{"id":1404,"date":"2021-11-01T18:17:04","date_gmt":"2021-11-01T18:17:04","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1404"},"modified":"2023-01-13T13:56:00","modified_gmt":"2023-01-13T13:56:00","slug":"1-3-5-multiplying-and-dividing-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/1-3-5-multiplying-and-dividing-fractions\/","title":{"raw":"1.3.4: Multiplying and Dividing Fractions","rendered":"1.3.4: Multiplying and Dividing 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>Find the reciprocal of a fraction<\/li>\r\n \t<li>Divide fractions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key words<\/h3>\r\n<ul>\r\n \t<li><strong>Reciprocal<\/strong>: the inversion of a fraction<\/li>\r\n \t<li><strong>Cancelling<\/strong>: dividing the numerator and denominator by the same common factor<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Multiplication<\/h2>\r\nA model may help us to understand multiplication of fractions. We will use fraction tiles to model [latex]\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]. To multiply [latex]\\frac{1}{2}[\/latex] and [latex]\\frac{3}{4}[\/latex], think [latex]\\frac{1}{2}[\/latex] of [latex]\\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]\\frac{1}{4}[\/latex] tiles evenly into two parts, we use equivalent 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]\\frac{6}{8}[\/latex] is equivalent to [latex]\\frac{3}{4}[\/latex]. Taking half of the six [latex]\\frac{1}{8}[\/latex] tiles gives us three [latex]\\frac{1}{8}[\/latex] tiles, which is [latex]\\frac{3}{8}[\/latex].\r\n\r\nTherefore,\r\n<p style=\"text-align: center;\">[latex]\\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]\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]\r\n\r\nSolution:\r\nFirst shade in [latex]\\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]\\frac{1}{2}[\/latex] of this [latex]\\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]\\frac{3}{8}[\/latex] of the rectangle is heavily shaded.\r\nTherefore, [latex]\\frac{1}{2}[\/latex] of [latex]\\frac{3}{4}[\/latex] is [latex]\\frac{3}{8}[\/latex], or [latex]{\\frac{1}{2}\\cdot \\frac{3}{4}}={\\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]\\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]\\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]\\frac{1\\cdot 3}{2\\cdot 4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\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 integers where [latex]b\\ne 0\\text{ and }d\\ne 0[\/latex], then,\r\n<p style=\"text-align: center;\">[latex]\\frac{a}{b}\\cdot \\frac{c}{d}=\\frac{ac}{bd}[\/latex]<\/p>\r\n\r\n<\/div>\r\nNotice that [latex]b, d\\ne 0[\/latex]. This because we cannot divide by [latex]0[\/latex], so if\u00a0[latex]b[\/latex] or [latex]d=0[\/latex] then the fractions [latex]\\frac{a}{b}[\/latex] or\u00a0[latex]\\frac{c}{d}[\/latex] would be undefined.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMultiply, and write the answer in simplified form: [latex]\\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><\/td>\r\n<td>[latex]\\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]\\frac{3\\cdot 1}{4\\cdot 5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\frac{3}{20}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThere are no common factors (other than 1), so the fraction is simplified.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nMultiply, and write the answer in simplified form: [latex]\\frac{6}{5}\\cdot \\frac{1}{8}[\/latex]\r\n\r\n[reveal-answer q=\"HM385\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"HM385\"]\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><\/td>\r\n<td>[latex]\\frac{6}{5}\\cdot \\frac{1}{8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the numerators; multiply the denominators.<\/td>\r\n<td>[latex]\\frac{6\\cdot 1}{5\\cdot 8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\frac{6}{40}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Look for common factors.<\/td>\r\n<td>[latex]\\frac{\\color{red}{2}\\cdot 3}{\\color{red}{2}\\cdot 20}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\frac{2}{20}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThere are no more common factors (other than 1), so the fraction is simplified.\r\n\r\n[\/hidden-answer]\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]\\frac{a}{b}\\cdot\\frac{c}{d}\\cdot\\frac{e}{f}=\\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]\\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 dividing out the 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{\\color{red}{\\cancel{2}}\\cdot 1\\cdot\\color{blue}{\\cancel{3}}}{\\color{blue}{\\cancel{3}}\\cdot (\\color{red}{\\cancel{2}}\\cdot 2)\\cdot 5}\\\\\\Large\\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>\r\nThis technique of dividing out a number on the numerator with the same number on the denominator is often referred to as\u00a0<em><strong>cancelling<\/strong><\/em>.\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]-\\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]-\\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]\\frac{5\\cdot 2}{8\\cdot 3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\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]\\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]\\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]-\\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]\\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]\\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]\\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]-\\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]-\\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]-\\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]-\\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]-\\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]-\\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]\\frac{a}{1}[\/latex]. For example, [latex]3=\\frac{3}{1}[\/latex].\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]{\\frac{1}{7}}\\cdot 56[\/latex]<\/li>\r\n \t<li>[latex]{\\frac{12}{5}}\\left(-20\\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]\\frac{1}{7}\\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]\\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]\\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]\\frac{12}{5}\\left(-20\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]\u221220x[\/latex] as a fraction.<\/td>\r\n<td>[latex]\\frac{12}{5}\\left(\\frac{-20}{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]-\\frac{12\\cdot 20}{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]-\\frac{12\\cdot 4\\cdot {\\color{red}{5}}}{\\color{red}{5}\\cdot1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply remaining factors; simplify.<\/td>\r\n<td>[latex]\u221248[\/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\nMultiply, and write the answer in simplified form: \u00a0 [latex]-\\frac{4}{15}\\cdot 20[\/latex]\r\n\r\n[reveal-answer q=\"820763\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"820763\"]\r\n\r\n[latex]-\\frac{16}{3}[\/latex]\r\n\r\n[\/hidden-answer]\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\r\n<h2>Finding the Reciprocal of a Number<\/h2>\r\nThe fractions [latex]\\frac{2}{3}[\/latex] and [latex]\\frac{3}{2}[\/latex] are related to each other in a special way. So are [latex]-\\frac{10}{7}[\/latex] and [latex]-\\frac{7}{10}[\/latex]. Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be [latex]1[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\frac{2}{3}\\cdot \\frac{3}{2}=1\\text{ and }-\\frac{10}{7}\\left(-\\frac{7}{10}\\right)=1[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Such pairs of numbers are called <em><strong>reciprocals<\/strong><\/em>.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Reciprocal<\/h3>\r\nThe reciprocal of the fraction [latex]\\frac{a}{b}[\/latex] is [latex]\\frac{b}{a}[\/latex], where [latex]a\\ne 0[\/latex] and [latex]b\\ne 0[\/latex].