{"id":1393,"date":"2021-11-01T17:23:18","date_gmt":"2021-11-01T17:23:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=1393"},"modified":"2023-01-12T13:14:55","modified_gmt":"2023-01-12T13:14:55","slug":"1-3-3-simplifying-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/1-3-3-simplifying-fractions\/","title":{"raw":"1.3.3: Rational Numbers","rendered":"1.3.3: Rational Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Identify a fraction as proper or improper<\/li>\r\n \t<li>Find equivalent fractions<\/li>\r\n \t<li>Simplify fractions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key words<\/h3>\r\n<ul>\r\n \t<li style=\"margin-top: 0.5em;\"><strong>Fraction<\/strong>: a rational number<\/li>\r\n \t<li><strong>Numerator<\/strong>: the integer on top of a fraction<\/li>\r\n \t<li><strong>Denominator<\/strong>: the integer on the bottom of a fraction<\/li>\r\n \t<li><strong>Proper fraction<\/strong>: a fraction whose value lies between 0 and 1<\/li>\r\n \t<li><strong>Improper fraction<\/strong>:\u00a0a fraction whose value is greater than or equal to 1<\/li>\r\n \t<li><strong>Equivalent fractions<\/strong>: two or more fractions with equal value<\/li>\r\n \t<li><strong>Simplified fraction<\/strong>: a fraction that has no common factors (other than 1) in the numerator and denominator<\/li>\r\n<\/ul>\r\n<\/div>\r\nOften in life, whole amounts are not exactly what we need. A baker must use a little more than a cup of milk or part of a teaspoon of salt. Similarly a carpenter might need less than a foot of wood and a painter might use part of a gallon of paint. In the next section, we will learn about numbers that describe parts of a whole. These numbers, called fractions, are very useful both in algebra and in everyday life. We have also seen these types of numbers in the real number system. They are called <em><strong>rational numbers<\/strong><\/em>. The set of rational numbers is the set of numbers that can be written as a fraction of two integers. [latex]\\mathbb{Q}=\\,\\left\\{\\dfrac{m}{n}\\normalsize \\;\\large\\vert\\;\\normalsize\\,m\\text{ and }{n}\\text{ are integers and }{n}\\ne{ 0 }\\right\\}[\/latex].\r\n<div><\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>RATIONAL NUMBERS<\/h3>\r\nA <strong>rational number <\/strong>(also called a fraction)\u00a0is written [latex]\\frac{a}{b}[\/latex], where [latex]a[\/latex] and\u00a0[latex]b[\/latex] are integers and [latex]b \\neq 0 [\/latex]. In a fraction,\u00a0[latex]a[\/latex] is called the numerator and\u00a0[latex]b[\/latex] is called the denominator.\r\n\r\n<\/div>\r\nRational numbers are also called fractions. The top number is referred to as the <em><strong>numerator<\/strong><\/em> of the fraction and the bottom number is the <em><strong>denominator<\/strong><\/em>. When the fraction is positive, and the numerator is smaller than the denominator, the fraction is a <em><strong>proper fraction<\/strong><\/em><em>. <\/em>This means that the fraction is less than [latex]1[\/latex] and represents part of a whole. For example, [latex]\\frac{2}{3}[\/latex] is a proper fraction since [latex]2 \\lt 3[\/latex].\u00a0[latex]\\frac{2}{3}[\/latex] is part of a whole. On the other hand, when the numerator is greater than or equal to the denominator, the fraction is said to be <em><strong>improper<\/strong><\/em>. The fractions [latex]\\frac{7}{4}[\/latex] and\u00a0[latex]\\frac{5}{5}[\/latex] are examples of improper fractions because their numerators are greater than or equal to their denominators. Improper fractions represent numbers that are greater than or equal to 1; they are a whole or larger.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nDetermine of the fraction is proper or improper:\r\n<ul>\r\n \t<li>[latex]\\frac{2}{5}[\/latex] \u00a0Proper since [latex]2\\lt 5[\/latex]<\/li>\r\n \t<li>[latex]\\frac{3}{4}[\/latex] \u00a0Proper since [latex]3\\lt 4[\/latex]<\/li>\r\n \t<li>[latex]\\frac{7}{2}[\/latex] \u00a0Improper since [latex]7\\geq 2[\/latex]<\/li>\r\n \t<li>[latex]\\frac{9}{9}[\/latex] \u00a0Improper since [latex]9\\geq 9[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nDetermine of the fraction is proper or improper:\r\n<ul>\r\n \t<li>[latex]\\frac{6}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{8}{9}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{7}{7}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{6}{9}[\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"595809\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"595809\"]\r\n<ul>\r\n \t<li>[latex]\\frac{6}{5}[\/latex] \u00a0Improper since [latex]6\\geq 5[\/latex]<\/li>\r\n \t<li>[latex]\\frac{8}{9}[\/latex] \u00a0Proper since [latex]8\\lt 9[\/latex]<\/li>\r\n \t<li>[latex]\\frac{7}{7}[\/latex] \u00a0Improper since [latex]7\\geq 7[\/latex]<\/li>\r\n \t<li>[latex]\\frac{6}{9}[\/latex] \u00a0Proper since [latex]6\\lt 9[\/latex]<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Equivalent Fractions<\/h2>\r\nIf Anwar eats [latex]\\frac{1}{2}[\/latex] of a pizza and Bobby eats [latex]\\frac{2}{4}[\/latex] of the pizza, have they eaten the same amount of pizza? In other words, does [latex]\\frac{1}{2}=\\frac{2}{4}[\/latex]? We can use fraction tiles to find out whether Anwar and Bobby have eaten <em>equivalent<\/em> parts of the pizza.\r\n<div class=\"textbox shaded\">\r\n<h3>Equivalent Fractions<\/h3>\r\nEquivalent fractions are fractions that have the same value.\r\n\r\n<\/div>\r\nFraction tiles serve as a useful model of equivalent fractions. You may want to use fraction tiles to do the following activity. Or you might make a copy of the fraction tiles shown earlier\u00a0and extend it to include eighths, tenths, and twelfths.\r\n\r\nStart with a [latex]\\frac{1}{2}[\/latex] tile. How many fourths equal one-half? How many of the [latex]\\frac{1}{4}[\/latex] tiles exactly cover the [latex]\\frac{1}{2}[\/latex] tile?\r\n<p style=\"text-align: center;\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220749\/CNX_BMath_Figure_04_01_037_img.png\" alt=\"One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into four pieces, each labeled as one fourth.\" \/>\r\nSince two [latex]\\frac{1}{4}[\/latex] tiles cover the [latex]\\frac{1}{2}[\/latex] tile, we see that [latex]\\frac{2}{4}[\/latex] is the same as [latex]\\frac{1}{2}[\/latex], or [latex]\\frac{2}{4}=\\frac{1}{2}[\/latex].<\/p>\r\nHow many of the [latex]\\frac{1}{6}[\/latex] tiles cover the [latex]\\frac{1}{2}[\/latex] tile?