{"id":70,"date":"2018-03-19T17:55:14","date_gmt":"2018-03-19T17:55:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/?post_type=chapter&#038;p=70"},"modified":"2024-04-26T22:01:48","modified_gmt":"2024-04-26T22:01:48","slug":"multiplying-and-dividing-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/chapter\/multiplying-and-dividing-fractions\/","title":{"raw":"Multiplying and Dividing Fractions","rendered":"Multiplying and Dividing Fractions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use multiplication and division when evaluating expressions with fractions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 data-type=\"title\">Fraction Multiplication<\/h2>\r\n<p data-type=\"title\">A model may help you understand multiplication of fractions. We will use fraction tiles to model [latex]\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex].<\/p>\r\n<p data-type=\"title\">To multiply [latex]\\frac{1}{2}[\/latex] and [latex]\\frac{3}{4}[\/latex], think \"I need to find [latex]\\frac{1}{2}[\/latex] of [latex]\\frac{3}{4}[\/latex].\"<\/p>\r\n<p data-type=\"title\">Start 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 exchange them for smaller tiles.<\/p>\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.\" data-media-type=\"image\/png\" \/>\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, [latex]\\frac{1}{2}\\cdot \\frac{3}{4}=\\frac{3}{8}[\/latex]\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse a diagram to model [latex]\\frac{1}{3}\\cdot \\frac{2}{5}[\/latex]\r\n\r\nSolution:\r\n\r\nYou want to find one-third of two-fifths.\r\n\r\nFirst shade in [latex]\\frac{2}{5}[\/latex] of the rectangle.\r\n\r\n<img class=\"wp-image-2128 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2985\/2018\/03\/25165040\/fractions_1.png\" alt=\"A long rectangle is divided into five equal sections with vertical lines. Two of the resulting boxes are shaded blue.\" width=\"383\" height=\"83\" \/>\r\nWe will take [latex]\\frac{1}{3}[\/latex] of this [latex]\\frac{2}{5}[\/latex], so we heavily shade [latex]\\frac{1}{3}[\/latex] of the shaded region.\r\n\r\n<img class=\"aligncenter wp-image-2129\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2985\/2018\/03\/25165142\/fractions_2.png\" alt=\"A long rectangle is divided into five equal sections with vertical lines and three equal sections with horizontal lines. There are fifteen resulting boxes and two are shaded dark blue to show the overlap.\" width=\"379\" height=\"82\" \/>\r\nNotice that [latex]2[\/latex] out of the [latex]15[\/latex] pieces are heavily shaded. This means that [latex]\\frac{2}{15}[\/latex] of the rectangle is heavily shaded.\r\nTherefore, [latex]\\frac{1}{3}[\/latex] of [latex]\\frac{2}{15}[\/latex] is [latex]\\frac{2}{15}[\/latex],\u00a0 \u00a0or [latex]\\frac{1}{3}\\cdot \\frac{2}{5}=\\frac{2}{15}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[ohm_question height=\"270\"]146020[\/ohm_question]\r\n\r\n<\/div>\r\nLook at the result we got from the examples above. We found that [latex]\\frac{1}{2}\\cdot \\frac{3}{4}=\\frac{3}{8}[\/latex] and [latex]\\frac{1}{3}\\cdot \\frac{2}{5}=\\frac{2}{15}[\/latex]. Do you notice that we could have gotten the same answers by multiplying the numerators and multiplying the denominators?\r\n<table id=\"eip-id1168468256450\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\" data-label=\"\">\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<td>[latex]\\frac{1}{3}\\cdot \\frac{2}{5}[\/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}{2}\\cdot \\frac{3}{4}=\\frac{1\\cdot 3}{2\\cdot 4}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{3}\\cdot \\frac{2}{5}=\\frac{1\\cdot 2}{3\\cdot 5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\frac{3}{8}[\/latex]<\/td>\r\n<td>[latex]\\frac{2}{15}[\/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\n<p style=\"padding-left: 30px;\">If [latex]a,b,c,\\text{ and }d[\/latex] are numbers where [latex]b\\ne 0\\text{ and }d\\ne 0[\/latex], then [latex]\\Large\\frac{a}{b}\\cdot \\Large\\frac{c}{d}=\\Large\\frac{ac}{bd}[\/latex]<\/p>\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{3}{4}\\cdot \\frac{1}{5}[\/latex]\r\n[reveal-answer q=\"56385\"]Show Answer[\/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=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\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, 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\n[ohm_question]146021[\/ohm_question]\r\n\r\n<\/div>\r\nNote that 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.\r\n\r\nThe following video provides more examples of how to multiply fractions, and simplify the result.\r\n\r\nhttps:\/\/youtu.be\/f_L-EFC8Z7c\r\n\r\nWhen multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, [latex]a[\/latex], can be written as [latex]\\large\\frac{a}{1}[\/latex]. So, [latex]3=\\frac{3}{1}[\/latex], for example.