{"id":4495,"date":"2017-06-07T18:49:52","date_gmt":"2017-06-07T18:49:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-multiplying-fractions\/"},"modified":"2017-08-15T12:03:10","modified_gmt":"2017-08-15T12:03:10","slug":"read-multiplying-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-multiplying-fractions\/","title":{"raw":"Multiplying Fractions","rendered":"Multiplying Fractions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Multiply fractions\r\n<ul>\r\n \t<li>Multiply two or more fractions<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Multiply Fractions<\/h2>\r\nJust as you add, subtract, multiply, and divide when working with whole numbers, you also use these operations when working with fractions. \u00a0\u00a0There are many times when it is necessary to multiply fractions. A model may help you understand multiplication of fractions.\r\n\r\nWhen you multiply a fraction by a fraction, you are finding a \u201cfraction of a fraction.\u201d Suppose you have [latex]\\frac{3}{4}[\/latex]\u00a0of a candy bar and you want to find [latex]\\frac{1}{2}[\/latex]\u00a0of the [latex]\\frac{3}{4}[\/latex]:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184948\/image109.gif\" alt=\"3 out of four boxes are shaded. This is 3\/4.\" width=\"208\" height=\"65\" \/>\r\n\r\nBy dividing each fourth in half, you can divide the candy bar into eighths.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184950\/image110.gif\" alt=\"Six of 8 boxes are shaded. This is 6\/8.\" width=\"208\" height=\"62\" \/>\r\n\r\nThen, choose half of those to get [latex]\\frac{3}{8}[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184951\/image112.gif\" alt=\"Six of 8 boxes are shaded, and of those six, three of them are shaded purple. The 3 purple boxes represent 3\/8.\" width=\"208\" height=\"54\" \/>\r\n\r\nIn both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying Two Fractions<\/h3>\r\n[latex] \\frac{a}{b}\\cdot \\frac{c}{d}=\\frac{a\\cdot c}{b\\cdot d}=\\frac{\\text{product of the numerators}}{\\text{product of the denominators}}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying More Than Two Fractions<\/h3>\r\n[latex] \\frac{a}{b}\\cdot \\frac{c}{d}\\cdot \\frac{e}{f}=\\frac{a\\cdot c\\cdot e}{b\\cdot d\\cdot f}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply [latex] \\frac{2}{3}\\cdot \\frac{4}{5}[\/latex].\r\n\r\n[reveal-answer q=\"368042\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"368042\"]Multiply the numerators and multiply the denominators.\r\n<p style=\"text-align: center\">[latex] \\frac{2\\cdot 4}{3\\cdot 5}[\/latex]<\/p>\r\nSimplify, if possible. This fraction is already in lowest terms.\r\n<p style=\"text-align: center\">[latex] \\frac{8}{15}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\frac{8}{15}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo review: if a fraction has\u00a0common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.\r\n<p id=\"fs-id2701635\">For example,<\/p>\r\n\r\n<ul id=\"fs-id1302300\">\r\n \t<li>Given [latex] \\frac{8}{15}[\/latex], the factors of 8 are: 1, 2, 4, 8 and the factors of 15 are: 1, 3, 5, 15. \u00a0[latex] \\frac{8}{15}[\/latex] is simplified because there are no common factors of <span style=\"font-size: 14px;line-height: normal\">8<\/span>\u00a0and 15<span id=\"MathJax-Element-8-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\"><span class=\"MJX_Assistive_MathML\">.<\/span><\/span><\/li>\r\n \t<li>Given [latex] \\frac{10}{15}[\/latex], the factors of 10 are: 1, 2, 5, 10 and the factors of15 are: 1, 3, 5, 15. [latex] \\frac{10}{15}[\/latex] is not simplified because <span id=\"MathJax-Element-10-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\"><span id=\"MathJax-Span-82\" class=\"math\"><span id=\"MathJax-Span-83\" class=\"mrow\"><span id=\"MathJax-Span-84\" class=\"semantics\"><span id=\"MathJax-Span-85\" class=\"mrow\"><span id=\"MathJax-Span-86\" class=\"mn\">5<\/span><\/span><\/span><\/span><\/span><\/span>\u00a0is a common factor of <span id=\"MathJax-Element-11-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\"><span id=\"MathJax-Span-87\" class=\"math\"><span id=\"MathJax-Span-88\" class=\"mrow\"><span id=\"MathJax-Span-89\" class=\"semantics\"><span id=\"MathJax-Span-90\" class=\"mrow\"><span id=\"MathJax-Span-91\" class=\"mrow\"><span id=\"MathJax-Span-92\" class=\"mn\">10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0and <span id=\"MathJax-Element-12-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\"><span id=\"MathJax-Span-93\" class=\"math\"><span id=\"MathJax-Span-94\" class=\"mrow\"><span id=\"MathJax-Span-95\" class=\"semantics\"><span id=\"MathJax-Span-96\" class=\"mrow\"><span id=\"MathJax-Span-97\" class=\"mrow\"><span id=\"MathJax-Span-98\" class=\"mn\">15<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\">.