\r\n<p style=\"text-align: center;\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220751\/CNX_BMath_Figure_04_01_038_img.png\" alt=\"One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into six pieces, each labeled as one sixth.\" \/>\r\nSince three [latex]\\frac{1}{6}[\/latex] tiles cover the [latex]\\frac{1}{2}[\/latex] tile, we see that [latex]\\frac{3}{6}[\/latex] is the same as [latex]\\frac{1}{2}[\/latex].<\/p>\r\nSo, [latex]\\frac{3}{6}=\\frac{1}{2}[\/latex]. The fractions are equivalent fractions.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse fraction tiles to find equivalent fractions. Show your result with a figure.\r\n<ol>\r\n \t<li>How many eighths [latex]\\left( \\frac{1}{8} \\right) [\/latex] equal one-half [latex]\\left( \\frac{1}{2} \\right) [\/latex]?<\/li>\r\n \t<li>How many tenths [latex]\\left( \\frac{1}{10} \\right) [\/latex] equal one-half [latex]\\left( \\frac{1}{2} \\right) [\/latex]?<\/li>\r\n \t<li>How many twelfths [latex]\\left( \\frac{1}{12} \\right) [\/latex] equal one-half [latex]\\left( \\frac{1}{2}\\right)[\/latex]?<\/li>\r\n<\/ol>\r\nSolution\r\n1. It takes four [latex]\\frac{1}{8}[\/latex] tiles to exactly cover the [latex]\\frac{1}{2}[\/latex] tile, so [latex]\\frac{4}{8}=\\frac{1}{2}[\/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\/24220752\/CNX_BMath_Figure_04_01_070_img.png\" alt=\"One long, undivided rectangle is shown, labeled 1. Below it is an identical rectangle divided vertically into two pieces, each labeled 1 half. Below that is an identical rectangle divided vertically into eight pieces, each labeled 1 eighth.\" \/>\r\n2. It takes five [latex]\\frac{1}{10}[\/latex] tiles to exactly cover the [latex]\\frac{1}{2}[\/latex] tile, so [latex]\\frac{5}{10}=\\frac{1}{2}[\/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\/24220753\/CNX_BMath_Figure_04_01_039_img.png\" alt=\"One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into ten pieces, each labeled as one tenth.\" \/>\r\n3. It takes six [latex]\\frac{1}{12}[\/latex] tiles to exactly cover the [latex]\\frac{1}{2}[\/latex] tile, so [latex]\\frac{6}{12}=\\frac{1}{2}[\/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\/24220755\/CNX_BMath_Figure_04_01_040_img.png\" alt=\"One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into twelve pieces, each labeled as one twelfth.\" \/>\r\n\r\n<\/div>\r\nSuppose you had tiles marked [latex]\\frac{1}{20}[\/latex]. How many of them would it take to equal [latex]\\frac{1}{2}[\/latex]? Are you thinking ten tiles? If you are, you\u2019re right, because [latex]\\frac{10}{20}=\\frac{1}{2}[\/latex].\r\n\r\nWe have shown that [latex]\\frac{1}{2},\\frac{2}{4},\\frac{3}{6},\\frac{4}{8},\\frac{5}{10},\\frac{6}{12}[\/latex], and [latex]\\frac{10}{20}[\/latex] are all equivalent fractions.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[ohm_question height=\"270\"]146001[\/ohm_question]\r\n\r\n<\/div>\r\n<h3>Finding Equivalent Fractions<\/h3>\r\nWe used fraction tiles to show that there are many fractions equivalent to [latex]\\frac{1}{2}[\/latex]. For example, [latex]\\frac{2}{4},\\frac{3}{6}[\/latex], and [latex]\\frac{4}{8}[\/latex] are all equivalent to [latex]\\frac{1}{2}[\/latex]. When we lined up the fraction tiles, it took four of the [latex]\\frac{1}{8}[\/latex] tiles to make the same length as a [latex]\\frac{1}{2}[\/latex] tile. This showed that [latex]\\frac{4}{8}=\\frac{1}{2}[\/latex]. See the previous example.\r\n\r\nWe can show this with pizzas, too. Image (a) shows a single pizza, cut into two equal pieces with [latex]{\\frac{1}{2}}[\/latex] shaded. Image (b) shows a second pizza of the same size, cut into eight pieces with [latex]{\\frac{4}{8}}[\/latex] shaded.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220757\/CNX_BMath_Figure_04_01_071_img.png\" alt=\"Two pizzas are shown. The pizza on the left is divided into 2 equal pieces. 1 piece is shaded. The pizza on the right is divided into 8 equal pieces. 4 pieces are shaded.\" \/>\r\nThis is another way to show that [latex]\\frac{1}{2}[\/latex] is equivalent to [latex]\\frac{4}{8}[\/latex].\r\n\r\nHow can we use mathematics to change [latex]\\frac{1}{2}[\/latex] into [latex]\\frac{4}{8}[\/latex]? How could you take a pizza that is cut into two pieces and cut it into eight pieces? You could cut each of the two larger pieces into four smaller pieces! The whole pizza would then be cut into eight pieces instead of just two. Mathematically, what we\u2019ve described could be written as:\r\n<p style=\"text-align: center;\">[latex]\\frac{1\\cdot\\color{blue}{4}}{2\\cdot\\color{blue}{4}}=\\frac{4}{8}[\/latex]<\/p>\r\nThese models lead to the <em><strong>Equivalent Fractions<\/strong> <strong>Property<\/strong><\/em>, which states that if we multiply the numerator and denominator of a fraction by the same number, the value of the fraction does not change.\r\n<div class=\"textbox shaded\">\r\n<h3>Equivalent Fractions Property<\/h3>\r\nIf [latex]a,b[\/latex], and [latex]c[\/latex] are numbers where [latex]b\\ne 0[\/latex] and [latex]c\\ne 0[\/latex], then\r\n\r\n[latex]\\frac{a}{b}=\\frac{a\\cdot c}{b\\cdot c}[\/latex]\r\n\r\n<\/div>\r\nWhen working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. For example, consider the fraction one-half.\r\n<p style=\"text-align: center;\">[latex]{\\frac{1\\cdot\\color{blue}{3}}{2\\cdot\\color{blue}{3}}}={\\frac{3}{6}}[\/latex] so [latex]{\\frac{1}{2}}={\\frac{3}{6}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]{\\frac{1\\cdot\\color{blue}{2}}{2\\cdot\\color{blue}{2}}}={\\frac{2}{4}}[\/latex] so [latex]{\\frac{1}{2}}={\\frac{2}{4}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]{\\frac{1\\cdot\\color{blue}{10}}{2\\cdot\\color{blue}{10}}}={\\frac{10}{20}}[\/latex] so [latex]{\\frac{1}{2}}={\\frac{10}{20}}[\/latex]<\/p>\r\nSo, we say that [latex]\\frac{1}{2},\\frac{2}{4},\\frac{3}{6}[\/latex], and [latex]\\frac{10}{20}[\/latex] are <em><strong>equivalent fractions<\/strong><\/em>.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind three fractions equivalent to [latex]{\\frac{2}{5}}[\/latex].\r\n[reveal-answer q=\"931791\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"931791\"]\r\n\r\nSolution\r\nTo find a fraction equivalent to [latex]{\\frac{2}{5}}[\/latex], we multiply the numerator and denominator by the same number (but not zero). Let us multiply them by [latex]2,3[\/latex], and [latex]5[\/latex].