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply, and write the answer in simplified form:\r\n<ol>\r\n \t<li>[latex]\\frac{1}{7}\\cdot 56[\/latex]<\/li>\r\n \t<li>[latex](-20)(\\large\\frac{12}{5})[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"597781\"]Show Answer[\/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=\".\" data-label=\"\">\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 \" data-label=\"\">\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](-20)(\\frac{12}{5})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]-20[\/latex] as a fraction.<\/td>\r\n<td>[latex]\\frac{-20}{1}(\\frac{12}{5})[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Determine the sign of the product; multiply.<\/td>\r\n<td>[latex]-\\frac{20\\cdot 12\\cdot}{1\\cdot 5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply and simplify.<\/td>\r\n<td>[latex]-\\frac{240}{5}=-\\frac{2\\cdot\\color{red}{5}\\cdot24}{\\color{red}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-48[\/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]156968[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video to see more examples of how to multiply a fraction and a whole number,\r\n\r\nhttps:\/\/youtu.be\/Rxz7OUzNyV0\r\n<h2 data-type=\"title\">Reciprocals<\/h2>\r\n<p data-type=\"title\">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>\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: center;\">Such pairs of numbers are called reciprocals.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Reciprocal<\/h3>\r\n<p style=\"padding-left: 30px;\">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]. A number and its reciprocal have a product of [latex]1[\/latex].<\/p>\r\n<p style=\"padding-left: 30px;\">[latex]\\frac{a}{b}\\cdot \\frac{b}{a}=1[\/latex]<\/p>\r\n\r\n<\/div>\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. 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=\"The equation a divided by b times b divided by a equals positive 1. Below this equation are two other equations. The first of which is titled both positive. 3 times one third equals 1. The second equation is titled both negative. Negative 3 times negative one third equals 1.\" width=\"289\" height=\"124\" data-media-type=\"image\/png\" \/>\r\nTo find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to [latex]1[\/latex]. Is there any number [latex]r[\/latex] so that [latex]0\\cdot r=1?[\/latex] No. So, the number [latex]0[\/latex] does not have a reciprocal.\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\" data-number-style=\"arabic\">\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: 85%;\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the reciprocal of [latex]\\frac{4}{9}[\/latex] .<\/td>\r\n<td>The reciprocal of [latex]\\frac{4}{9}[\/latex] is [latex]\\frac{9}{4}[\/latex] .<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the number and its reciprocal.<\/td>\r\n<td>[latex]\\frac{4}{9}\\cdot \\frac{9}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply numerators and denominators.<\/td>\r\n<td>[latex]\\frac{36}{36}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[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: 85%;\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the reciprocal of [latex]-\\frac{1}{6}[\/latex] .<\/td>\r\n<td>[latex]-\\frac{6}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]-\\frac{1}{6}\\cdot \\left(-6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[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: 85%;\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the reciprocal of [latex]-\\frac{14}{5}[\/latex] .<\/td>\r\n<td>[latex]-\\frac{5}{14}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]-\\frac{14}{5}\\cdot \\left(-\\frac{5}{14}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\frac{70}{70}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[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: 85%;\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>4.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Find the reciprocal of [latex]7[\/latex] .<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]7[\/latex] as a fraction.<\/td>\r\n<td>[latex]\\frac{7}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the reciprocal of [latex]\\frac{7}{1}[\/latex] .<\/td>\r\n<td>[latex]\\frac{1}{7}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check:<\/td>\r\n<td>[latex]7\\cdot \\left(\\frac{1}{7}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]1\\quad\\checkmark [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]141842[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we will show more examples of how to find the reciprocal of integers, fractions and mixed numbers.\r\n\r\nhttps:\/\/youtu.be\/IM991IqCi44\r\n<h2 data-type=\"title\">Dividing Fractions<\/h2>\r\n<p data-type=\"title\">Why is [latex]12\\div 3=4?[\/latex] We previously modeled this with counters. How many groups of [latex]3[\/latex] counters can be made from a group of [latex]12[\/latex] counters?