{"id":6267,"date":"2018-02-14T19:45:31","date_gmt":"2018-02-14T19:45:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-albany-chemistry\/chapter\/fractions\/"},"modified":"2018-06-05T15:29:20","modified_gmt":"2018-06-05T15:29:20","slug":"fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-albany-chemistry\/chapter\/fractions\/","title":{"raw":"0.1 Fractions","rendered":"0.1 Fractions"},"content":{"raw":"<a href=\"https:\/\/drive.google.com\/open?id=1fYsqmQ56okTusI26peGrrZ9gv6P_1Yf7\" target=\"_blank\" rel=\"noopener\">Chapter 0 Lecture Notes<\/a>\r\n<div class=\"textbox learning-objectives\">\r\n\r\n[embed]https:\/\/www.youtube.com\/watch?v=PIYCx6O1aL4&amp;t=0s&amp;list=PLrL0qe-gjHgkzaMAgazN-4jteNKVifyCL&amp;index=2[\/embed]\r\n<h3>QUICK REFERENCE<\/h3>\r\nAdding and Subtracting Fractions\r\n<ul>\r\n \t<li>Convert the fractions so they have common denominators.<\/li>\r\n \t<li>Perform the addition or subtraction on the numerator and keep the common denominator.<\/li>\r\n \t<li>Simplify the answer (write the fraction in the lowest terms).<\/li>\r\n<\/ul>\r\nMultiplying and Dividing Fractions\r\n<ul>\r\n \t<li>To multiply, multiply across the numerators and denominators.<\/li>\r\n \t<li>To divide, multiply the first number by the reciprocal of the second number.<\/li>\r\n \t<li>Simplify the answer (write the fraction in the lowest terms).<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Adding and Subtracting Fractions<\/h2>\r\nIn order to add or subtract fractions, you first must make sure that the fractions have the same denominator. The denominator tells you how many pieces the whole has been broken into, and the numerator tells you how many of those pieces you are considering.\r\n\r\nTo find a common denominator you will determine the\u00a0<strong>least common multiple. <\/strong>Remember that\u00a0if a number is a multiple of another, you can divide them and have no remainder.\r\n\r\nOne way to find the least common multiple of two or more numbers is to first multiply each\u00a0by 1, 2, 3, 4, etc. \u00a0For example, find the least common multiple of 2 and 5.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td scope=\"col\">First, list all the multiples of 2:<\/td>\r\n<td scope=\"col\">Then list all the multiples of 5:<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\cdot 1 = 2[\/latex]<\/td>\r\n<td>[latex]5\\cdot 1 = 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\cdot 2 = 4[\/latex]<\/td>\r\n<td>[latex]5\\cdot 2 = 10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\cdot 3 = 6[\/latex]<\/td>\r\n<td>[latex]5\\cdot 3 = 15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\cdot 4 = 8[\/latex]<\/td>\r\n<td>[latex]5\\cdot 4 = 20[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\cdot 5 = 10[\/latex]<\/td>\r\n<td>[latex]5\\cdot 5 = 25[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe smallest multiple they have in common will be the common denominator.\u00a0 The least common multiple of 2 and 5 is 10.\r\n<div class=\"textbox shaded\">\r\n<h3>Adding Fractions with Unlike Denominators<\/h3>\r\n<ol>\r\n \t<li>Find a common denominator.<\/li>\r\n \t<li>Rewrite each fraction using the common denominator.<\/li>\r\n \t<li>Add the numerators but keep the common denominator.<\/li>\r\n \t<li>Simplify by canceling out all common factors in the numerator and denominator.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>Simplifying a Fraction<\/h2>\r\nA common convention used in mathematics is writing a fraction in lowest terms.\u00a0 The process of simplifying a fraction is often called <em>reducing the fraction<\/em>. We can simplify by canceling (dividing) the common factors in a fraction's numerator and denominator. \u00a0This is possible because a fraction represents division (a part divided by the whole).