\r\n\r\n[latex]{\\frac{2\\cdot\\color{blue}{2}}{5\\cdot\\color{blue}{2}}}={\\frac{4}{10}}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]{\\frac{2\\cdot\\color{blue}{3}}{5\\cdot\\color{blue}{3}}}={\\frac{6}{15}}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]{\\frac{2\\cdot\\color{blue}{5}}{5\\cdot\\color{blue}{5}}}={\\frac{10}{25}}[\/latex]\r\n\r\nSo, [latex]{\\frac{4}{10},\\frac{6}{15}}[\/latex], and [latex]{\\frac{10}{25}}[\/latex] are equivalent to [latex]{\\frac{2}{5}}[\/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\nFind three fractions equivalent to [latex]{\\frac{3}{5}}[\/latex].\r\n[reveal-answer q=\"675004\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"675004\"]\r\n\r\nCorrect answers include [latex]{\\frac{6}{10},\\frac{9}{15}},\\text{and }{\\frac{12}{20}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nFind three fractions equivalent to [latex]{\\frac{4}{5}}[\/latex].\r\n[reveal-answer q=\"171774\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"171774\"]\r\n\r\nCorrect answers include [latex]{\\frac{8}{10},\\frac{12}{15}},\\text{and }{\\frac{16}{20}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWhen we add and subtract fractions, we often need to <em><strong>build<\/strong><\/em> fractions to create fractions with the same denominator. We build fractions by multiplying the numerator and denominator of the fraction by the same number (the equivalent fractions property).\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind a fraction with a denominator of [latex]21[\/latex] that is equivalent to [latex]{\\frac{2}{7}}[\/latex].\r\n[reveal-answer q=\"810854\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"810854\"]\r\n\r\nTo find equivalent fractions, we multiply the numerator and denominator by the same number. In this case, we need to multiply the denominator by a number that will result in [latex]21[\/latex].\r\n\r\nSince we can multiply [latex]7[\/latex] by [latex]3[\/latex] to get [latex]21[\/latex], we can find the equivalent fraction by multiplying both the numerator and denominator by [latex]3[\/latex].\r\n<p style=\"text-align: center;\">[latex]{\\frac{2}{7}}={\\frac{2\\cdot\\color{blue}{3}}{7\\cdot\\color{blue}{3}}}={\\frac{6}{21}}[\/latex]<\/p>\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\"]146005[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to find an equivalent fraction given a specific denominator.\r\n\r\nhttps:\/\/youtu.be\/8gJS0kvtGFU\r\n<h2><strong>Positive and Negative Fractions<\/strong><\/h2>\r\nRational numbers can be negative as well as positive. Being negative just means that they sit on the opposite side of zero from the positive version of the fraction.\r\n\r\nFor example, the fraction [latex]\\frac{1}{2}[\/latex] is located exactly half-way between [latex]0[\/latex] and [latex]1[\/latex] on the number line. The distance between\u00a0[latex]0[\/latex] and [latex]1[\/latex] is split into two equal lengths and\u00a0[latex]\\frac{1}{2}[\/latex] sits exactly one of the two equal lengths from zero.\r\n\r\n<img class=\"aligncenter wp-image-1598 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/11\/10173313\/FRACTIONS-ON-NUMBER-LINE-1024x160.png\" alt=\"Fractions on a number line\" width=\"1024\" height=\"160\" \/>\r\n\r\n[latex]-\\frac{1}{2}[\/latex] is the opposite of\u00a0[latex]\\frac{1}{2}[\/latex] and lies on the opposite side of zero.\r\n\r\n[latex]\\frac{3}{2}[\/latex] lies three half-units from zero, and\u00a0[latex]-\\frac{3}{2}[\/latex] lies three\u00a0half-units from zero on the negative side of the number line. Notice also, that\u00a0[latex]\\frac{1}{2}[\/latex] lies between\u00a0[latex]0[\/latex] and [latex]1[\/latex] so is a proper fraction, while\u00a0[latex]\\frac{3}{2}[\/latex] is greater than or equal to\u00a0[latex]1[\/latex] so is an improper fraction.\r\n<h2>Simplifying Fractions<\/h2>\r\nIn working with equivalent fractions, we saw that there are many ways to write fractions that have the same value, or represent the same part of the whole. How do we know which one to use? Most often, we\u2019ll use the fraction that is in <strong><em>simplified<\/em><\/strong> form.\r\n\r\nA fraction is considered simplified if there are no common factors, other than [latex]1[\/latex], in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can simplify the fraction to its simplified form by dividing the numerator and denominator by the common factors.\r\n<div class=\"textbox shaded\">\r\n<h3>Simplified Fraction<\/h3>\r\nA fraction is considered simplified if there are no common factors in the numerator and denominator.\r\n\r\n<\/div>\r\nFor example,\r\n<ul id=\"fs-id1302300\">\r\n \t<li>[latex]\\frac{2}{3}[\/latex] is simplified because, other than 1, there are no common factors of [latex]2[\/latex] and [latex]3[\/latex].<\/li>\r\n \t<li>[latex]\\frac{10}{15}[\/latex] is NOT simplified because [latex]5[\/latex] is a common factor of [latex]10[\/latex] and [latex]15[\/latex].<\/li>\r\n<\/ul>\r\n&nbsp;\r\n\r\nWe can use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.\r\n<div class=\"textbox shaded\">\r\n<h3>Equivalent Fractions Property<\/h3>\r\nIf [latex]a,b,c[\/latex] are integers where [latex]b\\ne 0,c\\ne 0[\/latex], then\r\n\r\n[latex]{\\frac{a}{b}}={\\frac{a\\cdot c}{b\\cdot c}}\\text{ and }{\\frac{a\\cdot c}{b\\cdot c}}={\\frac{a}{b}}[\/latex].\r\n\r\n<\/div>\r\nNotice that [latex]c[\/latex] is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, we can divide the numerator and denominator by the common factor to remove it.\r\n<div class=\"textbox shaded\">\r\n<h3>Simplifying a fraction.<\/h3>\r\n<ol id=\"eip-id1168467382990\" class=\"stepwise\">\r\n \t<li>Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.<\/li>\r\n \t<li>Simplify, using the equivalent fractions property, by dividing both the numerator and denominator by the common factor to remove it.<\/li>\r\n \t<li>Multiply any remaining factors.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h3><\/h3>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify: [latex]\\frac{10}{15}[\/latex]\r\n\r\nSolution:\r\nTo simplify the fraction, we look for any common factors in the numerator and the denominator.\r\n<table id=\"eip-id1168468231694\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td>Notice that [latex]5[\/latex] is a factor of both [latex]10[\/latex] and [latex]15[\/latex].<\/td>\r\n<td>[latex]\\frac{10}{15}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Factor the numerator and denominator.<\/td>\r\n<td>[latex]\\frac{2\\cdot5}{3\\cdot5}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5 is a common factor so divide the numerator and denominator by 5 to remove it.