\r\n\r\nA number and its reciprocal have a product of [latex]1[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\frac{a}{b}\\cdot \\frac{b}{a}=1[\/latex]<\/p>\r\n\r\n<\/div>\r\nHere are some examples of reciprocals:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Original number<\/th>\r\n<th>Reciprocal<\/th>\r\n<th>Product<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\dfrac{3}{4}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{4}{3}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{3}{4}\\cdot\\dfrac{4}{3}=\\dfrac{3\\cdot 4}{4\\cdot 3}=\\dfrac{12}{12}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\dfrac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{2}{1}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{1}{2}\\cdot\\dfrac{2}{1}=\\dfrac{1\\cdot2}{2\\cdot1}=\\dfrac{2}{2}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] 3=\\dfrac{3}{1}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{1}{3}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{3}{1}\\cdot\\dfrac{1}{3}=\\dfrac{3\\cdot 1}{1\\cdot 3}=\\dfrac{3}{3}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\dfrac{7}{3}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{3}{7}[\/latex]<\/td>\r\n<td>[latex]\\dfrac{7}{3}\\cdot\\dfrac{3}{7}=\\dfrac{7\\cdot3}{3\\cdot7}=\\dfrac{21}{21}= 1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.\u00a0 You can think of it as switching the numerator and denominator: swap the [latex]2[\/latex] with the [latex]5[\/latex] in [latex]\\dfrac{2}{5}[\/latex] to get the reciprocal [latex]\\dfrac{5}{2}[\/latex].\r\n\r\nMake sure that if it's a negative fraction, the reciprocal is also negative. This is because the product of two negative numbers will give you the positive one that you are looking for.\u00a0 To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220923\/CNX_BMath_Figure_04_02_035_img.png\" alt=\"\" \/>\r\nTo find the reciprocal, keep the same sign and invert the fraction.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the reciprocal of each number. Then check that the product of each number and its reciprocal is [latex]1[\/latex].\r\n<ol id=\"eip-id1168469776775\" class=\"circled\">\r\n \t<li>[latex]\\frac{4}{9}[\/latex]<\/li>\r\n \t<li>[latex]-\\frac{1}{6}[\/latex]<\/li>\r\n \t<li>[latex]-\\frac{14}{5}[\/latex]<\/li>\r\n \t<li>[latex]7[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\nTo find the reciprocals, we keep the sign and invert the fractions.\r\n<table id=\"eip-374\" class=\"unnumbered unstyled\" style=\"width: 98.25646204586712%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 47.46317512274959%;\">1.<\/td>\r\n<td style=\"width: 65.63011456628477%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.46317512274959%;\">Find the reciprocal of [latex]\\frac{4}{9}[\/latex]<\/td>\r\n<td style=\"width: 65.63011456628477%;\">The reciprocal of [latex]\\frac{4}{9}[\/latex] is [latex]\\frac{9}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.46317512274959%;\">Check:<\/td>\r\n<td style=\"width: 65.63011456628477%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.46317512274959%;\">Multiply the number and its reciprocal.<\/td>\r\n<td style=\"width: 65.63011456628477%;\">[latex]\\frac{4}{9}\\cdot \\frac{9}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.46317512274959%;\">Multiply numerators and denominators.<\/td>\r\n<td style=\"width: 65.63011456628477%;\">[latex]\\frac{36}{36}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.46317512274959%;\">Simplify.<\/td>\r\n<td style=\"width: 65.63011456628477%;\">[latex]1\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-37774\" class=\"unnumbered unstyled\" style=\"width: 98.3325237133434%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 47.42058026237508%;\">2.<\/td>\r\n<td style=\"width: 66.1313843001538%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.42058026237508%;\">Find the reciprocal of [latex]-\\frac{1}{6}[\/latex]<\/td>\r\n<td style=\"width: 66.1313843001538%;\">[latex]-\\frac{6}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.42058026237508%;\">Simplify.<\/td>\r\n<td style=\"width: 66.1313843001538%;\">[latex]-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.42058026237508%;\">Check:<\/td>\r\n<td style=\"width: 66.1313843001538%;\">[latex]-\\frac{1}{6}\\normalsize\\cdot \\left(-6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.42058026237508%;\"><\/td>\r\n<td style=\"width: 66.1313843001538%;\">[latex]1\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1170196522976\" class=\"unnumbered unstyled\" style=\"width: 98.61069209725667%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 47.13584288052373%;\">3.<\/td>\r\n<td style=\"width: 66.85795191994701%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.13584288052373%;\">Find the reciprocal of [latex]-\\frac{14}{5}[\/latex]<\/td>\r\n<td style=\"width: 66.85795191994701%;\">[latex]-\\frac{5}{14}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.13584288052373%;\">Check:<\/td>\r\n<td style=\"width: 66.85795191994701%;\">[latex]-\\frac{14}{5}\\cdot \\left(-\\frac{5}{14}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.13584288052373%;\"><\/td>\r\n<td style=\"width: 66.85795191994701%;\">[latex]\\frac{70}{70}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.13584288052373%;\"><\/td>\r\n<td style=\"width: 66.85795191994701%;\">[latex]1\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1170195508737\" class=\"unnumbered unstyled\" style=\"width: 98.88886048116994%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 47.04906070378735%;\">4.<\/td>\r\n<td style=\"width: 67.38656433462526%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.04906070378735%;\">Find the reciprocal of [latex]7[\/latex]<\/td>\r\n<td style=\"width: 67.38656433462526%;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.04906070378735%;\">Write [latex]7[\/latex] as a fraction.<\/td>\r\n<td style=\"width: 67.38656433462526%;\">[latex]\\frac{7}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.04906070378735%;\">Write the reciprocal of [latex]\\frac{7}{1}[\/latex]<\/td>\r\n<td style=\"width: 67.38656433462526%;\">[latex]\\frac{1}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.04906070378735%;\">Check:<\/td>\r\n<td style=\"width: 67.38656433462526%;\">[latex]7\\cdot\\left(\\frac{1}{7}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 47.04906070378735%;\"><\/td>\r\n<td style=\"width: 67.38656433462526%;\">[latex]1\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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=\"400\"]141842[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182614\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"62\" height=\"55\" \/>Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number a, [latex]\\dfrac{a}{0}[\/latex] is undefined. Additionally, the reciprocal of\u00a0[latex]\\dfrac{0}{a}[\/latex] will always be undefined.<\/div>\r\n<h2>Division by Zero<\/h2>\r\nYou know what it means to divide by [latex]2[\/latex] or divide by [latex]10[\/latex], but what does it mean to divide a quantity by [latex]0[\/latex]? Is this even possible? On the flip side, can you divide [latex]0[\/latex] by a number? Consider the\u00a0fraction\r\n<p style=\"text-align: center;\">[latex]\\dfrac{0}{8}[\/latex]<\/p>\r\nWe can read it as, \u201czero divided by eight.\u201d Since multiplication is the inverse of division, we could rewrite this as a multiplication problem. What number times [latex]8[\/latex] equals [latex]0[\/latex]?\r\n<p style=\"text-align: center;\">[latex]\\text{?}\\cdot{8}=0[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We can infer that the unknown must be [latex]0[\/latex] since that is the only number that will give a result of [latex]0[\/latex] when it is multiplied by [latex]8[\/latex].<\/p>\r\nNow let\u2019s consider the reciprocal of [latex]\\dfrac{0}{8}[\/latex] which would be [latex]\\dfrac{8}{0}[\/latex]. If we\u00a0rewrite this as a multiplication problem, we will have \"what times [latex]0[\/latex] equals [latex]8[\/latex]?\"\r\n<p style=\"text-align: center;\">[latex]\\text{?}\\cdot{0}=8[\/latex]<\/p>\r\nThis doesn't make any sense. There are no numbers that you can multiply by zero to get a result of 8. In fact, any number divided by [latex]0[\/latex] is impossible, or better stated, <strong>all division by zero is undefined<\/strong>.