\r\n<div class=\"page\" title=\"Page 150\">\r\n<div class=\"layoutArea\">\r\n<div class=\"column\">\r\n\r\nFor example, to simplify [latex]\\frac{6}{9}[\/latex] you can rewrite 6 and 9\u00a0using the smallest factors possible as follows:\r\n<p style=\"text-align: center\">[latex]\\frac{6}{9}=\\frac{2\\cdot3}{3\\cdot3}[\/latex]<\/p>\r\nSince there is a 3 in both the numerator and denominator, and fractions can be considered division, we can divide the 3 in the top by the 3 in the bottom to reduce to 1.\r\n<p style=\"text-align: center\">[latex]\\frac{6}{9}=\\frac{2\\cdot\\cancel{3}}{3\\cdot\\cancel{3}}=\\frac{2\\cdot1}{3}=\\frac{2}{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the next example you are shown how to add two\u00a0fractions with different denominators, then simplify the answer.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd [latex] \\frac{2}{3}+\\frac{1}{5}[\/latex].\u00a0Simplify the answer.\r\n\r\n[reveal-answer q=\"797488\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"797488\"]Since the denominators are not alike, find a common denominator by multiplying the denominators.\r\n<p style=\"text-align: center\">[latex]3\\cdot5=15[\/latex]<\/p>\r\nRewrite each fraction with a denominator of 15.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{2}{3}\\cdot \\frac{5}{5}=\\frac{10}{15}\\\\\\\\\\frac{1}{5}\\cdot \\frac{3}{3}=\\frac{3}{15}\\end{array}[\/latex]<\/p>\r\nAdd the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.\r\n<p style=\"text-align: center\">[latex] \\frac{10}{15}+\\frac{3}{15}=\\frac{13}{15}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\frac{2}{3}+\\frac{1}{5}=\\frac{13}{15}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd\u00a0[latex] \\frac{3}{7}+\\frac{2}{21}[\/latex]. Simplify the answer.\r\n\r\n[reveal-answer q=\"520906\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"520906\"]Since the denominators are not alike, find the least common denominator by finding the least common multiple (LCM) of 7 and 21.\r\n\r\nMultiples of 7: 7, 14, <strong>21<\/strong>\r\n\r\nMultiples of 21: <strong>21<\/strong>\r\n\r\nRewrite each fraction with a denominator of 21.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{3}{7}\\cdot \\frac{3}{3}=\\frac{9}{21}\\\\\\\\\\frac{2}{21}\\end{array}[\/latex]<\/p>\r\nAdd the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.\r\n<p style=\"text-align: center\">[latex] \\frac{9}{21}+\\frac{2}{21}=\\frac{11}{21}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\frac{3}{7}+\\frac{2}{21}=\\frac{11}{21}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Subtracting Fractions<\/h2>\r\nWhen you subtract fractions, you will still find a common denominator, but the numerators will be subtracted. Below are some examples of subtracting fractions whose denominators are not alike.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract\u00a0[latex]\\frac{1}{5}-\\frac{1}{6}[\/latex]. Simplify the answer.\r\n\r\n[reveal-answer q=\"155692\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"155692\"]The fractions have unlike denominators, so you need to find a common denominator. Recall that a common denominator can be found by multiplying the two denominators together.\r\n<p style=\"text-align: center\">[latex]5\\cdot6=30[\/latex]<\/p>\r\nRewrite each fraction as an equivalent fraction with a denominator of 30.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{1}{5}\\cdot \\frac{6}{6}=\\frac{6}{30}\\\\\\\\\\frac{1}{6}\\cdot \\frac{5}{5}=\\frac{5}{30}\\end{array}[\/latex]<\/p>\r\nSubtract the numerators. Simplify the answer if needed.\r\n<p style=\"text-align: center\">[latex] \\frac{6}{30}-\\frac{5}{30}=\\frac{1}{30}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\frac{1}{5}-\\frac{1}{6}=\\frac{1}{30}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract [latex]\\frac{5}{6}-\\frac{1}{4}[\/latex]. Simplify the answer.\r\n\r\n[reveal-answer q=\"984596\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"984596\"]Find the least common multiple of the denominators\u2014this is the least common denominator.\r\n\r\nMultiples of 6: 6, <strong>12<\/strong>, 18, 24\r\n\r\nMultiples of 4: 4, 8 <strong>12<\/strong>, 16, 20\r\n\r\n12 is the least common multiple of 6 and 4.