{"id":9487,"date":"2017-05-02T23:15:48","date_gmt":"2017-05-02T23:15:48","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=9487"},"modified":"2020-09-03T10:47:24","modified_gmt":"2020-09-03T10:47:24","slug":"multiplying-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/multiplying-fractions\/","title":{"raw":"2.2.b - Multiplying Fractions","rendered":"2.2.b &#8211; Multiplying Fractions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Multiply two or more fractions<\/li>\r\n \t<li>Multiply a fraction by a whole number<\/li>\r\n<\/ul>\r\n<\/div>\r\nA model may help you understand multiplication of fractions. We will use fraction tiles to model [latex]\\Large\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]. To multiply [latex]\\Large\\frac{1}{2}[\/latex] and [latex]\\Large\\frac{3}{4}[\/latex], think [latex]\\Large\\frac{1}{2}[\/latex] of [latex]\\Large\\frac{3}{4}[\/latex].\r\nStart with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three [latex]\\Large\\frac{1}{4}[\/latex] tiles evenly into two parts, we exchange them for smaller tiles.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220916\/CNX_BMath_Figure_04_02_010_img.png\" alt=\"A rectangle is divided vertically into three equal pieces. Each piece is labeled as one fourth. There is a an arrow pointing to an identical rectangle divided vertically into six equal pieces. Each piece is labeled as one eighth. There are braces showing that three of these rectangles represent three eighths.\" \/>\r\nWe see [latex]\\Large\\frac{6}{8}[\/latex] is equivalent to [latex]\\Large\\frac{3}{4}[\/latex]. Taking half of the six [latex]\\Large\\frac{1}{8}[\/latex] tiles gives us three [latex]\\Large\\frac{1}{8}[\/latex] tiles, which is [latex]\\Large\\frac{3}{8}[\/latex].\r\n\r\nTherefore,\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{1}{2}\\cdot \\frac{3}{4}=\\frac{3}{8}[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse a diagram to model [latex]\\Large\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]\r\n\r\nSolution:\r\nFirst shade in [latex]\\Large\\frac{3}{4}[\/latex] of the rectangle.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220917\/CNX_BMath_Figure_04_02_029_img.png\" alt=\"A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded.\" \/>\r\nWe will take [latex]\\Large\\frac{1}{2}[\/latex] of this [latex]\\Large\\frac{3}{4}[\/latex], so we heavily shade [latex]\\Large\\frac{1}{2}[\/latex] of the shaded region.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220919\/CNX_BMath_Figure_04_02_030_img.png\" alt=\"A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded. The rectangle is divided by a horizontal line, creating eight equal pieces. Three of the eight pieces are darkly shaded.\" \/>\r\nNotice that [latex]3[\/latex] out of the [latex]8[\/latex] pieces are heavily shaded. This means that [latex]\\Large\\frac{3}{8}[\/latex] of the rectangle is heavily shaded.\r\nTherefore, [latex]\\Large\\frac{1}{2}[\/latex] of [latex]\\Large\\frac{3}{4}[\/latex] is [latex]\\Large\\frac{3}{8}[\/latex], or [latex]{\\Large\\frac{1}{2}\\cdot \\frac{3}{4}}={\\Large\\frac{3}{8}}[\/latex].\r\n\r\n<\/div>\r\n<h3><\/h3>\r\nLook at the result we got from the model in the example above. We found that [latex]\\Large\\frac{1}{2}\\cdot \\frac{3}{4}=\\frac{3}{8}[\/latex]. Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?\r\n<table id=\"eip-id1168468256450\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the numerators, and multiply the denominators.<\/td>\r\n<td>[latex]\\Large\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{3}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.\r\n<div class=\"textbox shaded\">\r\n<h3>Fraction Multiplication<\/h3>\r\nIf [latex]a,b,c,\\text{ and }d[\/latex] are numbers where [latex]b\\ne 0\\text{ and }d\\ne 0[\/latex], then\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{a}{b}\\cdot \\frac{c}{d}=\\frac{ac}{bd}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMultiply, and write the answer in simplified form: [latex]\\Large\\frac{3}{4}\\cdot \\frac{1}{5}[\/latex]\r\n[reveal-answer q=\"56385\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"56385\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468398776\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\Large\\frac{3}{4}\\cdot \\frac{1}{5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the numerators; multiply the denominators.