<\/span><\/span><\/li>\r\n<\/ul>\r\nYou can\u00a0simplify first, before you multiply two fractions, to make your work easier. This allows you to work with smaller numbers when you multiply.\r\n\r\nIn the following video you will see an example of how to multiply two fractions, then simplify the answer.\r\n\r\nhttps:\/\/youtu.be\/f_L-EFC8Z7c\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nMultiply [latex] \\frac{2}{3}\\cdot \\frac{1}{4}\\cdot\\frac{3}{5}[\/latex]. Simplify the answer.\r\n\r\nWhat makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would multiply three fractions together.\r\n\r\n[practice-area rows=\"2\"][\/practice-area]\r\n\r\n[reveal-answer q=\"385641\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"385641\"]Multiply the numerators and multiply the denominators.\r\n<p style=\"text-align: center\">[latex] \\frac{2\\cdot 1\\cdot 3}{3\\cdot 4\\cdot 5}[\/latex]<\/p>\r\nSimplify first by canceling (dividing) the\u00a0common factors of 3 and 2. 3 divided by 3 is 1, and 2 divided by 2 is 1.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{2\\cdot 1\\cdot3}{3\\cdot (2\\cdot 2)\\cdot 5}\\\\\\frac{\\cancel{2}\\cdot 1\\cdot\\cancel{3}}{\\cancel{3}\\cdot (\\cancel{2}\\cdot 2)\\cdot 5}\\\\\\frac{1}{10}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\frac{2}{3}\\cdot \\frac{1}{4}\\cdot\\frac{3}{5}[\/latex] = [latex]\\frac{1}{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Multiply fractions\n<ul>\n<li>Multiply two or more fractions<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2>Multiply Fractions<\/h2>\n<p>Just as you add, subtract, multiply, and divide when working with whole numbers, you also use these operations when working with fractions. \u00a0\u00a0There are many times when it is necessary to multiply fractions. A model may help you understand multiplication of fractions.<\/p>\n<p>When you multiply a fraction by a fraction, you are finding a \u201cfraction of a fraction.\u201d Suppose you have [latex]\\frac{3}{4}[\/latex]\u00a0of a candy bar and you want to find [latex]\\frac{1}{2}[\/latex]\u00a0of the [latex]\\frac{3}{4}[\/latex]:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184948\/image109.gif\" alt=\"3 out of four boxes are shaded. This is 3\/4.\" width=\"208\" height=\"65\" \/><\/p>\n<p>By dividing each fourth in half, you can divide the candy bar into eighths.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184950\/image110.gif\" alt=\"Six of 8 boxes are shaded. This is 6\/8.\" width=\"208\" height=\"62\" \/><\/p>\n<p>Then, choose half of those to get [latex]\\frac{3}{8}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07184951\/image112.gif\" alt=\"Six of 8 boxes are shaded, and of those six, three of them are shaded purple. The 3 purple boxes represent 3\/8.\" width=\"208\" height=\"54\" \/><\/p>\n<p>In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplying Two Fractions<\/h3>\n<p>[latex]\\frac{a}{b}\\cdot \\frac{c}{d}=\\frac{a\\cdot c}{b\\cdot d}=\\frac{\\text{product of the numerators}}{\\text{product of the denominators}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Multiplying More Than Two Fractions<\/h3>\n<p>[latex]\\frac{a}{b}\\cdot \\frac{c}{d}\\cdot \\frac{e}{f}=\\frac{a\\cdot c\\cdot e}{b\\cdot d\\cdot f}[\/latex]<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply [latex]\\frac{2}{3}\\cdot \\frac{4}{5}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q368042\">Show Solution<\/span><\/p>\n<div id=\"q368042\" class=\"hidden-answer\" style=\"display: none\">Multiply the numerators and multiply the denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{2\\cdot 4}{3\\cdot 5}[\/latex]<\/p>\n<p>Simplify, if possible. This fraction is already in lowest terms.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{8}{15}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{8}{15}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To review: if a fraction has\u00a0common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.