<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220928\/CNX_BMath_Figure_04_02_015_img.png\" alt=\"Four red ovals are shown. Inside each oval are three grey circles.\" data-media-type=\"image\/png\" \/>\r\nThere are [latex]4[\/latex] groups of [latex]3[\/latex] counters. In other words, there are four [latex]3\\text{s}[\/latex] in [latex]12[\/latex]. So, [latex]12\\div 3=4[\/latex].\r\n\r\nWhat about dividing fractions? 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.\" data-media-type=\"image\/png\" \/>\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\r\n<img 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.\" data-media-type=\"image\/png\" \/>\r\nThere are two [latex]\\frac{1}{8}\\text{s}[\/latex] in [latex]\\frac{1}{4}[\/latex].\r\nSo, [latex]\\frac{1}{4}\\div \\frac{1}{8}=2[\/latex].\r\n\r\n<\/div>\r\nThe following video shows a whole number being divided by a fraction using a slightly different method.\r\n\r\nhttps:\/\/youtu.be\/JKsfdK1WT1s\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nModel: [latex]2\\div \\frac{1}{4}[\/latex]\r\n[reveal-answer q=\"391699\"]Show Answer[\/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\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.\" data-media-type=\"image\/png\" \/>\r\nBecause there are eight [latex]\\frac{1}{4}\\text{s}[\/latex] in [latex]2,2\\div \\frac{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\nModel: [latex]2\\div \\frac{1}{3}[\/latex]\r\n[reveal-answer q=\"73567\"]Show Answer[\/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.\" data-media-type=\"image\/png\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nModel: [latex]3\\div \\frac{1}{2}[\/latex]\r\n[reveal-answer q=\"354856\"]Show Answer[\/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.\" data-media-type=\"image\/png\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"270\"]117216[\/ohm_question]\r\n\r\n<\/div>\r\nLet\u2019s use money to model [latex]2\\div \\frac{1}{4}[\/latex] in another way. 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, [latex]2\\div \\frac{1}{4}=8[\/latex].\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.\r\n<div class=\"textbox shaded\">\r\n<h3>Fraction Division<\/h3>\r\n<p style=\"padding-left: 30px;\">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 [latex]\\frac{a}{b}\\div \\frac{c}{d}=\\frac{a}{b}\\cdot \\frac{d}{c}[\/latex]<\/p>\r\n<p style=\"padding-left: 30px;\">To divide fractions, multiply the first fraction by the reciprocal of the second.<\/p>\r\n<p style=\"padding-left: 30px;\">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>\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 Answer[\/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=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\frac{2}{5}\\div \\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 key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question height=\"270\"]146066[\/ohm_question]\r\n\r\n[ohm_question height=\"270\"]146067[\/ohm_question]\r\n\r\n<\/div>\r\nWatch this video for more examples of dividing fractions using a reciprocal.\r\n\r\nhttps:\/\/youtu.be\/fnaRnEXlUvs\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide, and write the answer in simplified form: [latex]\\frac{7}{18}\\div \\frac{14}{27}[\/latex]\r\n[reveal-answer q=\"987562\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"987562\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466022407\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"No Alt Text\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\frac{7}{18}\\div \\frac{14}{27}[\/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{7}{18}\\cdot \\frac{27}{14}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]\\frac{7\\cdot 27}{18\\cdot 14}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite showing common factors.<\/td>\r\n<td>[latex]\\frac{\\color{red}{7}\\cdot\\color{blue}{9}\\cdot3}{\\color{blue}{9}\\cdot2\\cdot\\color{red}{7}\\cdot2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors.<\/td>\r\n<td>[latex]\\frac{3}{2\\cdot 2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\frac{3}{4}[\/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\"]146091[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use multiplication and division when evaluating expressions with fractions<\/li>\n<\/ul>\n<\/div>\n<h2 data-type=\"title\">Fraction Multiplication<\/h2>\n<p data-type=\"title\">A model may help you understand multiplication of fractions. We will use fraction tiles to model [latex]\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex].<\/p>\n<p data-type=\"title\">To multiply [latex]\\frac{1}{2}[\/latex] and [latex]\\frac{3}{4}[\/latex], think &#8220;I need to find [latex]\\frac{1}{2}[\/latex] of [latex]\\frac{3}{4}[\/latex].&#8221;<\/p>\n<p data-type=\"title\">Start 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 exchange them for smaller tiles.