<\/td>\r\n<td>[latex]\\Large\\frac{3\\cdot 1}{4\\cdot 5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{3}{20}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThere are no common factors, so the fraction is simplified.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]146021[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video provides more examples of how to multiply fractions, and simplify the result.\r\nhttps:\/\/youtu.be\/f_L-EFC8Z7c\r\n\r\nTo multiply more than two fractions, we have a similar definition.\u00a0 We still multiply the numerators and multiply the denominators.\u00a0 Then we write the fraction in simplified form.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying More Than Two Fractions<\/h3>\r\nIf [latex]a,b,c,d,e \\text{ and }f[\/latex] are numbers where [latex]b\\ne 0,d\\ne 0\\text{ and }f\\ne 0[\/latex], then\r\n<p style=\"text-align: center\">[latex]\\Large\\frac{a}{b}\\cdot\\Large\\frac{c}{d}\\cdot\\Large\\frac{e}{f}=\\Large\\frac{a\\cdot c\\cdot e}{b\\cdot d\\cdot f}[\/latex]<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nMultiply [latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{1}{4}\\cdot\\Large\\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]\\Large\\frac{2\\cdot 1\\cdot 3}{3\\cdot 4\\cdot 5}[\/latex]<\/p>\r\nSimplify first by canceling (dividing) the\u00a0common factors of [latex]3[\/latex] and [latex]2[\/latex]. \u00a0[latex]3[\/latex] divided by \u00a0[latex]3[\/latex] is [latex]1[\/latex], and \u00a0[latex]2[\/latex] divided by \u00a0[latex]2[\/latex] is [latex]1[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{2\\cdot 1\\cdot3}{3\\cdot (2\\cdot 2)\\cdot 5}\\\\\\Large\\frac{\\cancel{2}\\cdot 1\\cdot\\cancel{3}}{\\cancel{3}\\cdot (\\cancel{2}\\cdot 2)\\cdot 5}\\\\\\Large\\frac{1}{10}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{1}{4}\\cdot\\Large\\frac{3}{5}[\/latex] = [latex]\\Large\\frac{1}{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWhen multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step. In the next example,\u00a0we will multiply two negatives, so the product will be positive.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMultiply, and write the answer in simplified form: [latex]\\Large-\\frac{5}{8}\\left(-\\frac{2}{3}\\right)[\/latex]\r\n[reveal-answer q=\"87955\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"87955\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466273465\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The problem is negative 5 eighths times negative 2 thirds. The second line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large-\\frac{5}{8}\\left(-\\frac{2}{3}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are the same, so the product is positive. Multiply the numerators, multiply the denominators.<\/td>\r\n<td>[latex]\\Large\\frac{5\\cdot 2}{8\\cdot 3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\Large\\frac{10}{24}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Look for common factors in the numerator and denominator. Rewrite showing common factors.<\/td>\r\n<td>[latex]\\Large\\frac{5\\cdot\\color{red}{2}}{12\\cdot\\color{red}{2}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors.<\/td>\r\n<td>[latex]\\Large\\frac{5}{12}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAnother way to find this product involves removing common factors earlier.\r\n<table id=\"eip-id1168466166066\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The problem is negative 5 eighths times negative 2 thirds. The second line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large-\\frac{5}{8}\\left(-\\frac{2}{3}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Determine the sign of the product. Multiply.<\/td>\r\n<td>[latex]\\Large\\frac{5\\cdot 2}{8\\cdot 3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Show common factors and then remove them.<\/td>\r\n<td>[latex]\\Large\\frac{5\\cdot\\color{red}{2}}{4\\cdot\\color{red}{2}\\cdot3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply remaining factors.<\/td>\r\n<td>[latex]\\Large\\frac{5}{12}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe get the same result.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[ohm_question height=\"270\"]146022[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMultiply, and write the answer in simplified form: [latex]\\Large-\\frac{14}{15}\\cdot \\frac{20}{21}[\/latex]\r\n[reveal-answer q=\"731970\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"731970\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466394330\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large-\\frac{14}{15}\\cdot \\frac{20}{21}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Determine the sign of the product; multiply.