\r\n\r\nRewrite each fraction with a denominator of 12.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{5}{6}\\cdot \\frac{2}{2}=\\frac{10}{12}\\\\\\\\\\frac{1}{4}\\cdot \\frac{3}{3}=\\frac{3}{12}\\end{array}[\/latex]<\/p>\r\nSubtract the fractions. Simplify the answer if needed.\r\n<p style=\"text-align: center\">[latex]\\frac{10}{12}-\\frac{3}{12}=\\frac{7}{12}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\frac{5}{6}-\\frac{1}{4}=\\frac{7}{12}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Multiplying Fractions<\/h2>\r\nWhen you multiply a fraction by a fraction, you are finding a \u201cfraction of a fraction.\u201d\u00a0 To multiply fractions you multiply across the numerators and denominators.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying Two Fractions<\/h3>\r\n[latex] \\frac{a}{b}\\cdot \\frac{c}{d}=\\frac{a\\cdot c}{b\\cdot d}=\\frac{\\text{product of the numerators}}{\\text{product of the denominators}}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplying More Than Two Fractions<\/h3>\r\n[latex] \\frac{a}{b}\\cdot \\frac{c}{d}\\cdot \\frac{e}{f}=\\frac{a\\cdot c\\cdot e}{b\\cdot d\\cdot f}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply [latex] \\frac{2}{3}\\cdot \\frac{4}{5}[\/latex].\r\n\r\n[reveal-answer q=\"368042\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"368042\"]Multiply the numerators and multiply the denominators.\r\n<p style=\"text-align: center\">[latex] \\frac{2\\cdot 4}{3\\cdot 5}[\/latex]<\/p>\r\nSimplify, if possible. This fraction is already in lowest terms.\r\n<p style=\"text-align: center\">[latex] \\frac{8}{15}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\frac{8}{15}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Dividing Fractions<\/h2>\r\nDividing fractions requires using the reciprocal of a number or fraction. If you multiply two numbers together and get 1 as a result, then the two numbers are reciprocals. Here 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] \\frac{3}{4}[\/latex]<\/td>\r\n<td>[latex] \\frac{4}{3}[\/latex]<\/td>\r\n<td>[latex] \\frac{3}{4}\\cdot \\frac{4}{3}=\\frac{3\\cdot 4}{4\\cdot 3}=\\frac{12}{12}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex] \\frac{2}{1}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{2}\\cdot\\frac{2}{1}=\\frac{1\\cdot}{2\\cdot1}=\\frac{2}{2}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] 3=\\frac{3}{1}[\/latex]<\/td>\r\n<td>[latex] \\frac{1}{3}[\/latex]<\/td>\r\n<td>[latex] \\frac{3}{1}\\cdot \\frac{1}{3}=\\frac{3\\cdot 1}{1\\cdot 3}=\\frac{3}{3}=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2\\frac{1}{3}=\\frac{7}{3}[\/latex]<\/td>\r\n<td>[latex] \\frac{3}{7}[\/latex]<\/td>\r\n<td>[latex]\\frac{7}{3}\\cdot\\frac{3}{7}=\\frac{7\\cdot3}{3\\cdot7}=\\frac{21}{21}=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSometimes we call\u00a0the reciprocal\u00a0the \u201cflip\u201d of the other number: flip [latex] \\frac{2}{5}[\/latex] to get the reciprocal [latex]\\frac{5}{2}[\/latex].\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\/2651\/2018\/02\/14194519\/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]\\frac{a}{0}[\/latex] is undefined. Additionally, the reciprocal of\u00a0[latex]\\frac{0}{a}[\/latex] will always be undefined.<\/div>\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\nThe same idea will work when the divisor (the thing being divided) is a fraction. If you have [latex] \\frac{3}{4}[\/latex] of a candy bar and need to divide it among 5 people, each person gets [latex] \\frac{1}{5}[\/latex] of the available candy:\r\n<p style=\"text-align: center\">[latex] \\frac{1}{5}\\text{ of }\\frac{3}{4}=\\frac{1}{5}\\cdot \\frac{3}{4}=\\frac{3}{20}[\/latex]<\/p>\r\n<p style=\"text-align: center\">Each person gets [latex]\\frac{3}{20}[\/latex]\u00a0of a whole candy bar.<\/p>\r\nIf you have a recipe that needs to be divided in half, you can divide each ingredient by 2, or you can multiply each ingredient by [latex]\\frac{1}{2}[\/latex]\u00a0to find the new amount.