will not buy so much next year. Calculate the amount of watermelon left over.<\/li>\n<li>Jeanette has a novel to read for English. She read [latex]\\tfrac{1}{2}[\/latex] of the book on the weekend, only had time to read [latex]\\tfrac{1}{8}[\/latex] of the book on Monday and another [latex]\\tfrac{1}{4}[\/latex] on Wednesday. How much of the book has she read?<\/li>\n<li>Dion walks around this route each day for exercise. How far does he walk each day? Is this a perimeter or area question?<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-181\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/4A-ex-17.f-1.png\" alt=\"Rectangle. 1 half kilometres by 1 and 2 thirds kilometres.\" width=\"342\" height=\"211\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/4A-ex-17.f-1.png 660w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/4A-ex-17.f-1-300x185.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/4A-ex-17.f-1-65x40.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/4A-ex-17.f-1-225x139.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/4A-ex-17.f-1-350x216.png 350w\" sizes=\"auto, (max-width: 342px) 100vw, 342px\" \/><\/li>\n<li>How many metres of baseboard are needed for a rectangular room [latex]4\\tfrac{1}{2}[\/latex] m by [latex]3\\tfrac{1}{5}[\/latex]m? Deduct 1 m for the doorway. (TIP: Draw a picture)<\/li>\n<li>Sana is going to frame a large piece of art with a wooden frame. The art piece is [latex]1\\tfrac{1}{10}[\/latex] m by [latex]\\tfrac{3}{5}[\/latex]\u00a0 m. How much framing material should she buy?<\/li>\n<li>Find the perimeter of the following rectangle.<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-373 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_i-300x183.jpg\" alt=\"\" width=\"300\" height=\"183\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_i-300x183.jpg 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_i-65x40.jpg 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_i-225x137.jpg 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_i-350x213.jpg 350w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_i.jpg 651w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/li>\n<li>Find the perimeter of a picture frame if one side is [latex]12 \\dfrac{1}{10}[\/latex] cm and the other side measures [latex]14 \\dfrac{1}{5}[\/latex]cm.<\/li>\n<li>Find the perimeter of this triangle.\n<div><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-370 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_k.jpg\" alt=\"a triangle\" width=\"233\" height=\"319\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_k.jpg 233w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_k-219x300.jpg 219w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_k-65x89.jpg 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2022\/10\/Unit-4_17_k-225x308.jpg 225w\" sizes=\"auto, (max-width: 233px) 100vw, 233px\" \/><\/div>\n<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 17<\/strong><\/p>\n<ol class=\"twocolumn\" type=\"a\">\n<li>[latex]1 \\dfrac{5}{24}[\/latex] bottles total<\/li>\n<li>[latex]5 \\dfrac{1}{2}[\/latex] sandwiches<\/li>\n<li>[latex]14 \\dfrac{1}{2}[\/latex] hours<\/li>\n<li>[latex]4 \\dfrac{7}{8}[\/latex] watermelons<\/li>\n<li>[latex]\\dfrac{7}{8}[\/latex] of the book<\/li>\n<li>He walks [latex]4 \\dfrac{1}{3}[\/latex] km each day, perimeter<\/li>\n<li>[latex]15 \\dfrac{2}{5}[\/latex] m of material<\/li>\n<li>[latex]3 \\dfrac{2}{5}[\/latex] m of material<\/li>\n<li>[latex]15 \\dfrac{2}{3}[\/latex] cm<\/li>\n<li>[latex]52 \\dfrac{3}{5}[\/latex] cm<\/li>\n<li>[latex]17 \\dfrac{11}{24}[\/latex] cm<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>Topic A: Self-Test<\/h1>\n<p><strong>Mark\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \/14\u00a0 \u00a0Aim 11\/14<\/strong><\/p>\n<ol type=\"A\">\n<li>Add and express the answers in lowest terms (6 marks).