<\/td>\r\n<td>[latex]\\Large-\\frac{14}{15}\\cdot \\frac{20}{21}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Are there any common factors in the numerator and the denominator?\r\n\r\nWe know that [latex]7[\/latex] is a factor of [latex]14[\/latex] and [latex]21[\/latex], and [latex]5[\/latex] is a factor of [latex]20[\/latex] and [latex]15[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite showing common factors.<\/td>\r\n<td>[latex]\\Large-\\frac{2\\cdot\\color{red}{7}\\cdot4\\cdot\\color{blue}{5}}{3\\cdot\\color{blue}{5}\\cdot3\\cdot\\color{red}{7}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove the common factors.<\/td>\r\n<td>[latex]\\Large-\\frac{2\\cdot 4}{3\\cdot 3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the remaining factors.<\/td>\r\n<td>[latex]\\Large-\\frac{8}{9}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[ohm_question height=\"270\"]146023[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video shows another example of multiplying fractions that are negative.\r\n\r\nhttps:\/\/youtu.be\/yUdJ46pTblo\r\n\r\nWhen multiplying a fraction by a whole number, it may be helpful to write the whole number as a fraction. Any whole number, [latex]a[\/latex], can be written as [latex]\\Large\\frac{a}{1}[\/latex]. So, [latex]3=\\Large\\frac{3}{1}[\/latex], for example.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply, and write the answer in simplified form:\r\n<ol>\r\n \t<li>[latex]\\Large{\\frac{1}{7}}\\normalsize\\cdot 56[\/latex]<\/li>\r\n \t<li>[latex]\\Large{\\frac{12}{5}}\\normalsize\\left(-20x\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"597781\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"597781\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466216346\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large\\frac{1}{7}\\normalsize\\cdot 56[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]56[\/latex] as a fraction.<\/td>\r\n<td>[latex]\\Large\\frac{1}{7}\\cdot \\frac{56}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Determine the sign of the product; multiply.<\/td>\r\n<td>[latex]\\Large\\frac{56}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466048420\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The first line says 12 fifths times negative 20 \">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\Large\\frac{12}{5}\\normalsize\\left(-20x\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]\u221220x[\/latex] as a fraction.<\/td>\r\n<td>[latex]\\Large\\frac{12}{5}\\left(\\frac{-20x}{1}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Determine the sign of the product; multiply.<\/td>\r\n<td>[latex]\\Large-\\frac{12\\cdot 20\\cdot x}{5\\cdot 1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Show common factors and then remove them.<\/td>\r\n<td>[latex]\\Large-\\frac{12\\cdot 4\\cdot {\\color{red}{5}}\\cdot x}{\\color{red}{5}\\cdot1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply remaining factors; simplify.<\/td>\r\n<td>[latex]\u221248x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[ohm_question height=\"270\"]146024[\/ohm_question]\r\n\r\n[ohm_question height=\"270\"]146025[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video to see more examples of how to multiply a fraction and a whole number.\r\n\r\nhttps:\/\/youtu.be\/Rxz7OUzNyV0","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply two or more fractions<\/li>\n<li>Multiply a fraction by a whole number<\/li>\n<\/ul>\n<\/div>\n<p>A model may help you understand multiplication of fractions. We will use fraction tiles to model [latex]\\Large\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]. To multiply [latex]\\Large\\frac{1}{2}[\/latex] and [latex]\\Large\\frac{3}{4}[\/latex], think [latex]\\Large\\frac{1}{2}[\/latex] of [latex]\\Large\\frac{3}{4}[\/latex].<br \/>\nStart with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three [latex]\\Large\\frac{1}{4}[\/latex] tiles evenly into two parts, we exchange them for smaller tiles.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220916\/CNX_BMath_Figure_04_02_010_img.png\" alt=\"A rectangle is divided vertically into three equal pieces. Each piece is labeled as one fourth. There is a an arrow pointing to an identical rectangle divided vertically into six equal pieces. Each piece is labeled as one eighth. There are braces showing that three of these rectangles represent three eighths.