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220916\/CNX_BMath_Figure_04_02_010_img.png\" alt=\"A rectangle is divided vertically into three equal pieces. Each piece is labeled as one fourth. There is a an arrow pointing to an identical rectangle divided vertically into six equal pieces. Each piece is labeled as one eighth. There are braces showing that three of these rectangles represent three eighths.\" data-media-type=\"image\/png\" \/><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, [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}{3}\\cdot \\frac{2}{5}[\/latex]<\/p>\n<p>Solution:<\/p>\n<p>You want to find one-third of two-fifths.<\/p>\n<p>First shade in [latex]\\frac{2}{5}[\/latex] of the rectangle.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2128 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2985\/2018\/03\/25165040\/fractions_1.png\" alt=\"A long rectangle is divided into five equal sections with vertical lines. Two of the resulting boxes are shaded blue.\" width=\"383\" height=\"83\" \/><br \/>\nWe will take [latex]\\frac{1}{3}[\/latex] of this [latex]\\frac{2}{5}[\/latex], so we heavily shade [latex]\\frac{1}{3}[\/latex] of the shaded region.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2129\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2985\/2018\/03\/25165142\/fractions_2.png\" alt=\"A long rectangle is divided into five equal sections with vertical lines and three equal sections with horizontal lines. There are fifteen resulting boxes and two are shaded dark blue to show the overlap.\" width=\"379\" height=\"82\" \/><br \/>\nNotice that [latex]2[\/latex] out of the [latex]15[\/latex] pieces are heavily shaded. This means that [latex]\\frac{2}{15}[\/latex] of the rectangle is heavily shaded.<br \/>\nTherefore, [latex]\\frac{1}{3}[\/latex] of [latex]\\frac{2}{15}[\/latex] is [latex]\\frac{2}{15}[\/latex],\u00a0 \u00a0or [latex]\\frac{1}{3}\\cdot \\frac{2}{5}=\\frac{2}{15}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146020\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146020&theme=oea&iframe_resize_id=ohm146020&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>Look at the result we got from the examples above. We found that [latex]\\frac{1}{2}\\cdot \\frac{3}{4}=\\frac{3}{8}[\/latex] and [latex]\\frac{1}{3}\\cdot \\frac{2}{5}=\\frac{2}{15}[\/latex]. Do you notice that we could have gotten the same answers by multiplying the numerators and multiplying the denominators?<\/p>\n<table id=\"eip-id1168468256450\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{1}{2}\\cdot \\frac{3}{4}[\/latex]<\/td>\n<td>[latex]\\frac{1}{3}\\cdot \\frac{2}{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the numerators, and multiply the denominators.<\/td>\n<td>[latex]\\frac{1}{2}\\cdot \\frac{3}{4}=\\frac{1\\cdot 3}{2\\cdot 4}[\/latex]<\/td>\n<td>[latex]\\frac{1}{3}\\cdot \\frac{2}{5}=\\frac{1\\cdot 2}{3\\cdot 5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\frac{3}{8}[\/latex]<\/td>\n<td>[latex]\\frac{2}{15}[\/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 style=\"padding-left: 30px;\">If [latex]a,b,c,\\text{ and }d[\/latex] are numbers where [latex]b\\ne 0\\text{ and }d\\ne 0[\/latex], then [latex]\\Large\\frac{a}{b}\\cdot \\Large\\frac{c}{d}=\\Large\\frac{ac}{bd}[\/latex]<\/p>\n<\/div>\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 Answer<\/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=\".\" data-label=\"\">\n<tbody>\n<tr>\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, 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><iframe loading=\"lazy\" id=\"ohm146021\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146021&theme=oea&iframe_resize_id=ohm146021&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Note that 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.<\/p>\n<p>The following video provides more examples of how to multiply fractions, and simplify the result.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 1: Multiply Fractions (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/f_L-EFC8Z7c?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, [latex]a[\/latex], can be written as [latex]\\large\\frac{a}{1}[\/latex]. So, [latex]3=\\frac{3}{1}[\/latex], for example.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply, and write the answer in simplified form:<\/p>\n<ol>\n<li>[latex]\\frac{1}{7}\\cdot 56[\/latex]<\/li>\n<li>[latex](-20)(\\large\\frac{12}{5})[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q597781\">Show Answer<\/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=\".\" data-label=\"\">\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\" data-label=\"\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex](-20)(\\frac{12}{5})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write [latex]-20[\/latex] as a fraction.<\/td>\n<td>[latex]\\frac{-20}{1}(\\frac{12}{5})[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Determine the sign of the product; multiply.<\/td>\n<td>[latex]-\\frac{20\\cdot 12\\cdot}{1\\cdot 5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply and simplify.