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind [latex] \\frac{2}{3}\\div 4[\/latex].\r\n\r\n[reveal-answer q=\"769187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"769187\"]Write your answer in lowest terms.\r\n\r\nDividing by 4 or [latex] \\frac{4}{1}[\/latex] is the same as multiplying by the reciprocal of 4, which is [latex] \\frac{1}{4}[\/latex].\r\n<p style=\"text-align: center\">[latex] \\frac{2}{3}\\div 4=\\frac{2}{3}\\cdot \\frac{1}{4}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center\">[latex] \\frac{2\\cdot 1}{3\\cdot 4}=\\frac{2}{12}[\/latex]<\/p>\r\nSimplify to lowest terms by dividing numerator and denominator by the common factor 4.\r\n<p style=\"text-align: center\">[latex] \\frac{1}{6}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{2}{3}\\div4=\\frac{1}{6}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide. [latex] 9\\div\\frac{1}{2}[\/latex].\r\n\r\n[reveal-answer q=\"269187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"269187\"]Write your answer in lowest terms.\r\n\r\nDividing by [latex]\\frac{1}{2}[\/latex] is the same as multiplying by the reciprocal of [latex]\\frac{1}{2}[\/latex], which is [latex] \\frac{2}{1}[\/latex].\r\n<p style=\"text-align: center\">[latex]9\\div\\frac{1}{2}=\\frac{9}{1}\\cdot\\frac{2}{1}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center\">[latex] \\frac{9\\cdot 2}{1\\cdot 1}=\\frac{18}{1}=18[\/latex]<\/p>\r\nThis answer is already simplified to lowest terms.\r\n<h4>Answer<\/h4>\r\n[latex]9\\div\\frac{1}{2}=18[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Dividing a Fraction by a Fraction<\/h2>\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 number that follows the division symbol.<\/li>\r\n \t<li>Multiply the first number (the one before the division symbol) by the reciprocal of the second number (the one 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=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide [latex] \\frac{2}{3}\\div \\frac{1}{6}[\/latex].\r\n\r\n[reveal-answer q=\"569112\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"569112\"]Multiply by the reciprocal.\r\n\r\n<strong>KEEP<\/strong> [latex] \\frac{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]\\frac{1}{6}[\/latex]\r\n<p style=\"text-align: center\">[latex] \\frac{2}{3}\\cdot \\frac{6}{1}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center\">[latex]\\frac{2\\cdot6}{3\\cdot1}=\\frac{12}{3}[\/latex]<\/p>\r\n&nbsp;\r\n\r\nSimplify.\r\n<p style=\"text-align: center\">[latex] \\frac{12}{3}=4[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\frac{2}{3}\\div \\frac{1}{6}=4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide [latex] \\frac{3}{5}\\div \\frac{2}{3}[\/latex].\r\n\r\n[reveal-answer q=\"950670\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"950670\"]Multiply by the reciprocal.\u00a0Keep [latex] \\frac{3}{5}[\/latex], change [latex] \\div [\/latex] to [latex]\\cdot[\/latex], and flip [latex] \\frac{2}{3}[\/latex].\r\n<p style=\"text-align: center\">[latex] \\frac{3}{5}\\cdot \\frac{3}{2}[\/latex]<\/p>\r\nMultiply numerators and multiply denominators.\r\n<p style=\"text-align: center\">[latex] \\frac{3\\cdot 3}{5\\cdot 2}=\\frac{9}{10}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<p><a href=\"https:\/\/drive.google.com\/open?id=1fYsqmQ56okTusI26peGrrZ9gv6P_1Yf7\" target=\"_blank\" rel=\"noopener\">Chapter 0 Lecture Notes<\/a><\/p>\n<div class=\"textbox learning-objectives\">\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"General Chemistry Lecture 0.1 Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/PIYCx6O1aL4?list=PLrL0qe-gjHgkzaMAgazN-4jteNKVifyCL\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>QUICK REFERENCE<\/h3>\n<p>Adding and Subtracting Fractions<\/p>\n<ul>\n<li>Convert the fractions so they have common denominators.<\/li>\n<li>Perform the addition or subtraction on the numerator and keep the common denominator.<\/li>\n<li>Simplify the answer (write the fraction in the lowest terms).