\r\n<h2>Divide Fractions<\/h2>\r\nThere are times when you need to use division to solve a problem. For example, if painting one coat of paint on the walls of a room requires [latex]3[\/latex] quarts of paint and you have\u00a0a bucket that contains [latex]6[\/latex] quarts of paint, how many coats of paint can you paint on the walls? You divide [latex]6[\/latex] by [latex]3[\/latex] for an answer of [latex]2[\/latex]\u00a0coats. There will also be times when you need to divide by a fraction. Suppose painting a closet with one coat only required [latex]\\dfrac{1}{2}[\/latex] quart of paint. How many coats could be painted with the 6 quarts of paint? To find the answer, you need to divide [latex]6[\/latex]\u00a0by the fraction, [latex]\\dfrac{1}{2}[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Dividing is Multiplying by the Reciprocal<\/h3>\r\nFor all division, you can turn the operation\u00a0into multiplication by using the reciprocal. Dividing is the same as multiplying by the reciprocal.\r\n\r\n<\/div>\r\nIf you have a recipe that needs to be divided in half, you can divide each ingredient by [latex]2[\/latex], or you can multiply each ingredient by [latex]\\dfrac{1}{2}[\/latex]\u00a0to find the new amount.\r\n\r\nIf you have [latex]\\dfrac{3}{4}[\/latex] of a candy bar and need to divide it among [latex]5[\/latex] people, each person gets [latex]\\dfrac{1}{5}[\/latex] of the available candy:\r\n<p style=\"text-align: center;\">[latex]\\dfrac{1}{5}\\text{ of }\\dfrac{3}{4}=\\dfrac{1}{5}\\cdot\\dfrac{3}{4}=\\dfrac{3}{20}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">Each person gets [latex]\\dfrac{3}{20}[\/latex]\u00a0of a whole candy bar.<\/p>\r\nIf you have [latex]\\dfrac{3}{2}[\/latex] of a pizza left over, how can you divide what is left (the red shaded region) among [latex]6[\/latex] people fairly?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182616\/image143.gif\" alt=\"Two pizzas divided into fourths. One pizza has all four pieces shaded, and the other pizza has two of the four slices shaded. 3\/2 divided by 6 is equal to 3\/2 times 1\/6. This is 3\/2 times 1\/6 equals 1\/4.\" width=\"360\" height=\"239\" \/>\r\n\r\nEach person gets one piece, so each person gets [latex]\\dfrac{1}{4}[\/latex] of a pizza.\r\n<h2>Divide a Whole Number by a Fraction<\/h2>\r\nLet\u2019s use money to model [latex]2\\div\\frac{1}{4}[\/latex]. We often read [latex]\\frac{1}{4}[\/latex] as a \u2018quarter\u2019, and we know that a quarter is one-fourth of a dollar as shown in the image below. So we can think of [latex]2\\div\\frac{1}{4}[\/latex] as, \"How many quarters are there in two dollars?\" One dollar is [latex]4[\/latex] quarters, so [latex]2[\/latex] dollars would be [latex]8[\/latex] quarters. So again,\u00a0[latex]2\\div\\dfrac{1}{4}=\\dfrac{2}{1}\\cdot\\dfrac{4}{1}=8[\/latex].\r\n\r\nLet's look at another way to model [latex]2\\div\\frac{1}{4}[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide: [latex]2\\div\\frac{1}{4}[\/latex]\r\n[reveal-answer q=\"391699\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"391699\"]\r\n\r\nSolution:\r\nWe are trying to determine how many [latex]\\frac{1}{4}\\text{s}[\/latex] there are in [latex]2[\/latex]. We can model this as shown.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220935\/CNX_BMath_Figure_04_02_020_img.png\" alt=\"Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into four pieces. Each of the eight pieces is labeled as one fourth.\" \/>\r\nWe can see that there are eight [latex]\\frac{1}{4}\\text{s}[\/latex] in [latex]2[\/latex].\u00a0 Because dividing by a fraction is the same as multiplying by its reciprocal, [latex]2\\div\\dfrac{1}{4}=\\dfrac{2}{1}\\cdot\\dfrac{4}{1}=8[\/latex].\r\n<h4>Answer<\/h4>\r\n[latex]2\\div\\dfrac{1}{4}= 8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDivide: [latex]2\\div\\frac{1}{3}[\/latex]\r\n[reveal-answer q=\"73567\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"73567\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220936\/CNX_BMath_Figure_04_02_021_img.png\" alt=\"Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into three pieces. Each of the six pieces is labeled as one third.\" \/>\r\n\r\n&nbsp;\r\n\r\nDividing by [latex]\\dfrac{1}{3}[\/latex] is the same as multiplying by the reciprocal of [latex]\\dfrac{1}{3}[\/latex], which is [latex]\\dfrac{3}{1}[\/latex].\r\n<p style=\"text-align: center;\">[latex]2\\div\\dfrac{1}{3}=\\dfrac{2}{1}\\cdot\\dfrac{3}{1}=6[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]2\\div\\dfrac{1}{3}= 6[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nDivide: [latex]3\\div\\frac{1}{2}[\/latex]\r\n[reveal-answer q=\"354856\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"354856\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220938\/CNX_BMath_Figure_04_02_022_img.png\" alt=\"Three rectangles are shown, each labeled as 1. Below are three identical rectangles, each split into 2 equal pieces. Each of these six pieces is labeled as one half.\" \/>\r\n\r\nDividing by [latex]\\dfrac{1}{2}[\/latex] is the same as multiplying by the reciprocal of [latex]\\dfrac{1}{2}[\/latex], which is [latex]\\dfrac{2}{1}[\/latex].\r\n<p style=\"text-align: center;\">[latex]3\\div\\dfrac{1}{2}=\\dfrac{3}{1}\\cdot\\dfrac{2}{1}=6[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]3\\div\\dfrac{1}{2}=\\normalsize 6[\/latex]\r\n\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=\"800\"]117216[\/ohm_question]\r\n\r\n<\/div>\r\nThe next video shows more examples of how to divide a whole number by a fraction.\r\n\r\nhttps:\/\/youtu.be\/JKsfdK1WT1s\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide. [latex] 9\\div\\dfrac{1}{2}[\/latex]\r\n\r\n[reveal-answer q=\"269187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"269187\"]\r\n\r\nWrite your answer in lowest terms.\r\n\r\nDividing by [latex]\\dfrac{1}{2}[\/latex] is the same as multiplying by the reciprocal of [latex]\\dfrac{1}{2}[\/latex], which is [latex]\\dfrac{2}{1}[\/latex].\r\n<p style=\"text-align: center;\">[latex]9\\div\\dfrac{1}{2}=\\dfrac{9}{1}\\cdot\\dfrac{2}{1}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{9\\cdot 2}{1\\cdot 1}=\\dfrac{18}{1}= 18[\/latex]<\/p>\r\nThis answer is already simplified to lowest terms.\r\n<h4>Answer<\/h4>\r\n[latex]9\\div\\dfrac{1}{2}= 18[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Divide a Fraction by a Fraction<\/h2>\r\nSometimes we need to solve a problem that requires dividing a fraction by a fraction. Suppose we want to find the quotient: [latex]\\frac{1}{2}\\div\\frac{1}{6}[\/latex]. We need to figure out how many [latex]\\frac{1}{6}\\text{s}[\/latex] there are in [latex]\\frac{1}{2}[\/latex]. We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown below. Notice, there are three [latex]\\frac{1}{6}[\/latex] tiles in [latex]\\frac{1}{2}[\/latex], so [latex]\\frac{1}{2}\\div\\frac{1}{6}=3[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220930\/CNX_BMath_Figure_04_02_016.png\" alt=\"A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth.\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nModel: [latex]\\frac{1}{4}\\div\\frac{1}{8}[\/latex]\r\n\r\nSolution:\r\nWe want to determine how many [latex]\\frac{1}{8}\\text{s}[\/latex] are in [latex]\\frac{1}{4}[\/latex]. Start with one [latex]\\frac{1}{4}[\/latex] tile. Line up [latex]\\frac{1}{8}[\/latex] tiles underneath the [latex]\\frac{1}{4}[\/latex] tile.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220931\/CNX_BMath_Figure_04_02_017_img.png\" alt=\"A rectangle is shown, labeled one fourth. Below it is an identical rectangle split into two equal pieces, each labeled as one eighth.\" \/>\r\nThere are two [latex]\\frac{1}{8}[\/latex]s in [latex]\\frac{1}{4}[\/latex].\r\nSo, [latex]\\frac{1}{4}\\div\\frac{1}{8}=2[\/latex].<\/p>\r\n\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nModel: [latex]\\frac{1}{3}\\div\\frac{1}{6}[\/latex]\r\n[reveal-answer q=\"218091\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"218091\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220932\/CNX_BMath_Figure_04_02_018_img.png\" alt=\"A rectangle is shown, labeled as one third. Below it is an identical rectangle split into two equal pieces, each labeled as one sixth.