<\/td>\r\n<td>[latex]\\frac{2\\cdot\\color{red}{5}}{3\\cdot\\color{red}{5}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\frac{2}{3}[\/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\nSimplify the fractions:\r\n\r\n1. [latex]\\frac{14}{21}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a02. [latex]\\frac{20}{50}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a03. [latex]\\frac{12}{24}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a04. [latex]\\frac{8}{28}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a05. [latex]\\frac{17}{17}[\/latex]\r\n\r\n[reveal-answer q=\"377520\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"377520\"]\r\n\r\n1. [latex]\\frac{14}{21}=\\frac{2}{3}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a02. [latex]\\frac{20}{50}=\\frac{2}{5}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a03. [latex]\\frac{12}{24}=\\frac{1}{2}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a04. [latex]\\frac{8}{28}=-\\frac{2}{7}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a05. [latex]\\frac{17}{17}=1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nTo simplify a negative fraction, we use the same process as in the previous example. Remember to keep the negative sign.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify: [latex]-\\frac{18}{24}[\/latex]\r\n[reveal-answer q=\"270732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"270732\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168469841089\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"We notice that 18 and 24 both have factors,\">\r\n<tbody>\r\n<tr>\r\n<td>We notice that 18 and 24 both have factors of 6.<\/td>\r\n<td>[latex]-\\frac{18}{24}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite the numerator and denominator showing the common factor.<\/td>\r\n<td>[latex]-\\frac{3\\cdot6}{4\\cdot6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors by division.<\/td>\r\n<td>[latex]-\\frac{3\\cdot\\color{red}{6}}{4\\cdot\\color{red}{6}}[\/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\nSimplify the fractions:\r\n\r\n1. [latex]-\\frac{14}{21}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a02. [latex]-\\frac{40}{50}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a03. [latex]-\\frac{48}{24}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a04. [latex]-\\frac{21}{28}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a05. [latex]-\\frac{23}{23}[\/latex]\r\n\r\n[reveal-answer q=\"HM7520\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"HM7520\"]\r\n\r\n1. [latex]-\\frac{14}{21}=-\\frac{2}{3}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a02. [latex]-\\frac{40}{50}=-\\frac{4}{5}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a03. [latex]-\\frac{48}{24}=-2[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a04. [latex]-\\frac{21}{28}=-\\frac{3}{4}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a05. [latex]-\\frac{23}{23}=-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see another example of how to simplify a fraction.\r\n\r\nhttps:\/\/youtu.be\/_2Wk7jXf3Ok\r\n\r\nAfter simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: <em>a fraction is considered simplified if there are no common factors in the numerator and denominator<\/em>.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify: [latex]-\\frac{56}{32}[\/latex]\r\n[reveal-answer q=\"877414\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"877414\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168466600046\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line shows negative 56 over 32. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-\\frac{56}{32}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite the numerator and denominator, showing the common factors, 8.<\/td>\r\n<td>[latex]-\\frac{7\\cdot8}{4\\cdot8}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove common factors by division.<\/td>\r\n<td>[latex]-\\frac{7\\cdot\\color{red}{8}}{4\\cdot\\color{red}{8}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-\\frac{7}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the Equivalent Fractions Property.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSimplify: [latex]\\frac{210}{385}[\/latex]\r\n[reveal-answer q=\"721590\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"721590\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168467251049\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The fraction 210 over 385 is shown. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td>Use factor trees to factor the numerator and denominator.<\/td>\r\n<td>[latex]\\frac{210}{385}[\/latex]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220913\/CNX_BMath_Figure_04_02_028_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite the numerator and denominator as the product of the primes.<\/td>\r\n<td>[latex]{\\frac{210}{385}}={\\frac{2\\cdot 3\\cdot 5\\cdot 7}{5\\cdot 7\\cdot 11}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Remove the common factors by division.<\/td>\r\n<td>[latex]\\frac{2\\cdot 3\\cdot\\color{blue}{5}\\cdot\\color{red}{7}}{\\color{blue}{5}\\cdot\\color{red}{7}\\cdot 11}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]\\frac{2\\cdot 3}{11}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply any remaining factors.<\/td>\r\n<td>[latex]\\frac{6}{11}[\/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\nSimplify [latex]\\frac{315}{675}[\/latex]\r\n\r\n[reveal-answer q=\"495207\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"495207\"]\r\n\r\n[latex]\\frac{315}{675}=\\frac{\\color{red}{3\\cdot 3\\cdot} \\color{red}{5\\cdot} 7}{\\color{red}{3\\cdot 3\\cdot} 3\\cdot \\color{red}{5\\cdot} 5 }=\\frac{7}{15}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><\/h2>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Identify a fraction as proper or improper<\/li>\n<li>Find equivalent fractions<\/li>\n<li>Simplify fractions<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li style=\"margin-top: 0.5em;\"><strong>Fraction<\/strong>: a rational number<\/li>\n<li><strong>Numerator<\/strong>: the integer on top of a fraction<\/li>\n<li><strong>Denominator<\/strong>: the integer on the bottom of a fraction<\/li>\n<li><strong>Proper fraction<\/strong>: a fraction whose value lies between 0 and 1<\/li>\n<li><strong>Improper fraction<\/strong>:\u00a0a fraction whose value is greater