\" \/><br \/>\nWe see [latex]\\Large\\frac{6}{8}[\/latex] is equivalent to [latex]\\Large\\frac{3}{4}[\/latex]. Taking half of the six [latex]\\Large\\frac{1}{8}[\/latex] tiles gives us three [latex]\\Large\\frac{1}{8}[\/latex] tiles, which is [latex]\\Large\\frac{3}{8}[\/latex].<\/p>\n<p>Therefore,<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{1}{2}\\cdot \\frac{3}{4}=\\frac{3}{8}[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use a diagram to model [latex]\\Large\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]<\/p>\n<p>Solution:<br \/>\nFirst shade in [latex]\\Large\\frac{3}{4}[\/latex] of the rectangle.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220917\/CNX_BMath_Figure_04_02_029_img.png\" alt=\"A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded.\" \/><br \/>\nWe will take [latex]\\Large\\frac{1}{2}[\/latex] of this [latex]\\Large\\frac{3}{4}[\/latex], so we heavily shade [latex]\\Large\\frac{1}{2}[\/latex] of the shaded region.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220919\/CNX_BMath_Figure_04_02_030_img.png\" alt=\"A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded. The rectangle is divided by a horizontal line, creating eight equal pieces. Three of the eight pieces are darkly shaded.\" \/><br \/>\nNotice that [latex]3[\/latex] out of the [latex]8[\/latex] pieces are heavily shaded. This means that [latex]\\Large\\frac{3}{8}[\/latex] of the rectangle is heavily shaded.<br \/>\nTherefore, [latex]\\Large\\frac{1}{2}[\/latex] of [latex]\\Large\\frac{3}{4}[\/latex] is [latex]\\Large\\frac{3}{8}[\/latex], or [latex]{\\Large\\frac{1}{2}\\cdot \\frac{3}{4}}={\\Large\\frac{3}{8}}[\/latex].<\/p>\n<\/div>\n<h3><\/h3>\n<p>Look at the result we got from the model in the example above. We found that [latex]\\Large\\frac{1}{2}\\cdot \\frac{3}{4}=\\frac{3}{8}[\/latex]. Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?<\/p>\n<table id=\"eip-id1168468256450\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the numerators, and multiply the denominators.<\/td>\n<td>[latex]\\Large\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{3}{8}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.<\/p>\n<div class=\"textbox shaded\">\n<h3>Fraction Multiplication<\/h3>\n<p>If [latex]a,b,c,\\text{ and }d[\/latex] are numbers where [latex]b\\ne 0\\text{ and }d\\ne 0[\/latex], then<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{a}{b}\\cdot \\frac{c}{d}=\\frac{ac}{bd}[\/latex]<\/p>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Multiply, and write the answer in simplified form: [latex]\\Large\\frac{3}{4}\\cdot \\frac{1}{5}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q56385\">Show Solution<\/span><\/p>\n<div id=\"q56385\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468398776\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>[latex]\\Large\\frac{3}{4}\\cdot \\frac{1}{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the numerators; multiply the denominators.<\/td>\n<td>[latex]\\Large\\frac{3\\cdot 1}{4\\cdot 5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{3}{20}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>There are no common factors, so the fraction is simplified.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146021\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146021&theme=oea&iframe_resize_id=ohm146021&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>The following video provides more examples of how to multiply fractions, and simplify the result.<br \/>\n<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<p>To multiply more than two fractions, we have a similar definition.\u00a0 We still multiply the numerators and multiply the denominators.\u00a0 Then we write the fraction in simplified form.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplying More Than Two Fractions<\/h3>\n<p>If [latex]a,b,c,d,e \\text{ and }f[\/latex] are numbers where [latex]b\\ne 0,d\\ne 0\\text{ and }f\\ne 0[\/latex], then<\/p>\n<p style=\"text-align: center\">[latex]\\Large\\frac{a}{b}\\cdot\\Large\\frac{c}{d}\\cdot\\Large\\frac{e}{f}=\\Large\\frac{a\\cdot c\\cdot e}{b\\cdot d\\cdot f}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>Multiply [latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{1}{4}\\cdot\\Large\\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]\\Large\\frac{2\\cdot 1\\cdot 3}{3\\cdot 4\\cdot 5}[\/latex]<\/p>\n<p>Simplify first by canceling (dividing) the\u00a0common factors of [latex]3[\/latex] and [latex]2[\/latex]. \u00a0[latex]3[\/latex] divided by \u00a0[latex]3[\/latex] is [latex]1[\/latex], and \u00a0[latex]2[\/latex] divided by \u00a0[latex]2[\/latex] is [latex]1[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\Large\\frac{2\\cdot 1\\cdot3}{3\\cdot (2\\cdot 2)\\cdot 5}\\\\\\Large\\frac{\\cancel{2}\\cdot 1\\cdot\\cancel{3}}{\\cancel{3}\\cdot (\\cancel{2}\\cdot 2)\\cdot 5}\\\\\\Large\\frac{1}{10}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\Large\\frac{2}{3}\\cdot\\Large\\frac{1}{4}\\cdot\\Large\\frac{3}{5}[\/latex] = [latex]\\Large\\frac{1}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>When multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step. In the next example,\u00a0we will multiply two negatives, so the product will be positive.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Multiply, and write the answer in simplified form: [latex]\\Large-\\frac{5}{8}\\left(-\\frac{2}{3}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q87955\">Show Solution<\/span><\/p>\n<div id=\"q87955\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466273465\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The problem is negative 5 eighths times negative 2 thirds. The second line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large-\\frac{5}{8}\\left(-\\frac{2}{3}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The signs are the same, so the product is positive. Multiply the numerators, multiply the denominators.<\/td>\n<td>[latex]\\Large\\frac{5\\cdot 2}{8\\cdot 3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\Large\\frac{10}{24}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Look for common factors in the numerator and denominator. Rewrite showing common factors.<\/td>\n<td>[latex]\\Large\\frac{5\\cdot\\color{red}{2}}{12\\cdot\\color{red}{2}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors.<\/td>\n<td>[latex]\\Large\\frac{5}{12}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Another way to find this product involves removing common factors earlier.<\/p>\n<table id=\"eip-id1168466166066\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The problem is negative 5 eighths times negative 2 thirds. The second line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large-\\frac{5}{8}\\left(-\\frac{2}{3}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Determine the sign of the product. Multiply.<\/td>\n<td>[latex]\\Large\\frac{5\\cdot 2}{8\\cdot 3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Show common factors and then remove them.<\/td>\n<td>[latex]\\Large\\frac{5\\cdot\\color{red}{2}}{4\\cdot\\color{red}{2}\\cdot3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply remaining factors.<\/td>\n<td>[latex]\\Large\\frac{5}{12}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We get the same result.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146022\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146022&theme=oea&iframe_resize_id=ohm146022&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Multiply, and write the answer in simplified form: [latex]\\Large-\\frac{14}{15}\\cdot \\frac{20}{21}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q731970\">Show Solution<\/span><\/p>\n<div id=\"q731970\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466394330\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\Large-\\frac{14}{15}\\cdot \\frac{20}{21}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Determine the sign of the product; multiply.<\/td>\n<td>[latex]\\Large-\\frac{14}{15}\\cdot \\frac{20}{21}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Are there any common factors in the numerator and the denominator?<\/p>\n<p>We know that [latex]7[\/latex] is a factor of [latex]14[\/latex] and [latex]21[\/latex], and [latex]5[\/latex] is a factor of [latex]20[\/latex] and [latex]15[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Rewrite showing common factors.<\/td>\n<td>[latex]\\Large-\\frac{2\\cdot\\color{red}{7}\\cdot4\\cdot\\color{blue}{5}}{3\\cdot\\color{blue}{5}\\cdot3\\cdot\\color{red}{7}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove the common factors.<\/td>\n<td>[latex]\\Large-\\frac{2\\cdot 4}{3\\cdot 3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the remaining factors.<\/td>\n<td>[latex]\\Large-\\frac{8}{9}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146023\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146023&theme=oea&iframe_resize_id=ohm146023&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>The following video shows another example of multiplying fractions that are negative.