<\/td>\n<td>[latex]-\\frac{240}{5}=-\\frac{2\\cdot\\color{red}{5}\\cdot24}{\\color{red}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-48[\/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=\"ohm156968\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=156968&theme=oea&iframe_resize_id=ohm156968&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video to see more examples of how to multiply a fraction and a whole number,<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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 data-type=\"title\">Reciprocals<\/h2>\n<p data-type=\"title\">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: center;\">Such pairs of numbers are called reciprocals.<\/p>\n<div class=\"textbox shaded\">\n<h3>Reciprocal<\/h3>\n<p style=\"padding-left: 30px;\">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]. A number and its reciprocal have a product of [latex]1[\/latex].<\/p>\n<p style=\"padding-left: 30px;\">[latex]\\frac{a}{b}\\cdot \\frac{b}{a}=1[\/latex]<\/p>\n<\/div>\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. 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 loading=\"lazy\" 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=\"The equation a divided by b times b divided by a equals positive 1. Below this equation are two other equations. The first of which is titled both positive. 3 times one third equals 1. The second equation is titled both negative. Negative 3 times negative one third equals 1.\" width=\"289\" height=\"124\" data-media-type=\"image\/png\" \/><br \/>\nTo find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to [latex]1[\/latex]. Is there any number [latex]r[\/latex] so that [latex]0\\cdot r=1?[\/latex] No. So, the number [latex]0[\/latex] does not have a reciprocal.<\/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\" data-number-style=\"arabic\">\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: 85%;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Find the reciprocal of [latex]\\frac{4}{9}[\/latex] .<\/td>\n<td>The reciprocal of [latex]\\frac{4}{9}[\/latex] is [latex]\\frac{9}{4}[\/latex] .<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Multiply the number and its reciprocal.<\/td>\n<td>[latex]\\frac{4}{9}\\cdot \\frac{9}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply numerators and denominators.<\/td>\n<td>[latex]\\frac{36}{36}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]1\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-37774\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Find the reciprocal of [latex]-\\frac{1}{6}[\/latex] .<\/td>\n<td>[latex]-\\frac{6}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]-\\frac{1}{6}\\cdot \\left(-6\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]1\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1170196522976\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Find the reciprocal of [latex]-\\frac{14}{5}[\/latex] .<\/td>\n<td>[latex]-\\frac{5}{14}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]-\\frac{14}{5}\\cdot \\left(-\\frac{5}{14}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{70}{70}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]1\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1170195508737\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>4.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Find the reciprocal of [latex]7[\/latex] .<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write [latex]7[\/latex] as a fraction.<\/td>\n<td>[latex]\\frac{7}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write the reciprocal of [latex]\\frac{7}{1}[\/latex] .<\/td>\n<td>[latex]\\frac{1}{7}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check:<\/td>\n<td>[latex]7\\cdot \\left(\\frac{1}{7}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]1\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\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=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we will show more examples of how to find the reciprocal of integers, fractions and mixed numbers.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Determine the Reciprocal of Integers, Fractions, and Mixed Numbers\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/IM991IqCi44?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 data-type=\"title\">Dividing Fractions<\/h2>\n<p data-type=\"title\">Why is [latex]12\\div 3=4?[\/latex] We previously modeled this with counters. How many groups of [latex]3[\/latex] counters can be made from a group of [latex]12[\/latex] counters?<\/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\/24220928\/CNX_BMath_Figure_04_02_015_img.png\" alt=\"Four red ovals are shown. Inside each oval are three grey circles.\" data-media-type=\"image\/png\" \/><br \/>\nThere are [latex]4[\/latex] groups of [latex]3[\/latex] counters. In other words, there are four [latex]3\\text{s}[\/latex] in [latex]12[\/latex]. So, [latex]12\\div 3=4[\/latex].<\/p>\n<p>What about dividing fractions? 