\n<ol class=\"twocolumn space\" type=\"a\">\n<li>\u00a0[latex]\\begin{array}{rr}&\\dfrac{1}{4}\\\\+&\\dfrac{3}{4}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>\u00a0[latex]\\begin{array}{rr}&1\\dfrac{3}{5}\\\\+&3\\dfrac{4}{5}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&\\dfrac{3}{8}\\\\+&\\dfrac{3}{4}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&2\\dfrac{1}{6}\\\\+&3\\dfrac{5}{12}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&6\\dfrac{3}{4}\\\\+&2\\dfrac{1}{2}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{rr}&6\\dfrac{7}{8}\\\\+&9\\dfrac{1}{3}\\\\ \\hline&\\end{array}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Word Problems (8 marks).\n<ol type=\"a\">\n<li>The flight from Vancouver to Sandspit took [latex]1\\dfrac{1}{4}[\/latex] hours. The wait in Sandspit was [latex]1\\dfrac{1}{2}[\/latex] hours and the flight from there to Ketchican, Alaska was [latex]\\dfrac{3}{4}[\/latex] of an hour. How long did it take to make the trip from Vancouver, BC to Ketchican, Alaska?<\/li>\n<li>Dave built\u00a0[latex]\\dfrac{1}{8}[\/latex] of the fence around his house on Monday, [latex]\\dfrac{1}{4}[\/latex] of it on Tuesday and another [latex]\\dfrac{1}{4}[\/latex] on Wednesday. How much of the fence has he built?<\/li>\n<li>John bought snacks in bulk for the class party. His items weighed [latex]\\dfrac{2}{5}[\/latex] kg of chips, [latex]\\dfrac{3}{5}[\/latex] kg of peanuts, [latex]\\dfrac{1}{2}[\/latex] kg of cheese and [latex]1\\dfrac{1}{4}[\/latex] kg of fresh veggies. How much did all his snacks weigh?<\/li>\n<li>Clarence is making a frame for his favourite photo. The frame needs to be [latex]\\dfrac{1}{8}[\/latex] m by [latex]\\dfrac{5}{6}[\/latex] m. How much material should he buy?<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><strong>Answers to Topic A Self-Test<\/strong><\/p>\n<ol type=\"A\">\n<li>\n<ol class=\"threecolumn\" type=\"a\">\n<li>1<\/li>\n<li>[latex]5\\dfrac{2}{5}[\/latex]<\/li>\n<li>[latex]1\\dfrac{1}{8}[\/latex]<\/li>\n<li>[latex]5\\dfrac{7}{12}[\/latex]<\/li>\n<li>[latex]9\\dfrac{1}{4}[\/latex]<\/li>\n<li>[latex]16\\dfrac{5}{24}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>\n<ol class=\"twocolumn\" type=\"a\">\n<li>[latex]3 \\dfrac{1}{2}[\/latex] hr<\/li>\n<li>[latex]\\dfrac{5}{8}[\/latex] of the fence<\/li>\n<li>[latex]2 \\dfrac{3}{4}[\/latex] kg of food<\/li>\n<li>[latex]1 \\dfrac{11}{12}[\/latex] m of material<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_182_269\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_182_269\"><div tabindex=\"-1\"><p>The result of an addition question, the answer to an addition question.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_182_282\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_182_282\"><div tabindex=\"-1\"><p>A common fraction with a value less than one.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_182_278\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_182_278\"><div tabindex=\"-1\"><p>The numbers to be added together in an addition question. In 3 + 5 = 8, the addends are 3 and 5.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_182_280\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_182_280\"><div tabindex=\"-1\"><p>The bottom number in a common fraction; the denominator tells into how many equal parts the whole thing has been divided.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":999,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-182","chapter","type-chapter","status-publish","hentry"],"part":159,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/chapters\/182","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/wp\/v2\/users\/999"}],"version-history":[{"count":14,"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/chapters\/182\/revisions"}],"predecessor-version":[{"id":369,"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/chapters\/182\/revisions\/369"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/parts\/159"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/chapters\/182\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/wp\/v2\/media?parent=182"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/chapter-type?post=182"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/wp\/v2\/contributor?post=182"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/wp\/v2\/license?post=182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}