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Multiplying Signed Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/yUdJ46pTblo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>When multiplying a fraction by a whole number, it may be helpful to write the whole number as a fraction. Any whole number, [latex]a[\/latex], can be written as [latex]\\Large\\frac{a}{1}[\/latex]. So, [latex]3=\\Large\\frac{3}{1}[\/latex], for example.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply, and write the answer in simplified form:<\/p>\n<ol>\n<li>[latex]\\Large{\\frac{1}{7}}\\normalsize\\cdot 56[\/latex]<\/li>\n<li>[latex]\\Large{\\frac{12}{5}}\\normalsize\\left(-20x\\right)[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q597781\">Show Solution<\/span><\/p>\n<div id=\"q597781\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466216346\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{1}{7}\\normalsize\\cdot 56[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write [latex]56[\/latex] as a fraction.<\/td>\n<td>[latex]\\Large\\frac{1}{7}\\cdot \\frac{56}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Determine the sign of the product; multiply.<\/td>\n<td>[latex]\\Large\\frac{56}{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466048420\" class=\"unnumbered unstyled\" style=\"width: 85%\" summary=\"The first line says 12 fifths times negative 20\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\Large\\frac{12}{5}\\normalsize\\left(-20x\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write [latex]\u221220x[\/latex] as a fraction.<\/td>\n<td>[latex]\\Large\\frac{12}{5}\\left(\\frac{-20x}{1}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Determine the sign of the product; multiply.<\/td>\n<td>[latex]\\Large-\\frac{12\\cdot 20\\cdot x}{5\\cdot 1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Show common factors and then remove them.<\/td>\n<td>[latex]\\Large-\\frac{12\\cdot 4\\cdot {\\color{red}{5}}\\cdot x}{\\color{red}{5}\\cdot1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply remaining factors; simplify.<\/td>\n<td>[latex]\u221248x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146024\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146024&theme=oea&iframe_resize_id=ohm146024&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146025\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146025&theme=oea&iframe_resize_id=ohm146025&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video to see more examples of how to multiply a fraction and a whole number.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex 2: Multiply Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Rxz7OUzNyV0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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-9487\">\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>Ex 2: Multiply Fractions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Rxz7OUzNyV0\">https:\/\/youtu.be\/Rxz7OUzNyV0<\/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>Question ID: 146020, 146021, 146022, 146023, 146024, 146025. <strong>Authored by<\/strong>: Alyson Day. <strong>Provided by<\/strong>: #. <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>: IMathAS Community License CC-BY + GPL<\/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>Ex: Multiplying Signed Fractions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/yUdJ46pTblo\">https:\/\/youtu.be\/yUdJ46pTblo<\/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>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <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@9.757<\/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":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"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\",\"description\":\"Ex: Multiplying Signed Fractions\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/yUdJ46pTblo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Ex 2: Multiply Fractions\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Rxz7OUzNyV0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID: 146020, 146021, 146022, 146023, 146024, 146025\",\"author\":\"Alyson Day\",\"organization\":\"#\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"0678e17bee3a4b39a1b7badfae5b722b, 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