than or equal to 1<\/li>\n<li><strong>Equivalent fractions<\/strong>: two or more fractions with equal value<\/li>\n<li><strong>Simplified fraction<\/strong>: a fraction that has no common factors (other than 1) in the numerator and denominator<\/li>\n<\/ul>\n<\/div>\n<p>Often in life, whole amounts are not exactly what we need. A baker must use a little more than a cup of milk or part of a teaspoon of salt. Similarly a carpenter might need less than a foot of wood and a painter might use part of a gallon of paint. In the next section, we will learn about numbers that describe parts of a whole. These numbers, called fractions, are very useful both in algebra and in everyday life. We have also seen these types of numbers in the real number system. They are called <em><strong>rational numbers<\/strong><\/em>. The set of rational numbers is the set of numbers that can be written as a fraction of two integers. [latex]\\mathbb{Q}=\\,\\left\\{\\dfrac{m}{n}\\normalsize \\;\\large\\vert\\;\\normalsize\\,m\\text{ and }{n}\\text{ are integers and }{n}\\ne{ 0 }\\right\\}[\/latex].<\/p>\n<div><\/div>\n<div class=\"textbox shaded\">\n<h3>RATIONAL NUMBERS<\/h3>\n<p>A <strong>rational number <\/strong>(also called a fraction)\u00a0is written [latex]\\frac{a}{b}[\/latex], where [latex]a[\/latex] and\u00a0[latex]b[\/latex] are integers and [latex]b \\neq 0[\/latex]. In a fraction,\u00a0[latex]a[\/latex] is called the numerator and\u00a0[latex]b[\/latex] is called the denominator.<\/p>\n<\/div>\n<p>Rational numbers are also called fractions. The top number is referred to as the <em><strong>numerator<\/strong><\/em> of the fraction and the bottom number is the <em><strong>denominator<\/strong><\/em>. When the fraction is positive, and the numerator is smaller than the denominator, the fraction is a <em><strong>proper fraction<\/strong><\/em><em>. <\/em>This means that the fraction is less than [latex]1[\/latex] and represents part of a whole. For example, [latex]\\frac{2}{3}[\/latex] is a proper fraction since [latex]2 \\lt 3[\/latex].\u00a0[latex]\\frac{2}{3}[\/latex] is part of a whole. On the other hand, when the numerator is greater than or equal to the denominator, the fraction is said to be <em><strong>improper<\/strong><\/em>. The fractions [latex]\\frac{7}{4}[\/latex] and\u00a0[latex]\\frac{5}{5}[\/latex] are examples of improper fractions because their numerators are greater than or equal to their denominators. Improper fractions represent numbers that are greater than or equal to 1; they are a whole or larger.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Determine of the fraction is proper or improper:<\/p>\n<ul>\n<li>[latex]\\frac{2}{5}[\/latex] \u00a0Proper since [latex]2\\lt 5[\/latex]<\/li>\n<li>[latex]\\frac{3}{4}[\/latex] \u00a0Proper since [latex]3\\lt 4[\/latex]<\/li>\n<li>[latex]\\frac{7}{2}[\/latex] \u00a0Improper since [latex]7\\geq 2[\/latex]<\/li>\n<li>[latex]\\frac{9}{9}[\/latex] \u00a0Improper since [latex]9\\geq 9[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Determine of the fraction is proper or improper:<\/p>\n<ul>\n<li>[latex]\\frac{6}{5}[\/latex]<\/li>\n<li>[latex]\\frac{8}{9}[\/latex]<\/li>\n<li>[latex]\\frac{7}{7}[\/latex]<\/li>\n<li>[latex]\\frac{6}{9}[\/latex]<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q595809\">Show Answer<\/span><\/p>\n<div id=\"q595809\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>[latex]\\frac{6}{5}[\/latex] \u00a0Improper since [latex]6\\geq 5[\/latex]<\/li>\n<li>[latex]\\frac{8}{9}[\/latex] \u00a0Proper since [latex]8\\lt 9[\/latex]<\/li>\n<li>[latex]\\frac{7}{7}[\/latex] \u00a0Improper since [latex]7\\geq 7[\/latex]<\/li>\n<li>[latex]\\frac{6}{9}[\/latex] \u00a0Proper since [latex]6\\lt 9[\/latex]<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Equivalent Fractions<\/h2>\n<p>If Anwar eats [latex]\\frac{1}{2}[\/latex] of a pizza and Bobby eats [latex]\\frac{2}{4}[\/latex] of the pizza, have they eaten the same amount of pizza? In other words, does [latex]\\frac{1}{2}=\\frac{2}{4}[\/latex]? We can use fraction tiles to find out whether Anwar and Bobby have eaten <em>equivalent<\/em> parts of the pizza.<\/p>\n<div class=\"textbox shaded\">\n<h3>Equivalent Fractions<\/h3>\n<p>Equivalent fractions are fractions that have the same value.<\/p>\n<\/div>\n<p>Fraction tiles serve as a useful model of equivalent fractions. You may want to use fraction tiles to do the following activity. Or you might make a copy of the fraction tiles shown earlier\u00a0and extend it to include eighths, tenths, and twelfths.<\/p>\n<p>Start with a [latex]\\frac{1}{2}[\/latex] tile. How many fourths equal one-half? How many of the [latex]\\frac{1}{4}[\/latex] tiles exactly cover the [latex]\\frac{1}{2}[\/latex] tile?<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220749\/CNX_BMath_Figure_04_01_037_img.png\" alt=\"One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into four pieces, each labeled as one fourth.\" \/><br \/>\nSince two [latex]\\frac{1}{4}[\/latex] tiles cover the [latex]\\frac{1}{2}[\/latex] tile, we see that [latex]\\frac{2}{4}[\/latex] is the same as [latex]\\frac{1}{2}[\/latex], or [latex]\\frac{2}{4}=\\frac{1}{2}[\/latex].<\/p>\n<p>How many of the [latex]\\frac{1}{6}[\/latex] tiles cover the [latex]\\frac{1}{2}[\/latex] tile?<\/p>\n<p style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220751\/CNX_BMath_Figure_04_01_038_img.png\" alt=\"One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into six pieces, each labeled as one sixth.\" \/><br \/>\nSince three [latex]\\frac{1}{6}[\/latex] tiles cover the [latex]\\frac{1}{2}[\/latex] tile, we see that [latex]\\frac{3}{6}[\/latex] is the same as [latex]\\frac{1}{2}[\/latex].<\/p>\n<p>So, [latex]\\frac{3}{6}=\\frac{1}{2}[\/latex]. The fractions are equivalent fractions.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use fraction tiles to find equivalent fractions. Show your result with a figure.<\/p>\n<ol>\n<li>How many eighths [latex]\\left( \\frac{1}{8} \\right)[\/latex] equal one-half [latex]\\left( \\frac{1}{2} \\right)[\/latex]?<\/li>\n<li>How many tenths [latex]\\left( \\frac{1}{10} \\right)[\/latex] equal one-half [latex]\\left( \\frac{1}{2} \\right)[\/latex]?<\/li>\n<li>How many twelfths [latex]\\left( \\frac{1}{12} \\right)[\/latex] equal one-half [latex]\\left( \\frac{1}{2}\\right)[\/latex]?