<\/li>\n<\/ul>\n<p>Multiplying and Dividing Fractions<\/p>\n<ul>\n<li>To multiply, multiply across the numerators and denominators.<\/li>\n<li>To divide, multiply the first number by the reciprocal of the second number.<\/li>\n<li>Simplify the answer (write the fraction in the lowest terms).<\/li>\n<\/ul>\n<\/div>\n<h2>Adding and Subtracting Fractions<\/h2>\n<p>In order to add or subtract fractions, you first must make sure that the fractions have the same denominator. The denominator tells you how many pieces the whole has been broken into, and the numerator tells you how many of those pieces you are considering.<\/p>\n<p>To find a common denominator you will determine the\u00a0<strong>least common multiple. <\/strong>Remember that\u00a0if a number is a multiple of another, you can divide them and have no remainder.<\/p>\n<p>One way to find the least common multiple of two or more numbers is to first multiply each\u00a0by 1, 2, 3, 4, etc. \u00a0For example, find the least common multiple of 2 and 5.<\/p>\n<table>\n<tbody>\n<tr>\n<td scope=\"col\">First, list all the multiples of 2:<\/td>\n<td scope=\"col\">Then list all the multiples of 5:<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\cdot 1 = 2[\/latex]<\/td>\n<td>[latex]5\\cdot 1 = 5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\cdot 2 = 4[\/latex]<\/td>\n<td>[latex]5\\cdot 2 = 10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\cdot 3 = 6[\/latex]<\/td>\n<td>[latex]5\\cdot 3 = 15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\cdot 4 = 8[\/latex]<\/td>\n<td>[latex]5\\cdot 4 = 20[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\cdot 5 = 10[\/latex]<\/td>\n<td>[latex]5\\cdot 5 = 25[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The smallest multiple they have in common will be the common denominator.\u00a0 The least common multiple of 2 and 5 is 10.<\/p>\n<div class=\"textbox shaded\">\n<h3>Adding Fractions with Unlike Denominators<\/h3>\n<ol>\n<li>Find a common denominator.<\/li>\n<li>Rewrite each fraction using the common denominator.<\/li>\n<li>Add the numerators but keep the common denominator.<\/li>\n<li>Simplify by canceling out all common factors in the numerator and denominator.<\/li>\n<\/ol>\n<\/div>\n<h2>Simplifying a Fraction<\/h2>\n<p>A common convention used in mathematics is writing a fraction in lowest terms.\u00a0 The process of simplifying a fraction is often called <em>reducing the fraction<\/em>. We can simplify by canceling (dividing) the common factors in a fraction&#8217;s numerator and denominator. \u00a0This is possible because a fraction represents division (a part divided by the whole).<\/p>\n<div class=\"page\" title=\"Page 150\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>For example, to simplify [latex]\\frac{6}{9}[\/latex] you can rewrite 6 and 9\u00a0using the smallest factors possible as follows:<\/p>\n<p style=\"text-align: center\">[latex]\\frac{6}{9}=\\frac{2\\cdot3}{3\\cdot3}[\/latex]<\/p>\n<p>Since there is a 3 in both the numerator and denominator, and fractions can be considered division, we can divide the 3 in the top by the 3 in the bottom to reduce to 1.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{6}{9}=\\frac{2\\cdot\\cancel{3}}{3\\cdot\\cancel{3}}=\\frac{2\\cdot1}{3}=\\frac{2}{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example you are shown how to add two\u00a0fractions with different denominators, then simplify the answer.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add [latex]\\frac{2}{3}+\\frac{1}{5}[\/latex].\u00a0Simplify the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q797488\">Show Solution<\/span><\/p>\n<div id=\"q797488\" class=\"hidden-answer\" style=\"display: none\">Since the denominators are not alike, find a common denominator by multiplying the denominators.<\/p>\n<p style=\"text-align: center\">[latex]3\\cdot5=15[\/latex]<\/p>\n<p>Rewrite each fraction with a denominator of 15.