\" \/>\r\n\r\n[latex]2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nModel: [latex]\\frac{1}{2}\\div\\frac{1}{4}[\/latex]\r\n[reveal-answer q=\"763601\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"763601\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220934\/CNX_BMath_Figure_04_02_019_img.png\" alt=\"A rectangle is shown, labeled as one half. Below it is an identical rectangle split into two equal pieces, each labeled as one fourth.\" \/>\r\n\r\n[latex]2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows another way to model division of two fractions.\r\n\r\nhttps:\/\/youtu.be\/pk-K5JF9iMo\r\n\r\nUsing fraction tiles, we showed that [latex]\\frac{1}{2}\\div\\frac{1}{6}=3[\/latex]. Notice that [latex]\\frac{1}{2}\\cdot \\frac{6}{1}=3[\/latex] also. How are [latex]\\frac{1}{6}[\/latex] and [latex]\\frac{6}{1}[\/latex] related? They are reciprocals. This leads us to the procedure for fraction division.\u00a0 Suppose we have a pizza that is already cut into [latex]4[\/latex] slices. How many [latex]\\dfrac{1}{2}[\/latex] slices are there?\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182618\/image146.gif\" alt=\"A pizza divided into four equal pieces. There are four slices.\" width=\"180\" height=\"179\" \/><\/td>\r\n<td><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182619\/image147.gif\" alt=\"A pizza divided into four equal slices. Each slice is then divided in half. There are now 8 slices.\" width=\"180\" height=\"179\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThere are [latex]8[\/latex] slices. You can see that dividing [latex]4[\/latex] by [latex]\\dfrac{1}{2}[\/latex] gives the same result as multiplying [latex]4[\/latex] by [latex]2[\/latex].\r\n\r\nWhat would happen if you needed to divide each slice into thirds?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182621\/image148.gif\" alt=\"A pizza divided into four equal slice. Each slice is divided into thirds. There are now 12 slices.\" width=\"180\" height=\"179\" \/>\r\n\r\nYou would have [latex]12[\/latex] slices, which is the same as multiplying [latex]4[\/latex] by [latex]3[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Fraction Division<\/h3>\r\nIf [latex]a,b,c,\\text{ and }d[\/latex] are numbers where [latex]b\\ne 0,c\\ne 0,\\text{ and }d\\ne 0[\/latex], then\r\n<p style=\"text-align: center;\">[latex]\\frac{a}{b}\\div\\frac{c}{d}=\\frac{a}{b}\\cdot \\frac{d}{c}[\/latex]<\/p>\r\nTo divide fractions, multiply the first fraction by the reciprocal of the second.\r\n\r\nWe need to say [latex]b\\ne 0,c\\ne 0\\text{ and }d\\ne 0[\/latex] to be sure we don\u2019t divide by zero.\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Dividing with Fractions<\/h3>\r\n<ol>\r\n \t<li>Find the reciprocal of the divisor (the number that follows the division symbol).<\/li>\r\n \t<li>Multiply the dividend (the number before the division symbol) by the reciprocal of the divisor (the number after the division symbol).<\/li>\r\n<\/ol>\r\n<\/div>\r\nAny easy way to remember how to divide fractions is the phrase \u201ckeep, change, flip.\u201d This means to <strong>KEEP<\/strong> the first number, <strong>CHANGE<\/strong> the division sign to multiplication, and then <strong>FLIP<\/strong> (use the reciprocal) of the second number.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide [latex]\\dfrac{2}{3}\\div\\dfrac{1}{6}[\/latex]\r\n\r\n[reveal-answer q=\"569112\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"569112\"]\r\n\r\nMultiply by the reciprocal.\r\n\r\n<strong>KEEP<\/strong> [latex]\\dfrac{2}{3}[\/latex]\r\n\r\n<strong>CHANGE<\/strong>\u00a0 [latex] \\div [\/latex] to \u00a0[latex]\\cdot[\/latex]\r\n\r\n<strong>FLIP\u00a0<\/strong> [latex]\\dfrac{1}{6}[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\dfrac{2}{3}\\cdot\\dfrac{6}{1}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{2\\cdot6}{3\\cdot1}=\\dfrac{12}{3}[\/latex]<\/p>\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex]\\dfrac{12}{3}=\\normalsize 4[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\dfrac{2}{3}\\div\\dfrac{1}{6}=4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide [latex]-\\dfrac{3}{5}\\div\\dfrac{2}{3}[\/latex]\r\n\r\n[reveal-answer q=\"950676\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"950676\"]\r\n\r\nMultiply by the reciprocal.\u00a0Keep [latex]-\\dfrac{3}{5}[\/latex], change [latex] \\div [\/latex] to [latex]\\cdot[\/latex], and flip [latex]\\dfrac{2}{3}[\/latex].\r\n<p style=\"text-align: center;\">[latex]-\\dfrac{3}{5}\\cdot\\dfrac{3}{2}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center;\">[latex]-\\dfrac{3\\cdot 3}{5\\cdot 2}=-\\dfrac{9}{10}[\/latex]<\/p>\r\nThere are no common factors (other than 1), so the fraction is simplified.\r\n<h4>Answer<\/h4>\r\n[latex]-\\dfrac{3}{5}\\div\\dfrac{2}{3}=-\\dfrac{9}{10}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWhen solving a division problem by multiplying by the reciprocal, remember to write all whole numbers as improper fractions before doing calculations\u00a0 [latex](\\text{i.e. } 5=\\dfrac{5}{1}[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]263630[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide, and write the answer in simplified form: [latex]\\frac{2}{5}\\div\\left(-\\frac{3}{7}\\right)[\/latex]\r\n[reveal-answer q=\"261121\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"261121\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168468274991\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\frac{2}{5}\\normalsize\\div\\Large\\left(-\\frac{3}{7}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td>[latex]\\frac{2}{5}\\left(-\\frac{7}{3}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply. The product is negative.<\/td>\r\n<td>[latex]-\\frac{14}{15}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide, and write the answer in simplified form: [latex]-\\frac{3}{4}\\div\\left(-\\frac{7}{8}\\right)[\/latex]\r\n[reveal-answer q=\"873547\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"873547\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467263034\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 55.64648117839607%;\"><\/td>\r\n<td style=\"width: 44.189852700491%;\">[latex]-\\frac{3}{4}\\div\\left(-\\frac{7}{8}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 55.64648117839607%;\">Multiply the first fraction by the reciprocal of the second.<\/td>\r\n<td style=\"width: 44.189852700491%;\">[latex]-\\frac{3}{4}\\cdot \\left(-\\frac{8}{7}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 55.64648117839607%;\">Multiply. Remember to determine the sign first.<\/td>\r\n<td style=\"width: 44.189852700491%;\">[latex]\\frac{3\\cdot 8}{4\\cdot 7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 55.64648117839607%;\">Rewrite to show common factors.<\/td>\r\n<td style=\"width: 44.189852700491%;\">[latex]\\frac{3\\cdot 4\\cdot 2}{4\\cdot 7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 55.64648117839607%;\">Remove common factors and simplify.<\/td>\r\n<td style=\"width: 44.189852700491%;\">[latex]\\frac{6}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]146066[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video shows more examples of dividing fractions that are negative.\r\n\r\nhttps:\/\/youtu.be\/OPHdadhDJoI","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply two or more fractions<\/li>\n<li>Find the reciprocal of a fraction<\/li>\n<li>Divide fractions<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li><strong>Reciprocal<\/strong>: the inversion of a fraction<\/li>\n<li><strong>Cancelling<\/strong>: dividing the numerator and denominator by the same common factor<\/li>\n<\/ul>\n<\/div>\n<h2>Multiplication<\/h2>\n<p>A model may help us to understand multiplication of fractions. We will use fraction tiles to model [latex]\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]. To multiply [latex]\\frac{1}{2}[\/latex] and [latex]\\frac{3}{4}[\/latex], think [latex]\\frac{1}{2}[\/latex] of [latex]\\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]\\frac{1}{4}[\/latex] tiles evenly into two parts, we use equivalent 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]\\frac{6}{8}[\/latex] is equivalent to [latex]\\frac{3}{4}[\/latex]. Taking half of the six [latex]\\frac{1}{8}[\/latex] tiles gives us three [latex]\\frac{1}{8}[\/latex] tiles, which is [latex]\\frac{3}{8}[\/latex].