<\/li>\n<\/ol>\n<p>Solution<br \/>\n1. It takes four [latex]\\frac{1}{8}[\/latex] tiles to exactly cover the [latex]\\frac{1}{2}[\/latex] tile, so [latex]\\frac{4}{8}=\\frac{1}{2}[\/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\/24220752\/CNX_BMath_Figure_04_01_070_img.png\" alt=\"One long, undivided rectangle is shown, labeled 1. Below it is an identical rectangle divided vertically into two pieces, each labeled 1 half. Below that is an identical rectangle divided vertically into eight pieces, each labeled 1 eighth.\" \/><br \/>\n2. It takes five [latex]\\frac{1}{10}[\/latex] tiles to exactly cover the [latex]\\frac{1}{2}[\/latex] tile, so [latex]\\frac{5}{10}=\\frac{1}{2}[\/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\/24220753\/CNX_BMath_Figure_04_01_039_img.png\" alt=\"One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into ten pieces, each labeled as one tenth.\" \/><br \/>\n3. It takes six [latex]\\frac{1}{12}[\/latex] tiles to exactly cover the [latex]\\frac{1}{2}[\/latex] tile, so [latex]\\frac{6}{12}=\\frac{1}{2}[\/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\/24220755\/CNX_BMath_Figure_04_01_040_img.png\" alt=\"One long, undivided rectangle is shown. Below it is a rectangle divided vertically into two pieces, each labeled as one half. Below that is a rectangle divided vertically into twelve pieces, each labeled as one twelfth.\" \/><\/p>\n<\/div>\n<p>Suppose you had tiles marked [latex]\\frac{1}{20}[\/latex]. How many of them would it take to equal [latex]\\frac{1}{2}[\/latex]? Are you thinking ten tiles? If you are, you\u2019re right, because [latex]\\frac{10}{20}=\\frac{1}{2}[\/latex].<\/p>\n<p>We have shown that [latex]\\frac{1}{2},\\frac{2}{4},\\frac{3}{6},\\frac{4}{8},\\frac{5}{10},\\frac{6}{12}[\/latex], and [latex]\\frac{10}{20}[\/latex] are all equivalent fractions.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146001\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146001&theme=oea&iframe_resize_id=ohm146001&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<h3>Finding Equivalent Fractions<\/h3>\n<p>We used fraction tiles to show that there are many fractions equivalent to [latex]\\frac{1}{2}[\/latex]. For example, [latex]\\frac{2}{4},\\frac{3}{6}[\/latex], and [latex]\\frac{4}{8}[\/latex] are all equivalent to [latex]\\frac{1}{2}[\/latex]. When we lined up the fraction tiles, it took four of the [latex]\\frac{1}{8}[\/latex] tiles to make the same length as a [latex]\\frac{1}{2}[\/latex] tile. This showed that [latex]\\frac{4}{8}=\\frac{1}{2}[\/latex]. See the previous example.<\/p>\n<p>We can show this with pizzas, too. Image (a) shows a single pizza, cut into two equal pieces with [latex]{\\frac{1}{2}}[\/latex] shaded. Image (b) shows a second pizza of the same size, cut into eight pieces with [latex]{\\frac{4}{8}}[\/latex] shaded.<\/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\/24220757\/CNX_BMath_Figure_04_01_071_img.png\" alt=\"Two pizzas are shown. The pizza on the left is divided into 2 equal pieces. 1 piece is shaded. The pizza on the right is divided into 8 equal pieces. 4 pieces are shaded.\" \/><br \/>\nThis is another way to show that [latex]\\frac{1}{2}[\/latex] is equivalent to [latex]\\frac{4}{8}[\/latex].<\/p>\n<p>How can we use mathematics to change [latex]\\frac{1}{2}[\/latex] into [latex]\\frac{4}{8}[\/latex]? How could you take a pizza that is cut into two pieces and cut it into eight pieces? You could cut each of the two larger pieces into four smaller pieces! The whole pizza would then be cut into eight pieces instead of just two. Mathematically, what we\u2019ve described could be written as:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{1\\cdot\\color{blue}{4}}{2\\cdot\\color{blue}{4}}=\\frac{4}{8}[\/latex]<\/p>\n<p>These models lead to the <em><strong>Equivalent Fractions<\/strong> <strong>Property<\/strong><\/em>, which states that if we multiply the numerator and denominator of a fraction by the same number, the value of the fraction does not change.<\/p>\n<div class=\"textbox shaded\">\n<h3>Equivalent Fractions Property<\/h3>\n<p>If [latex]a,b[\/latex], and [latex]c[\/latex] are numbers where [latex]b\\ne 0[\/latex] and [latex]c\\ne 0[\/latex], then<\/p>\n<p>[latex]\\frac{a}{b}=\\frac{a\\cdot c}{b\\cdot c}[\/latex]<\/p>\n<\/div>\n<p>When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. For example, consider the fraction one-half.<\/p>\n<p style=\"text-align: center;\">[latex]{\\frac{1\\cdot\\color{blue}{3}}{2\\cdot\\color{blue}{3}}}={\\frac{3}{6}}[\/latex] so [latex]{\\frac{1}{2}}={\\frac{3}{6}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]{\\frac{1\\cdot\\color{blue}{2}}{2\\cdot\\color{blue}{2}}}={\\frac{2}{4}}[\/latex] so [latex]{\\frac{1}{2}}={\\frac{2}{4}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]{\\frac{1\\cdot\\color{blue}{10}}{2\\cdot\\color{blue}{10}}}={\\frac{10}{20}}[\/latex] so [latex]{\\frac{1}{2}}={\\frac{10}{20}}[\/latex]<\/p>\n<p>So, we say that [latex]\\frac{1}{2},\\frac{2}{4},\\frac{3}{6}[\/latex], and [latex]\\frac{10}{20}[\/latex] are <em><strong>equivalent fractions<\/strong><\/em>.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find three fractions equivalent to [latex]{\\frac{2}{5}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q931791\">Show Solution<\/span><\/p>\n<div id=\"q931791\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nTo find a fraction equivalent to [latex]{\\frac{2}{5}}[\/latex], we multiply the numerator and denominator by the same number (but not zero). Let us multiply them by [latex]2,3[\/latex], and [latex]5[\/latex].<\/p>\n<p>[latex]{\\frac{2\\cdot\\color{blue}{2}}{5\\cdot\\color{blue}{2}}}={\\frac{4}{10}}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]{\\frac{2\\cdot\\color{blue}{3}}{5\\cdot\\color{blue}{3}}}={\\frac{6}{15}}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]{\\frac{2\\cdot\\color{blue}{5}}{5\\cdot\\color{blue}{5}}}={\\frac{10}{25}}[\/latex]<\/p>\n<p>So, [latex]{\\frac{4}{10},\\frac{6}{15}}[\/latex], and [latex]{\\frac{10}{25}}[\/latex] are equivalent to [latex]{\\frac{2}{5}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>Find three fractions equivalent to [latex]{\\frac{3}{5}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q675004\">Show Solution<\/span><\/p>\n<div id=\"q675004\" class=\"hidden-answer\" style=\"display: none\">\n<p>Correct answers include [latex]{\\frac{6}{10},\\frac{9}{15}},\\text{and }{\\frac{12}{20}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Find three fractions equivalent to [latex]{\\frac{4}{5}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q171774\">Show Solution<\/span><\/p>\n<div id=\"q171774\" class=\"hidden-answer\" style=\"display: none\">\n<p>Correct answers include [latex]{\\frac{8}{10},\\frac{12}{15}},\\text{and }{\\frac{16}{20}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>When we add and subtract fractions, we often need to <em><strong>build<\/strong><\/em> fractions to create fractions with the same denominator. We build fractions by multiplying the numerator and denominator of the fraction by the same number (the equivalent fractions property).