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{2}{3}\\cdot \\frac{5}{5}=\\frac{10}{15}\\\\\\\\\\frac{1}{5}\\cdot \\frac{3}{3}=\\frac{3}{15}\\end{array}[\/latex]<\/p>\n<p>Add the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{10}{15}+\\frac{3}{15}=\\frac{13}{15}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{2}{3}+\\frac{1}{5}=\\frac{13}{15}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add\u00a0[latex]\\frac{3}{7}+\\frac{2}{21}[\/latex]. Simplify the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q520906\">Show Solution<\/span><\/p>\n<div id=\"q520906\" class=\"hidden-answer\" style=\"display: none\">Since the denominators are not alike, find the least common denominator by finding the least common multiple (LCM) of 7 and 21.<\/p>\n<p>Multiples of 7: 7, 14, <strong>21<\/strong><\/p>\n<p>Multiples of 21: <strong>21<\/strong><\/p>\n<p>Rewrite each fraction with a denominator of 21.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{3}{7}\\cdot \\frac{3}{3}=\\frac{9}{21}\\\\\\\\\\frac{2}{21}\\end{array}[\/latex]<\/p>\n<p>Add the fractions by adding the numerators and keeping the denominator the same. Make sure the fraction cannot be simplified.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{9}{21}+\\frac{2}{21}=\\frac{11}{21}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{3}{7}+\\frac{2}{21}=\\frac{11}{21}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Subtracting Fractions<\/h2>\n<p>When you subtract fractions, you will still find a common denominator, but the numerators will be subtracted. Below are some examples of subtracting fractions whose denominators are not alike.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract\u00a0[latex]\\frac{1}{5}-\\frac{1}{6}[\/latex]. Simplify the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q155692\">Show Solution<\/span><\/p>\n<div id=\"q155692\" class=\"hidden-answer\" style=\"display: none\">The fractions have unlike denominators, so you need to find a common denominator. Recall that a common denominator can be found by multiplying the two denominators together.<\/p>\n<p style=\"text-align: center\">[latex]5\\cdot6=30[\/latex]<\/p>\n<p>Rewrite each fraction as an equivalent fraction with a denominator of 30.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{1}{5}\\cdot \\frac{6}{6}=\\frac{6}{30}\\\\\\\\\\frac{1}{6}\\cdot \\frac{5}{5}=\\frac{5}{30}\\end{array}[\/latex]<\/p>\n<p>Subtract the numerators. Simplify the answer if needed.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{6}{30}-\\frac{5}{30}=\\frac{1}{30}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{1}{5}-\\frac{1}{6}=\\frac{1}{30}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract [latex]\\frac{5}{6}-\\frac{1}{4}[\/latex]. Simplify the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q984596\">Show Solution<\/span><\/p>\n<div id=\"q984596\" class=\"hidden-answer\" style=\"display: none\">Find the least common multiple of the denominators\u2014this is the least common denominator.<\/p>\n<p>Multiples of 6: 6, <strong>12<\/strong>, 18, 24<\/p>\n<p>Multiples of 4: 4, 8 <strong>12<\/strong>, 16, 20<\/p>\n<p>12 is the least common multiple of 6 and 4.<\/p>\n<p>Rewrite each fraction with a denominator of 12.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\frac{5}{6}\\cdot \\frac{2}{2}=\\frac{10}{12}\\\\\\\\\\frac{1}{4}\\cdot \\frac{3}{3}=\\frac{3}{12}\\end{array}[\/latex]<\/p>\n<p>Subtract the fractions. Simplify the answer if needed.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{10}{12}-\\frac{3}{12}=\\frac{7}{12}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{5}{6}-\\frac{1}{4}=\\frac{7}{12}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Multiplying Fractions<\/h2>\n<p>When you multiply a fraction by a fraction, you are finding a \u201cfraction of a fraction.\u201d\u00a0 To multiply fractions you multiply across the numerators and denominators.