<\/p>\n<p id=\"fs-id2701635\">For example,<\/p>\n<ul id=\"fs-id1302300\">\n<li>Given [latex]\\frac{8}{15}[\/latex], the factors of 8 are: 1, 2, 4, 8 and the factors of 15 are: 1, 3, 5, 15. \u00a0[latex]\\frac{8}{15}[\/latex] is simplified because there are no common factors of <span style=\"font-size: 14px;line-height: normal\">8<\/span>\u00a0and 15<span id=\"MathJax-Element-8-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\"><span class=\"MJX_Assistive_MathML\">.<\/span><\/span><\/li>\n<li>Given [latex]\\frac{10}{15}[\/latex], the factors of 10 are: 1, 2, 5, 10 and the factors of15 are: 1, 3, 5, 15. [latex]\\frac{10}{15}[\/latex] is not simplified because <span id=\"MathJax-Element-10-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\"><span id=\"MathJax-Span-82\" class=\"math\"><span id=\"MathJax-Span-83\" class=\"mrow\"><span id=\"MathJax-Span-84\" class=\"semantics\"><span id=\"MathJax-Span-85\" class=\"mrow\"><span id=\"MathJax-Span-86\" class=\"mn\">5<\/span><\/span><\/span><\/span><\/span><\/span>\u00a0is a common factor of <span id=\"MathJax-Element-11-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\"><span id=\"MathJax-Span-87\" class=\"math\"><span id=\"MathJax-Span-88\" class=\"mrow\"><span id=\"MathJax-Span-89\" class=\"semantics\"><span id=\"MathJax-Span-90\" class=\"mrow\"><span id=\"MathJax-Span-91\" class=\"mrow\"><span id=\"MathJax-Span-92\" class=\"mn\">10<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0and <span id=\"MathJax-Element-12-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\"><span id=\"MathJax-Span-93\" class=\"math\"><span id=\"MathJax-Span-94\" class=\"mrow\"><span id=\"MathJax-Span-95\" class=\"semantics\"><span id=\"MathJax-Span-96\" class=\"mrow\"><span id=\"MathJax-Span-97\" class=\"mrow\"><span id=\"MathJax-Span-98\" class=\"mn\">15<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\">.<\/span><\/span><\/li>\n<\/ul>\n<p>You can\u00a0simplify first, before you multiply two fractions, to make your work easier. This allows you to work with smaller numbers when you multiply.<\/p>\n<p>In the following video you will see an example of how to multiply two fractions, then simplify the answer.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1: Multiply Fractions (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/f_L-EFC8Z7c?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>Multiply [latex]\\frac{2}{3}\\cdot \\frac{1}{4}\\cdot\\frac{3}{5}[\/latex]. Simplify the answer.<\/p>\n<p>What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would multiply three fractions together.<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"2\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q385641\">Show Solution<\/span><\/p>\n<div id=\"q385641\" class=\"hidden-answer\" style=\"display: none\">Multiply the numerators and multiply the denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{2\\cdot 1\\cdot 3}{3\\cdot 4\\cdot 5}[\/latex]<\/p>\n<p>Simplify first by canceling (dividing) the\u00a0common factors of 3 and 2. 3 divided by 3 is 1, and 2 divided by 2 is 1.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{2\\cdot 1\\cdot3}{3\\cdot (2\\cdot 2)\\cdot 5}\\\\\\frac{\\cancel{2}\\cdot 1\\cdot\\cancel{3}}{\\cancel{3}\\cdot (\\cancel{2}\\cdot 2)\\cdot 5}\\\\\\frac{1}{10}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{2}{3}\\cdot \\frac{1}{4}\\cdot\\frac{3}{5}[\/latex] = [latex]\\frac{1}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4495\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Adaptation and Revision. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex 1: Multiply Fractions (Basic) . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/f_L-EFC8Z7c\">https:\/\/youtu.be\/f_L-EFC8Z7c<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 2: Fractions and Mixed Numbers, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li> Multiply and Divide Fractions. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/yqV9q0HH@7.3:s7ku6WX5@2\/Multiply-and-Divide-Fractions\">http:\/\/cnx.org\/contents\/yqV9q0HH@7.3:s7ku6WX5@2\/Multiply-and-Divide-Fractions<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@7.3<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex 1: Multiply Fractions (Basic) \",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/f_L-EFC8Z7c\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc-attribution\",\"description\":\" Multiply and Divide 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