<\/p>\n<p>Therefore,<\/p>\n<p style=\"text-align: center;\">[latex]\\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]\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]<\/p>\n<p>Solution:<br \/>\nFirst shade in [latex]\\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]\\frac{1}{2}[\/latex] of this [latex]\\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]\\frac{3}{8}[\/latex] of the rectangle is heavily shaded.<br \/>\nTherefore, [latex]\\frac{1}{2}[\/latex] of [latex]\\frac{3}{4}[\/latex] is [latex]\\frac{3}{8}[\/latex], or [latex]{\\frac{1}{2}\\cdot \\frac{3}{4}}={\\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]\\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]\\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]\\frac{1\\cdot 3}{2\\cdot 4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\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 integers where [latex]b\\ne 0\\text{ and }d\\ne 0[\/latex], then,<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{a}{b}\\cdot \\frac{c}{d}=\\frac{ac}{bd}[\/latex]<\/p>\n<\/div>\n<p>Notice that [latex]b, d\\ne 0[\/latex]. This because we cannot divide by [latex]0[\/latex], so if\u00a0[latex]b[\/latex] or [latex]d=0[\/latex] then the fractions [latex]\\frac{a}{b}[\/latex] or\u00a0[latex]\\frac{c}{d}[\/latex] would be undefined.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Multiply, and write the answer in simplified form: [latex]\\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><\/td>\n<td>[latex]\\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]\\frac{3\\cdot 1}{4\\cdot 5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\frac{3}{20}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>There are no common factors (other than 1), so the fraction is simplified.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Multiply, and write the answer in simplified form: [latex]\\frac{6}{5}\\cdot \\frac{1}{8}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qHM385\">Show Solution<\/span><\/p>\n<div id=\"qHM385\" 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><\/td>\n<td>[latex]\\frac{6}{5}\\cdot \\frac{1}{8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the numerators; multiply the denominators.<\/td>\n<td>[latex]\\frac{6\\cdot 1}{5\\cdot 8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\frac{6}{40}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Look for common factors.<\/td>\n<td>[latex]\\frac{\\color{red}{2}\\cdot 3}{\\color{red}{2}\\cdot 20}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\frac{2}{20}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>There are no more common factors (other than 1), so the fraction is simplified.<\/p>\n<\/div>\n<\/div>\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]\\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<p>&nbsp;<\/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 dividing out the 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{\\color{red}{\\cancel{2}}\\cdot 1\\cdot\\color{blue}{\\cancel{3}}}{\\color{blue}{\\cancel{3}}\\cdot (\\color{red}{\\cancel{2}}\\cdot 2)\\cdot 5}\\\\\\Large\\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<p>This technique of dividing out a number on the numerator with the same number on the denominator is often referred to as\u00a0<em><strong>cancelling<\/strong><\/em>.<\/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]-\\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]-\\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]\\frac{5\\cdot 2}{8\\cdot 3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\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]\\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]\\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]-\\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]\\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]\\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]\\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]-\\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]-\\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]-\\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]-\\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]-\\frac{2\\cdot 4}{3\\cdot 3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the remaining factors.<\/td>\n<td>[latex]-\\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]\\frac{a}{1}[\/latex]. For example, [latex]3=\\frac{3}{1}[\/latex].<\/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]{\\frac{1}{7}}\\cdot 56[\/latex]<\/li>\n<li>[latex]{\\frac{12}{5}}\\left(-20\\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]\\frac{1}{7}\\cdot 56[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write [latex]56[\/latex] as a fraction.<\/td>\n<td>[latex]\\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]\\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]\\frac{12}{5}\\left(-20\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write [latex]\u221220x[\/latex] as a fraction.<\/td>\n<td>[latex]\\frac{12}{5}\\left(\\frac{-20}{1}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Determine the sign of the product; multiply.<\/td>\n<td>[latex]-\\frac{12\\cdot 20}{5\\cdot 1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Show common factors and then remove them.<\/td>\n<td>[latex]-\\frac{12\\cdot 4\\cdot {\\color{red}{5}}}{\\color{red}{5}\\cdot1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply remaining factors; simplify.<\/td>\n<td>[latex]\u221248[\/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>Multiply, and write the answer in simplified form: \u00a0 [latex]-\\frac{4}{15}\\cdot 20[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q820763\">Show Answer<\/span><\/p>\n<div id=\"q820763\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-\\frac{16}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\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<h2>Finding the Reciprocal of a Number<\/h2>\n<p>The fractions [latex]\\frac{2}{3}[\/latex] and [latex]\\frac{3}{2}[\/latex] are related to each other in a special way. So are [latex]-\\frac{10}{7}[\/latex] and [latex]-\\frac{7}{10}[\/latex]. Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be [latex]1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2}{3}\\cdot \\frac{3}{2}=1\\text{ and }-\\frac{10}{7}\\left(-\\frac{7}{10}\\right)=1[\/latex]<\/p>\n<p style=\"text-align: left;\">Such pairs of numbers are called <em><strong>reciprocals<\/strong><\/em>.<\/p>\n<div class=\"textbox shaded\">\n<h3>Reciprocal<\/h3>\n<p>The reciprocal of the fraction [latex]\\frac{a}{b}[\/latex] is [latex]\\frac{b}{a}[\/latex], where [latex]a\\ne 0[\/latex] and [latex]b\\ne 0[\/latex].<\/p>\n<p>A number and its reciprocal have a product of [latex]1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{a}{b}\\cdot \\frac{b}{a}=1[\/latex]<\/p>\n<\/div>\n<p>Here are some examples of reciprocals:<\/p>\n<table>\n<thead>\n<tr>\n<th>Original number<\/th>\n<th>Reciprocal<\/th>\n<th>Product<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\dfrac{3}{4}[\/latex]<\/td>\n<td>[latex]\\dfrac{4}{3}[\/latex]<\/td>\n<td>[latex]\\dfrac{3}{4}\\cdot\\dfrac{4}{3}=\\dfrac{3\\cdot 4}{4\\cdot 3}=\\dfrac{12}{12}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\dfrac{1}{2}[\/latex]<\/td>\n<td>[latex]\\dfrac{2}{1}[\/latex]<\/td>\n<td>[latex]\\dfrac{1}{2}\\cdot\\dfrac{2}{1}=\\dfrac{1\\cdot2}{2\\cdot1}=\\dfrac{2}{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3=\\dfrac{3}{1}[\/latex]<\/td>\n<td>[latex]\\dfrac{1}{3}[\/latex]<\/td>\n<td>[latex]\\dfrac{3}{1}\\cdot\\dfrac{1}{3}=\\dfrac{3\\cdot 1}{1\\cdot 3}=\\dfrac{3}{3}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\dfrac{7}{3}[\/latex]<\/td>\n<td>[latex]\\dfrac{3}{7}[\/latex]<\/td>\n<td>[latex]\\dfrac{7}{3}\\cdot\\dfrac{3}{7}=\\dfrac{7\\cdot3}{3\\cdot7}=\\dfrac{21}{21}= 1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.\u00a0 You can think of it as switching the numerator and denominator: swap the [latex]2[\/latex] with the [latex]5[\/latex] in [latex]\\dfrac{2}{5}[\/latex] to get the reciprocal [latex]\\dfrac{5}{2}[\/latex].<\/p>\n<p>Make sure that if it&#8217;s a negative fraction, the reciprocal is also negative. This is because the product of two negative numbers will give you the positive one that you are looking for.