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiplying Two Fractions<\/h3>\n<p>[latex]\\frac{a}{b}\\cdot \\frac{c}{d}=\\frac{a\\cdot c}{b\\cdot d}=\\frac{\\text{product of the numerators}}{\\text{product of the denominators}}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Multiplying More Than Two Fractions<\/h3>\n<p>[latex]\\frac{a}{b}\\cdot \\frac{c}{d}\\cdot \\frac{e}{f}=\\frac{a\\cdot c\\cdot e}{b\\cdot d\\cdot f}[\/latex]<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply [latex]\\frac{2}{3}\\cdot \\frac{4}{5}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q368042\">Show Solution<\/span><\/p>\n<div id=\"q368042\" class=\"hidden-answer\" style=\"display: none\">Multiply the numerators and multiply the denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{2\\cdot 4}{3\\cdot 5}[\/latex]<\/p>\n<p>Simplify, if possible. This fraction is already in lowest terms.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{8}{15}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{8}{15}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Dividing Fractions<\/h2>\n<p>Dividing fractions requires using the reciprocal of a number or fraction. If you multiply two numbers together and get 1 as a result, then the two numbers are reciprocals. 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]\\frac{3}{4}[\/latex]<\/td>\n<td>[latex]\\frac{4}{3}[\/latex]<\/td>\n<td>[latex]\\frac{3}{4}\\cdot \\frac{4}{3}=\\frac{3\\cdot 4}{4\\cdot 3}=\\frac{12}{12}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{2}{1}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}\\cdot\\frac{2}{1}=\\frac{1\\cdot}{2\\cdot1}=\\frac{2}{2}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3=\\frac{3}{1}[\/latex]<\/td>\n<td>[latex]\\frac{1}{3}[\/latex]<\/td>\n<td>[latex]\\frac{3}{1}\\cdot \\frac{1}{3}=\\frac{3\\cdot 1}{1\\cdot 3}=\\frac{3}{3}=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2\\frac{1}{3}=\\frac{7}{3}[\/latex]<\/td>\n<td>[latex]\\frac{3}{7}[\/latex]<\/td>\n<td>[latex]\\frac{7}{3}\\cdot\\frac{3}{7}=\\frac{7\\cdot3}{3\\cdot7}=\\frac{21}{21}=1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Sometimes we call\u00a0the reciprocal\u00a0the \u201cflip\u201d of the other number: flip [latex]\\frac{2}{5}[\/latex] to get the reciprocal [latex]\\frac{5}{2}[\/latex].<\/p>\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\/2651\/2018\/02\/14194519\/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]\\frac{a}{0}[\/latex] is undefined. Additionally, the reciprocal of\u00a0[latex]\\frac{0}{a}[\/latex] will always be undefined.<\/div>\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>The same idea will work when the divisor (the thing being divided) is a fraction. If you have [latex]\\frac{3}{4}[\/latex] of a candy bar and need to divide it among 5 people, each person gets [latex]\\frac{1}{5}[\/latex] of the available candy:<\/p>\n<p style=\"text-align: center\">[latex]\\frac{1}{5}\\text{ of }\\frac{3}{4}=\\frac{1}{5}\\cdot \\frac{3}{4}=\\frac{3}{20}[\/latex]<\/p>\n<p style=\"text-align: center\">Each person gets [latex]\\frac{3}{20}[\/latex]\u00a0of a whole candy bar.<\/p>\n<p>If you have a recipe that needs to be divided in half, you can divide each ingredient by 2, or you can multiply each ingredient by [latex]\\frac{1}{2}[\/latex]\u00a0to find the new amount.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find [latex]\\frac{2}{3}\\div 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q769187\">Show Solution<\/span><\/p>\n<div id=\"q769187\" class=\"hidden-answer\" style=\"display: none\">Write your answer in lowest terms.<\/p>\n<p>Dividing by 4 or [latex]\\frac{4}{1}[\/latex] is the same as multiplying by the reciprocal of 4, which is [latex]\\frac{1}{4}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\frac{2}{3}\\div 4=\\frac{2}{3}\\cdot \\frac{1}{4}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{2\\cdot 1}{3\\cdot 4}=\\frac{2}{12}[\/latex]<\/p>\n<p>Simplify to lowest terms by dividing numerator and denominator by the common factor 4.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{1}{6}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{2}{3}\\div4=\\frac{1}{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide. [latex]9\\div\\frac{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\">Write your answer in lowest terms.