<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find a fraction with a denominator of [latex]21[\/latex] that is equivalent to [latex]{\\frac{2}{7}}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q810854\">Show Solution<\/span><\/p>\n<div id=\"q810854\" class=\"hidden-answer\" style=\"display: none\">\n<p>To find equivalent fractions, we multiply the numerator and denominator by the same number. In this case, we need to multiply the denominator by a number that will result in [latex]21[\/latex].<\/p>\n<p>Since we can multiply [latex]7[\/latex] by [latex]3[\/latex] to get [latex]21[\/latex], we can find the equivalent fraction by multiplying both the numerator and denominator by [latex]3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]{\\frac{2}{7}}={\\frac{2\\cdot\\color{blue}{3}}{7\\cdot\\color{blue}{3}}}={\\frac{6}{21}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146005\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146005&theme=oea&iframe_resize_id=ohm146005&show_question_numbers\" width=\"100%\" height=\"270\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to find an equivalent fraction given a specific denominator.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Determine Equivalent Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/8gJS0kvtGFU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2><strong>Positive and Negative Fractions<\/strong><\/h2>\n<p>Rational numbers can be negative as well as positive. Being negative just means that they sit on the opposite side of zero from the positive version of the fraction.<\/p>\n<p>For example, the fraction [latex]\\frac{1}{2}[\/latex] is located exactly half-way between [latex]0[\/latex] and [latex]1[\/latex] on the number line. The distance between\u00a0[latex]0[\/latex] and [latex]1[\/latex] is split into two equal lengths and\u00a0[latex]\\frac{1}{2}[\/latex] sits exactly one of the two equal lengths from zero.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1598 size-large\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5676\/2021\/11\/10173313\/FRACTIONS-ON-NUMBER-LINE-1024x160.png\" alt=\"Fractions on a number line\" width=\"1024\" height=\"160\" \/><\/p>\n<p>[latex]-\\frac{1}{2}[\/latex] is the opposite of\u00a0[latex]\\frac{1}{2}[\/latex] and lies on the opposite side of zero.<\/p>\n<p>[latex]\\frac{3}{2}[\/latex] lies three half-units from zero, and\u00a0[latex]-\\frac{3}{2}[\/latex] lies three\u00a0half-units from zero on the negative side of the number line. Notice also, that\u00a0[latex]\\frac{1}{2}[\/latex] lies between\u00a0[latex]0[\/latex] and [latex]1[\/latex] so is a proper fraction, while\u00a0[latex]\\frac{3}{2}[\/latex] is greater than or equal to\u00a0[latex]1[\/latex] so is an improper fraction.<\/p>\n<h2>Simplifying Fractions<\/h2>\n<p>In working with equivalent fractions, we saw that there are many ways to write fractions that have the same value, or represent the same part of the whole. How do we know which one to use? Most often, we\u2019ll use the fraction that is in <strong><em>simplified<\/em><\/strong> form.<\/p>\n<p>A fraction is considered simplified if there are no common factors, other than [latex]1[\/latex], in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can simplify the fraction to its simplified form by dividing the numerator and denominator by the common factors.<\/p>\n<div class=\"textbox shaded\">\n<h3>Simplified Fraction<\/h3>\n<p>A fraction is considered simplified if there are no common factors in the numerator and denominator.<\/p>\n<\/div>\n<p>For example,<\/p>\n<ul id=\"fs-id1302300\">\n<li>[latex]\\frac{2}{3}[\/latex] is simplified because, other than 1, there are no common factors of [latex]2[\/latex] and [latex]3[\/latex].<\/li>\n<li>[latex]\\frac{10}{15}[\/latex] is NOT simplified because [latex]5[\/latex] is a common factor of [latex]10[\/latex] and [latex]15[\/latex].<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>We can use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.<\/p>\n<div class=\"textbox shaded\">\n<h3>Equivalent Fractions Property<\/h3>\n<p>If [latex]a,b,c[\/latex] are integers where [latex]b\\ne 0,c\\ne 0[\/latex], then<\/p>\n<p>[latex]{\\frac{a}{b}}={\\frac{a\\cdot c}{b\\cdot c}}\\text{ and }{\\frac{a\\cdot c}{b\\cdot c}}={\\frac{a}{b}}[\/latex].<\/p>\n<\/div>\n<p>Notice that [latex]c[\/latex] is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, we can divide the numerator and denominator by the common factor to remove it.<\/p>\n<div class=\"textbox shaded\">\n<h3>Simplifying a fraction.<\/h3>\n<ol id=\"eip-id1168467382990\" class=\"stepwise\">\n<li>Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.<\/li>\n<li>Simplify, using the equivalent fractions property, by dividing both the numerator and denominator by the common factor to remove it.<\/li>\n<li>Multiply any remaining factors.<\/li>\n<\/ol>\n<\/div>\n<h3><\/h3>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify: [latex]\\frac{10}{15}[\/latex]<\/p>\n<p>Solution:<br \/>\nTo simplify the fraction, we look for any common factors in the numerator and the denominator.<\/p>\n<table id=\"eip-id1168468231694\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td>Notice that [latex]5[\/latex] is a factor of both [latex]10[\/latex] and [latex]15[\/latex].<\/td>\n<td>[latex]\\frac{10}{15}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Factor the numerator and denominator.<\/td>\n<td>[latex]\\frac{2\\cdot5}{3\\cdot5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>5 is a common factor so divide the numerator and denominator by 5 to remove it.