\u00a0 To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.<\/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\/24220923\/CNX_BMath_Figure_04_02_035_img.png\" alt=\"\" \/><br \/>\nTo find the reciprocal, keep the same sign and invert the fraction.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the reciprocal of each number. Then check that the product of each number and its reciprocal is [latex]1[\/latex].<\/p>\n<ol id=\"eip-id1168469776775\" class=\"circled\">\n<li>[latex]\\frac{4}{9}[\/latex]<\/li>\n<li>[latex]-\\frac{1}{6}[\/latex]<\/li>\n<li>[latex]-\\frac{14}{5}[\/latex]<\/li>\n<li>[latex]7[\/latex]<\/li>\n<\/ol>\n<p>Solution:<br \/>\nTo find the reciprocals, we keep the sign and invert the fractions.<\/p>\n<table id=\"eip-374\" class=\"unnumbered unstyled\" style=\"width: 98.25646204586712%;\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 47.46317512274959%;\">1.<\/td>\n<td style=\"width: 65.63011456628477%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.46317512274959%;\">Find the reciprocal of [latex]\\frac{4}{9}[\/latex]<\/td>\n<td style=\"width: 65.63011456628477%;\">The reciprocal of [latex]\\frac{4}{9}[\/latex] is [latex]\\frac{9}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.46317512274959%;\">Check:<\/td>\n<td style=\"width: 65.63011456628477%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.46317512274959%;\">Multiply the number and its reciprocal.<\/td>\n<td style=\"width: 65.63011456628477%;\">[latex]\\frac{4}{9}\\cdot \\frac{9}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.46317512274959%;\">Multiply numerators and denominators.<\/td>\n<td style=\"width: 65.63011456628477%;\">[latex]\\frac{36}{36}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.46317512274959%;\">Simplify.<\/td>\n<td style=\"width: 65.63011456628477%;\">[latex]1\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-37774\" class=\"unnumbered unstyled\" style=\"width: 98.3325237133434%;\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 47.42058026237508%;\">2.<\/td>\n<td style=\"width: 66.1313843001538%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.42058026237508%;\">Find the reciprocal of [latex]-\\frac{1}{6}[\/latex]<\/td>\n<td style=\"width: 66.1313843001538%;\">[latex]-\\frac{6}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.42058026237508%;\">Simplify.<\/td>\n<td style=\"width: 66.1313843001538%;\">[latex]-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.42058026237508%;\">Check:<\/td>\n<td style=\"width: 66.1313843001538%;\">[latex]-\\frac{1}{6}\\normalsize\\cdot \\left(-6\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.42058026237508%;\"><\/td>\n<td style=\"width: 66.1313843001538%;\">[latex]1\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1170196522976\" class=\"unnumbered unstyled\" style=\"width: 98.61069209725667%;\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 47.13584288052373%;\">3.<\/td>\n<td style=\"width: 66.85795191994701%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.13584288052373%;\">Find the reciprocal of [latex]-\\frac{14}{5}[\/latex]<\/td>\n<td style=\"width: 66.85795191994701%;\">[latex]-\\frac{5}{14}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.13584288052373%;\">Check:<\/td>\n<td style=\"width: 66.85795191994701%;\">[latex]-\\frac{14}{5}\\cdot \\left(-\\frac{5}{14}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.13584288052373%;\"><\/td>\n<td style=\"width: 66.85795191994701%;\">[latex]\\frac{70}{70}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.13584288052373%;\"><\/td>\n<td style=\"width: 66.85795191994701%;\">[latex]1\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1170195508737\" class=\"unnumbered unstyled\" style=\"width: 98.88886048116994%;\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 47.04906070378735%;\">4.<\/td>\n<td style=\"width: 67.38656433462526%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.04906070378735%;\">Find the reciprocal of [latex]7[\/latex]<\/td>\n<td style=\"width: 67.38656433462526%;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.04906070378735%;\">Write [latex]7[\/latex] as a fraction.<\/td>\n<td style=\"width: 67.38656433462526%;\">[latex]\\frac{7}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.04906070378735%;\">Write the reciprocal of [latex]\\frac{7}{1}[\/latex]<\/td>\n<td style=\"width: 67.38656433462526%;\">[latex]\\frac{1}{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.04906070378735%;\">Check:<\/td>\n<td style=\"width: 67.38656433462526%;\">[latex]7\\cdot\\left(\\frac{1}{7}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 47.04906070378735%;\"><\/td>\n<td style=\"width: 67.38656433462526%;\">[latex]1\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm141842\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=141842&theme=oea&iframe_resize_id=ohm141842&show_question_numbers\" width=\"100%\" height=\"400\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182614\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"62\" height=\"55\" \/>Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number a, [latex]\\dfrac{a}{0}[\/latex] is undefined. Additionally, the reciprocal of\u00a0[latex]\\dfrac{0}{a}[\/latex] will always be undefined.<\/div>\n<h2>Division by Zero<\/h2>\n<p>You know what it means to divide by [latex]2[\/latex] or divide by [latex]10[\/latex], but what does it mean to divide a quantity by [latex]0[\/latex]? Is this even possible? On the flip side, can you divide [latex]0[\/latex] by a number? Consider the\u00a0fraction<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{0}{8}[\/latex]<\/p>\n<p>We can read it as, \u201czero divided by eight.\u201d Since multiplication is the inverse of division, we could rewrite this as a multiplication problem. What number times [latex]8[\/latex] equals [latex]0[\/latex]?<\/p>\n<p style=\"text-align: center;\">[latex]\\text{?}\\cdot{8}=0[\/latex]<\/p>\n<p style=\"text-align: left;\">We can infer that the unknown must be [latex]0[\/latex] since that is the only number that will give a result of [latex]0[\/latex] when it is multiplied by [latex]8[\/latex].<\/p>\n<p>Now let\u2019s consider the reciprocal of [latex]\\dfrac{0}{8}[\/latex] which would be [latex]\\dfrac{8}{0}[\/latex]. If we\u00a0rewrite this as a multiplication problem, we will have &#8220;what times [latex]0[\/latex] equals [latex]8[\/latex]?&#8221;<\/p>\n<p style=\"text-align: center;\">[latex]\\text{?}\\cdot{0}=8[\/latex]<\/p>\n<p>This doesn&#8217;t make any sense. There are no numbers that you can multiply by zero to get a result of 8. In fact, any number divided by [latex]0[\/latex] is impossible, or better stated, <strong>all division by zero is undefined<\/strong>.<\/p>\n<h2>Divide Fractions<\/h2>\n<p>There are times when you need to use division to solve a problem. For example, if painting one coat of paint on the walls of a room requires [latex]3[\/latex] quarts of paint and you have\u00a0a bucket that contains [latex]6[\/latex] quarts of paint, how many coats of paint can you paint on the walls? You divide [latex]6[\/latex] by [latex]3[\/latex] for an answer of [latex]2[\/latex]\u00a0coats. There will also be times when you need to divide by a fraction. Suppose painting a closet with one coat only required [latex]\\dfrac{1}{2}[\/latex] quart of paint. How many coats could be painted with the 6 quarts of paint? To find the answer, you need to divide [latex]6[\/latex]\u00a0by the fraction, [latex]\\dfrac{1}{2}[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Dividing is Multiplying by the Reciprocal<\/h3>\n<p>For all division, you can turn the operation\u00a0into multiplication by using the reciprocal. Dividing is the same as multiplying by the reciprocal.<\/p>\n<\/div>\n<p>If you have a recipe that needs to be divided in half, you can divide each ingredient by [latex]2[\/latex], or you can multiply each ingredient by [latex]\\dfrac{1}{2}[\/latex]\u00a0to find the new amount.<\/p>\n<p>If you have [latex]\\dfrac{3}{4}[\/latex] of a candy bar and need to divide it among [latex]5[\/latex] people, each person gets [latex]\\dfrac{1}{5}[\/latex] of the available candy:<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{1}{5}\\text{ of }\\dfrac{3}{4}=\\dfrac{1}{5}\\cdot\\dfrac{3}{4}=\\dfrac{3}{20}[\/latex]<\/p>\n<p style=\"text-align: center;\">Each person gets [latex]\\dfrac{3}{20}[\/latex]\u00a0of a whole candy bar.