<\/p>\n<p>Dividing by [latex]\\frac{1}{2}[\/latex] is the same as multiplying by the reciprocal of [latex]\\frac{1}{2}[\/latex], which is [latex]\\frac{2}{1}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]9\\div\\frac{1}{2}=\\frac{9}{1}\\cdot\\frac{2}{1}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{9\\cdot 2}{1\\cdot 1}=\\frac{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\\frac{1}{2}=18[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Dividing a Fraction by a Fraction<\/h2>\n<div class=\"textbox shaded\">\n<h3>Dividing with Fractions<\/h3>\n<ol>\n<li>Find the reciprocal of the number that follows the division symbol.<\/li>\n<li>Multiply the first number (the one before the division symbol) by the reciprocal of the second number (the one 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=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide [latex]\\frac{2}{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=\"q569112\">Show Solution<\/span><\/p>\n<div id=\"q569112\" class=\"hidden-answer\" style=\"display: none\">Multiply by the reciprocal.<\/p>\n<p><strong>KEEP<\/strong> [latex]\\frac{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]\\frac{1}{6}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\frac{2}{3}\\cdot \\frac{6}{1}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{2\\cdot6}{3\\cdot1}=\\frac{12}{3}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{12}{3}=4[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{2}{3}\\div \\frac{1}{6}=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide [latex]\\frac{3}{5}\\div \\frac{2}{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q950670\">Show Solution<\/span><\/p>\n<div id=\"q950670\" class=\"hidden-answer\" style=\"display: none\">Multiply by the reciprocal.\u00a0Keep [latex]\\frac{3}{5}[\/latex], change [latex]\\div[\/latex] to [latex]\\cdot[\/latex], and flip [latex]\\frac{2}{3}[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\frac{3}{5}\\cdot \\frac{3}{2}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center\">[latex]\\frac{3\\cdot 3}{5\\cdot 2}=\\frac{9}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/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-6267\">\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>Revision and Adaptiation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Add Fractions with Unlike Denominators (Basic with Model). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/zV4q7j1-89I\">https:\/\/youtu.be\/zV4q7j1-89I<\/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: Subtract Fractions with Unlike Denominators (Basic with Model) Mathispower4u . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/RpHtOMjeI7g\">https:\/\/youtu.be\/RpHtOMjeI7g<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 2: Fractions and Mixed Numbers, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/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><li>College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/yqV9q0HH@7.3:s7ku6WX5@2\/Multiply-and-Divide-Fractions\">http:\/\/cnx.org\/contents\/yqV9q0HH@7.3:s7ku6WX5@2\/Multiply-and-Divide-Fractions<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@7.3<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Multiply and Divide Fractions. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/yqV9q0HH@7.3:s7ku6WX5@2\/Multiply-and-Divide-Fractions\">http:\/\/cnx.org\/contents\/yqV9q0HH@7.3:s7ku6WX5@2\/Multiply-and-Divide-Fractions<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@7.3<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":23485,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Add Fractions with Unlike Denominators (Basic with Model)\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/zV4q7j1-89I\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Subtract Fractions with Unlike Denominators 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