<\/td>\n<td>[latex]\\frac{2\\cdot\\color{red}{5}}{3\\cdot\\color{red}{5}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\frac{2}{3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Simplify the fractions:<\/p>\n<p>1. [latex]\\frac{14}{21}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a02. [latex]\\frac{20}{50}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a03. [latex]\\frac{12}{24}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a04. [latex]\\frac{8}{28}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a05. [latex]\\frac{17}{17}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q377520\">Show Answer<\/span><\/p>\n<div id=\"q377520\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]\\frac{14}{21}=\\frac{2}{3}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a02. [latex]\\frac{20}{50}=\\frac{2}{5}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a03. [latex]\\frac{12}{24}=\\frac{1}{2}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a04. [latex]\\frac{8}{28}=-\\frac{2}{7}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a05. [latex]\\frac{17}{17}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To simplify a negative fraction, we use the same process as in the previous example. Remember to keep the negative sign.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify: [latex]-\\frac{18}{24}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q270732\">Show Solution<\/span><\/p>\n<div id=\"q270732\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168469841089\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"We notice that 18 and 24 both have factors,\">\n<tbody>\n<tr>\n<td>We notice that 18 and 24 both have factors of 6.<\/td>\n<td>[latex]-\\frac{18}{24}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite the numerator and denominator showing the common factor.<\/td>\n<td>[latex]-\\frac{3\\cdot6}{4\\cdot6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors by division.<\/td>\n<td>[latex]-\\frac{3\\cdot\\color{red}{6}}{4\\cdot\\color{red}{6}}[\/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>Simplify the fractions:<\/p>\n<p>1. [latex]-\\frac{14}{21}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a02. [latex]-\\frac{40}{50}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a03. [latex]-\\frac{48}{24}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a04. [latex]-\\frac{21}{28}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a05. [latex]-\\frac{23}{23}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qHM7520\">Show Answer<\/span><\/p>\n<div id=\"qHM7520\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. [latex]-\\frac{14}{21}=-\\frac{2}{3}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a02. [latex]-\\frac{40}{50}=-\\frac{4}{5}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a03. [latex]-\\frac{48}{24}=-2[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a04. [latex]-\\frac{21}{28}=-\\frac{3}{4}[\/latex] \u00a0 \u00a0 \u00a0 \u00a0 \u00a05. [latex]-\\frac{23}{23}=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see another example of how to simplify a fraction.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex 1:  Simplify Fractions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_2Wk7jXf3Ok?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: <em>a fraction is considered simplified if there are no common factors in the numerator and denominator<\/em>.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify: [latex]-\\frac{56}{32}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q877414\">Show Solution<\/span><\/p>\n<div id=\"q877414\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168466600046\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The first line shows negative 56 over 32. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-\\frac{56}{32}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Rewrite the numerator and denominator, showing the common factors, 8.<\/td>\n<td>[latex]-\\frac{7\\cdot8}{4\\cdot8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove common factors by division.<\/td>\n<td>[latex]-\\frac{7\\cdot\\color{red}{8}}{4\\cdot\\color{red}{8}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-\\frac{7}{4}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Sometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the Equivalent Fractions Property.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Simplify: [latex]\\frac{210}{385}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q721590\">Show Solution<\/span><\/p>\n<div id=\"q721590\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168467251049\" class=\"unnumbered unstyled\" style=\"width: 85%;\" summary=\"The fraction 210 over 385 is shown. The next line says,\">\n<tbody>\n<tr>\n<td>Use factor trees to factor the numerator and denominator.<\/td>\n<td>[latex]\\frac{210}{385}[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220913\/CNX_BMath_Figure_04_02_028_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite the numerator and denominator as the product of the primes.<\/td>\n<td>[latex]{\\frac{210}{385}}={\\frac{2\\cdot 3\\cdot 5\\cdot 7}{5\\cdot 7\\cdot 11}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Remove the common factors by division.<\/td>\n<td>[latex]\\frac{2\\cdot 3\\cdot\\color{blue}{5}\\cdot\\color{red}{7}}{\\color{blue}{5}\\cdot\\color{red}{7}\\cdot 11}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]\\frac{2\\cdot 3}{11}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply any remaining factors.<\/td>\n<td>[latex]\\frac{6}{11}[\/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>Simplify [latex]\\frac{315}{675}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q495207\">Show Answer<\/span><\/p>\n<div id=\"q495207\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{315}{675}=\\frac{\\color{red}{3\\cdot 3\\cdot} \\color{red}{5\\cdot} 7}{\\color{red}{3\\cdot 3\\cdot} 3\\cdot \\color{red}{5\\cdot} 5 }=\\frac{7}{15}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\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-1393\">\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>Question ID: 146014, 146015, 146017, 146018, 146019. <strong>Authored by<\/strong>: Alyson Day. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Ex 1: Simplifying Fractions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/tLgfPeecGe0\">https:\/\/youtu.be\/tLgfPeecGe0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Authored 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":422605,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"OpenStax\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Question ID: 146014, 146015, 146017, 146018, 146019\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Ex 1: Simplifying Fractions\",\"author\":\"James Sousa 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