<\/p>\n<p>If you have [latex]\\dfrac{3}{2}[\/latex] of a pizza left over, how can you divide what is left (the red shaded region) among [latex]6[\/latex] people fairly?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182616\/image143.gif\" alt=\"Two pizzas divided into fourths. One pizza has all four pieces shaded, and the other pizza has two of the four slices shaded. 3\/2 divided by 6 is equal to 3\/2 times 1\/6. This is 3\/2 times 1\/6 equals 1\/4.\" width=\"360\" height=\"239\" \/><\/p>\n<p>Each person gets one piece, so each person gets [latex]\\dfrac{1}{4}[\/latex] of a pizza.<\/p>\n<h2>Divide a Whole Number by a Fraction<\/h2>\n<p>Let\u2019s use money to model [latex]2\\div\\frac{1}{4}[\/latex]. We often read [latex]\\frac{1}{4}[\/latex] as a \u2018quarter\u2019, and we know that a quarter is one-fourth of a dollar as shown in the image below. So we can think of [latex]2\\div\\frac{1}{4}[\/latex] as, &#8220;How many quarters are there in two dollars?&#8221; One dollar is [latex]4[\/latex] quarters, so [latex]2[\/latex] dollars would be [latex]8[\/latex] quarters. So again,\u00a0[latex]2\\div\\dfrac{1}{4}=\\dfrac{2}{1}\\cdot\\dfrac{4}{1}=8[\/latex].<\/p>\n<p>Let&#8217;s look at another way to model [latex]2\\div\\frac{1}{4}[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide: [latex]2\\div\\frac{1}{4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q391699\">Show Solution<\/span><\/p>\n<div id=\"q391699\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<br \/>\nWe are trying to determine how many [latex]\\frac{1}{4}\\text{s}[\/latex] there are in [latex]2[\/latex]. We can model this as shown.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220935\/CNX_BMath_Figure_04_02_020_img.png\" alt=\"Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into four pieces. Each of the eight pieces is labeled as one fourth.\" \/><br \/>\nWe can see that there are eight [latex]\\frac{1}{4}\\text{s}[\/latex] in [latex]2[\/latex].\u00a0 Because dividing by a fraction is the same as multiplying by its reciprocal, [latex]2\\div\\dfrac{1}{4}=\\dfrac{2}{1}\\cdot\\dfrac{4}{1}=8[\/latex].<\/p>\n<h4>Answer<\/h4>\n<p>[latex]2\\div\\dfrac{1}{4}= 8[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Divide: [latex]2\\div\\frac{1}{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q73567\">Show Solution<\/span><\/p>\n<div id=\"q73567\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220936\/CNX_BMath_Figure_04_02_021_img.png\" alt=\"Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into three pieces. Each of the six pieces is labeled as one third.\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Dividing by [latex]\\dfrac{1}{3}[\/latex] is the same as multiplying by the reciprocal of [latex]\\dfrac{1}{3}[\/latex], which is [latex]\\dfrac{3}{1}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]2\\div\\dfrac{1}{3}=\\dfrac{2}{1}\\cdot\\dfrac{3}{1}=6[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]2\\div\\dfrac{1}{3}= 6[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Divide: [latex]3\\div\\frac{1}{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q354856\">Show Solution<\/span><\/p>\n<div id=\"q354856\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220938\/CNX_BMath_Figure_04_02_022_img.png\" alt=\"Three rectangles are shown, each labeled as 1. Below are three identical rectangles, each split into 2 equal pieces. Each of these six pieces is labeled as one half.\" \/><\/p>\n<p>Dividing by [latex]\\dfrac{1}{2}[\/latex] is the same as multiplying by the reciprocal of [latex]\\dfrac{1}{2}[\/latex], which is [latex]\\dfrac{2}{1}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3\\div\\dfrac{1}{2}=\\dfrac{3}{1}\\cdot\\dfrac{2}{1}=6[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3\\div\\dfrac{1}{2}=\\normalsize 6[\/latex]<\/p>\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=\"ohm117216\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=117216&theme=oea&iframe_resize_id=ohm117216&show_question_numbers\" width=\"100%\" height=\"800\"><\/iframe><\/p>\n<\/div>\n<p>The next video shows more examples of how to divide a whole number by a fraction.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex: Find the Quotient of a Whole Number and Fraction using Fraction Strips\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/JKsfdK1WT1s?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide. [latex]9\\div\\dfrac{1}{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q269187\">Show Solution<\/span><\/p>\n<div id=\"q269187\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write your answer in lowest terms.<\/p>\n<p>Dividing by [latex]\\dfrac{1}{2}[\/latex] is the same as multiplying by the reciprocal of [latex]\\dfrac{1}{2}[\/latex], which is [latex]\\dfrac{2}{1}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]9\\div\\dfrac{1}{2}=\\dfrac{9}{1}\\cdot\\dfrac{2}{1}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{9\\cdot 2}{1\\cdot 1}=\\dfrac{18}{1}= 18[\/latex]<\/p>\n<p>This answer is already simplified to lowest terms.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]9\\div\\dfrac{1}{2}= 18[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Divide a Fraction by a Fraction<\/h2>\n<p>Sometimes we need to solve a problem that requires dividing a fraction by a fraction. Suppose we want to find the quotient: [latex]\\frac{1}{2}\\div\\frac{1}{6}[\/latex]. We need to figure out how many [latex]\\frac{1}{6}\\text{s}[\/latex] there are in [latex]\\frac{1}{2}[\/latex]. We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown below. Notice, there are three [latex]\\frac{1}{6}[\/latex] tiles in [latex]\\frac{1}{2}[\/latex], so [latex]\\frac{1}{2}\\div\\frac{1}{6}=3[\/latex].<\/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\/24220930\/CNX_BMath_Figure_04_02_016.png\" alt=\"A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth.\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Model: [latex]\\frac{1}{4}\\div\\frac{1}{8}[\/latex]<\/p>\n<p>Solution:<br \/>\nWe want to determine how many [latex]\\frac{1}{8}\\text{s}[\/latex] are in [latex]\\frac{1}{4}[\/latex]. Start with one [latex]\\frac{1}{4}[\/latex] tile. Line up [latex]\\frac{1}{8}[\/latex] tiles underneath the [latex]\\frac{1}{4}[\/latex] tile.<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220931\/CNX_BMath_Figure_04_02_017_img.png\" alt=\"A rectangle is shown, labeled one fourth. Below it is an identical rectangle split into two equal pieces, each labeled as one eighth.\" \/><br \/>\nThere are two [latex]\\frac{1}{8}[\/latex]s in [latex]\\frac{1}{4}[\/latex].<br \/>\nSo, [latex]\\frac{1}{4}\\div\\frac{1}{8}=2[\/latex].<\/p>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Model: [latex]\\frac{1}{3}\\div\\frac{1}{6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q218091\">Show Solution<\/span><\/p>\n<div id=\"q218091\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220932\/CNX_BMath_Figure_04_02_018_img.png\" alt=\"A rectangle is shown, labeled as one third. Below it is an identical rectangle split into two equal pieces, each labeled as one sixth.\" \/><\/p>\n<p>[latex]2[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Model: [latex]\\frac{1}{2}\\div\\frac{1}{4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q763601\">Show Solution<\/span><\/p>\n<div id=\"q763601\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220934\/CNX_BMath_Figure_04_02_019_img.png\" alt=\"A rectangle is shown, labeled as one half. Below it is an identical rectangle split into two equal pieces, each labeled as one fourth.\" \/><\/p>\n<p>[latex]2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows another way to model division of two fractions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex: Using a Fraction Wall to Find the Quotient of Two Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/pk-K5JF9iMo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Using fraction tiles, we showed that [latex]\\frac{1}{2}\\div\\frac{1}{6}=3[\/latex]. Notice that [latex]\\frac{1}{2}\\cdot \\frac{6}{1}=3[\/latex] also. How are [latex]\\frac{1}{6}[\/latex] and [latex]\\frac{6}{1}[\/latex] related? They are reciprocals. This leads us to the procedure for fraction division.