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.\" data-media-type=\"image\/png\" \/><\/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><img decoding=\"async\" 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.\" data-media-type=\"image\/png\" \/><br \/>\nThere are two [latex]\\frac{1}{8}\\text{s}[\/latex] in [latex]\\frac{1}{4}[\/latex].<br \/>\nSo, [latex]\\frac{1}{4}\\div \\frac{1}{8}=2[\/latex].<\/p>\n<\/div>\n<p>The following video shows a whole number being divided by a fraction using a slightly different method.<\/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>Model: [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 Answer<\/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.<\/p>\n<p><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.\" data-media-type=\"image\/png\" \/><br \/>\nBecause there are eight [latex]\\frac{1}{4}\\text{s}[\/latex] in [latex]2,2\\div \\frac{1}{4}=8[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Model: [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 Answer<\/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.\" data-media-type=\"image\/png\" \/><\/p>\n<\/div>\n<\/div>\n<p>Model: [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 Answer<\/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.\" data-media-type=\"image\/png\" \/><\/p>\n<\/div>\n<\/div>\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=\"270\"><\/iframe><\/p>\n<\/div>\n<p>Let\u2019s use money to model [latex]2\\div \\frac{1}{4}[\/latex] in another way. 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, [latex]2\\div \\frac{1}{4}=8[\/latex].<\/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.<\/p>\n<div class=\"textbox shaded\">\n<h3>Fraction Division<\/h3>\n<p style=\"padding-left: 30px;\">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 [latex]\\frac{a}{b}\\div \\frac{c}{d}=\\frac{a}{b}\\cdot \\frac{d}{c}[\/latex]<\/p>\n<p style=\"padding-left: 30px;\">To divide fractions, multiply the first fraction by the reciprocal of the second.<\/p>\n<p style=\"padding-left: 30px;\">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<\/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 Answer<\/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=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td>[latex]\\frac{2}{5}\\div \\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 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<p><iframe loading=\"lazy\" id=\"ohm146067\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146067&theme=oea&iframe_resize_id=ohm146067&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video for more examples of dividing fractions using a reciprocal.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Ex 2: Divide Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/fnaRnEXlUvs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide, and write the answer in simplified form: [latex]\\frac{7}{18}\\div \\frac{14}{27}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q987562\">Show Answer<\/span><\/p>\n<div id=\"q987562\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466022407\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"No Alt Text\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{7}{18}\\div \\frac{14}{27}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the first fraction by the reciprocal of the second.<\/td>\n<td>[latex]\\frac{7}{18}\\cdot \\frac{27}{14}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]\\frac{7\\cdot 27}{18\\cdot 14}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite showing common factors.<\/td>\n<td>[latex]\\frac{\\color{red}{7}\\cdot\\color{blue}{9}\\cdot3}{\\color{blue}{9}\\cdot2\\cdot\\color{red}{7}\\cdot2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors.<\/td>\n<td>[latex]\\frac{3}{2\\cdot 2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\frac{3}{4}[\/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=\"ohm146091\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146091&theme=oea&iframe_resize_id=ohm146091&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-70\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","CANDELA_OUTCOMES_GUID":"1f1dd0a7-dca2-43f3-9757-33d4e41a9363, 7146b4cd-d4f5-428b-a10b-2982b9f0caf0","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-70","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/70","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":19,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/70\/revisions"}],"predecessor-version":[{"id":3983,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/70\/revisions\/3983"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/70\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/media?parent=70"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapter-type?post=70"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/contributor?post=70"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/license?post=70"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}