\u00a0 Suppose we have a pizza that is already cut into [latex]4[\/latex] slices. How many [latex]\\dfrac{1}{2}[\/latex] slices are there?<\/p>\n<table>\n<tbody>\n<tr>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182618\/image146.gif\" alt=\"A pizza divided into four equal pieces. There are four slices.\" width=\"180\" height=\"179\" \/><\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182619\/image147.gif\" alt=\"A pizza divided into four equal slices. Each slice is then divided in half. There are now 8 slices.\" width=\"180\" height=\"179\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>There are [latex]8[\/latex] slices. You can see that dividing [latex]4[\/latex] by [latex]\\dfrac{1}{2}[\/latex] gives the same result as multiplying [latex]4[\/latex] by [latex]2[\/latex].<\/p>\n<p>What would happen if you needed to divide each slice into thirds?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182621\/image148.gif\" alt=\"A pizza divided into four equal slice. Each slice is divided into thirds. There are now 12 slices.\" width=\"180\" height=\"179\" \/><\/p>\n<p>You would have [latex]12[\/latex] slices, which is the same as multiplying [latex]4[\/latex] by [latex]3[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Fraction Division<\/h3>\n<p>If [latex]a,b,c,\\text{ and }d[\/latex] are numbers where [latex]b\\ne 0,c\\ne 0,\\text{ and }d\\ne 0[\/latex], then<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{a}{b}\\div\\frac{c}{d}=\\frac{a}{b}\\cdot \\frac{d}{c}[\/latex]<\/p>\n<p>To divide fractions, multiply the first fraction by the reciprocal of the second.<\/p>\n<p>We need to say [latex]b\\ne 0,c\\ne 0\\text{ and }d\\ne 0[\/latex] to be sure we don\u2019t divide by zero.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Dividing with Fractions<\/h3>\n<ol>\n<li>Find the reciprocal of the divisor (the number that follows the division symbol).<\/li>\n<li>Multiply the dividend (the number before the division symbol) by the reciprocal of the divisor (the number after the division symbol).<\/li>\n<\/ol>\n<\/div>\n<p>Any easy way to remember how to divide fractions is the phrase \u201ckeep, change, flip.\u201d This means to <strong>KEEP<\/strong> the first number, <strong>CHANGE<\/strong> the division sign to multiplication, and then <strong>FLIP<\/strong> (use the reciprocal) of the second number.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide [latex]\\dfrac{2}{3}\\div\\dfrac{1}{6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q569112\">Show Solution<\/span><\/p>\n<div id=\"q569112\" class=\"hidden-answer\" style=\"display: none\">\n<p>Multiply by the reciprocal.<\/p>\n<p><strong>KEEP<\/strong> [latex]\\dfrac{2}{3}[\/latex]<\/p>\n<p><strong>CHANGE<\/strong>\u00a0 [latex]\\div[\/latex] to \u00a0[latex]\\cdot[\/latex]<\/p>\n<p><strong>FLIP\u00a0<\/strong> [latex]\\dfrac{1}{6}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{2}{3}\\cdot\\dfrac{6}{1}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{2\\cdot6}{3\\cdot1}=\\dfrac{12}{3}[\/latex]<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\dfrac{12}{3}=\\normalsize 4[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\dfrac{2}{3}\\div\\dfrac{1}{6}=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide [latex]-\\dfrac{3}{5}\\div\\dfrac{2}{3}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q950676\">Show Solution<\/span><\/p>\n<div id=\"q950676\" class=\"hidden-answer\" style=\"display: none\">\n<p>Multiply by the reciprocal.\u00a0Keep [latex]-\\dfrac{3}{5}[\/latex], change [latex]\\div[\/latex] to [latex]\\cdot[\/latex], and flip [latex]\\dfrac{2}{3}[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]-\\dfrac{3}{5}\\cdot\\dfrac{3}{2}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center;\">[latex]-\\dfrac{3\\cdot 3}{5\\cdot 2}=-\\dfrac{9}{10}[\/latex]<\/p>\n<p>There are no common factors (other than 1), so the fraction is simplified.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]-\\dfrac{3}{5}\\div\\dfrac{2}{3}=-\\dfrac{9}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>When solving a division problem by multiplying by the reciprocal, remember to write all whole numbers as improper fractions before doing calculations\u00a0 [latex](\\text{i.e. } 5=\\dfrac{5}{1}[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm263630\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=263630&theme=oea&iframe_resize_id=ohm263630&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide, and write the answer in simplified form: [latex]\\frac{2}{5}\\div\\left(-\\frac{3}{7}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q261121\">Show Solution<\/span><\/p>\n<div id=\"q261121\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168468274991\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td>[latex]\\frac{2}{5}\\normalsize\\div\\Large\\left(-\\frac{3}{7}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<td>[latex]\\frac{2}{5}\\left(-\\frac{7}{3}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply. The product is negative.<\/td>\n<td>[latex]-\\frac{14}{15}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide, and write the answer in simplified form: [latex]-\\frac{3}{4}\\div\\left(-\\frac{7}{8}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q873547\">Show Solution<\/span><\/p>\n<div id=\"q873547\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467263034\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\">\n<tbody>\n<tr>\n<td style=\"width: 55.64648117839607%;\"><\/td>\n<td style=\"width: 44.189852700491%;\">[latex]-\\frac{3}{4}\\div\\left(-\\frac{7}{8}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 55.64648117839607%;\">Multiply the first fraction by the reciprocal of the second.<\/td>\n<td style=\"width: 44.189852700491%;\">[latex]-\\frac{3}{4}\\cdot \\left(-\\frac{8}{7}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 55.64648117839607%;\">Multiply. Remember to determine the sign first.<\/td>\n<td style=\"width: 44.189852700491%;\">[latex]\\frac{3\\cdot 8}{4\\cdot 7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 55.64648117839607%;\">Rewrite to show common factors.<\/td>\n<td style=\"width: 44.189852700491%;\">[latex]\\frac{3\\cdot 4\\cdot 2}{4\\cdot 7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 55.64648117839607%;\">Remove common factors and simplify.<\/td>\n<td style=\"width: 44.189852700491%;\">[latex]\\frac{6}{7}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146066\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146066&theme=oea&iframe_resize_id=ohm146066&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>The following video shows more examples of dividing fractions that are negative.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Ex 1:  Dividing Signed Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/OPHdadhDJoI?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-1404\">\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, 141842, 146026. <strong>Authored by<\/strong>: Alyson Day. <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><li>Revision and Adaptation. <strong>Authored by<\/strong>: Roxanne Brinkerhoff. <strong>Provided by<\/strong>: Utah Valley University. <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>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><li>Ex: Determine the Reciprocal of Integers, Fractions, and Mixed Numbers. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/IM991IqCi44\">https:\/\/youtu.be\/IM991IqCi44<\/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 2: Determine the Absolute Value of an Integer. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/lY5ksjix5Kg\">https:\/\/youtu.be\/lY5ksjix5Kg<\/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 1: Divide Fractions (Basic). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/F5YSNLel3n8\">https:\/\/youtu.be\/F5YSNLel3n8<\/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>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422605,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Ex 2: 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