{"id":4625,"date":"2020-04-21T00:19:11","date_gmt":"2020-04-21T00:19:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/modeling-and-finding-equivalent-fractions\/"},"modified":"2023-03-23T00:02:04","modified_gmt":"2023-03-23T00:02:04","slug":"modeling-and-finding-equivalent-fractions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/modeling-and-finding-equivalent-fractions\/","title":{"raw":"Modeling and Finding Equivalent Fractions","rendered":"Modeling and Finding Equivalent Fractions"},"content":{"raw":"\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use fraction tiles or visual aids to create equivalent fractions<\/li>\r\n \t<li>Find an equivalent fraction given a fraction<\/li>\r\n<\/ul>\r\n<\/div>\r\nLet\u2019s think about Andy and Bobby and their favorite food again. If Andy eats [latex]{\\Large\\frac{1}{2}}[\/latex] of a pizza and Bobby eats [latex]{\\Large\\frac{2}{4}}[\/latex] of the pizza, have they eaten the same amount of pizza? In other words, does [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{2}{4}}?[\/latex] We can use fraction tiles to find out whether Andy 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&nbsp;and extend it to include eighths, tenths, and twelfths.\r\n\r\nStart with a [latex]{\\Large\\frac{1}{2}}[\/latex] tile. How many fourths equal one-half? How many of the [latex]{\\Large\\frac{1}{4}}[\/latex] tiles exactly cover the [latex]{\\Large\\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]{\\Large\\frac{1}{4}}[\/latex] tiles cover the [latex]{\\Large\\frac{1}{2}}[\/latex] tile, we see that [latex]{\\Large\\frac{2}{4}}[\/latex] is the same as [latex]{\\Large\\frac{1}{2}}[\/latex], or [latex]{\\Large\\frac{2}{4}}={\\Large\\frac{1}{2}}[\/latex].<\/p>\r\nHow many of the [latex]{\\Large\\frac{1}{6}}[\/latex] tiles cover the [latex]{\\Large\\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]{\\Large\\frac{1}{6}}[\/latex] tiles cover the [latex]{\\Large\\frac{1}{2}}[\/latex] tile, we see that [latex]{\\Large\\frac{3}{6}}[\/latex] is the same as [latex]{\\Large\\frac{1}{2}}[\/latex].<\/p>\r\nSo, [latex]{\\Large\\frac{3}{6}}={\\Large\\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]{\\Large\\frac{1}{8}}[\/latex]) equal one-half ([latex]{\\Large\\frac{1}{2}}[\/latex])?<\/li>\r\n \t<li>How many tenths ([latex]{\\Large\\frac{1}{10}}[\/latex]) equal one-half ([latex]{\\Large\\frac{1}{2}}[\/latex])?<\/li>\r\n \t<li>How many twelfths ([latex]{\\Large\\frac{1}{12}}[\/latex]) equal one-half ([latex]{\\Large\\frac{1}{2}}[\/latex])?<\/li>\r\n<\/ol>\r\nSolution\r\n1. It takes four [latex]{\\Large\\frac{1}{8}}[\/latex] tiles to exactly cover the [latex]{\\Large\\frac{1}{2}}[\/latex] tile, so [latex]{\\Large\\frac{4}{8}}={\\Large\\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]{\\Large\\frac{1}{10}}[\/latex] tiles to exactly cover the [latex]{\\Large\\frac{1}{2}}[\/latex] tile, so [latex]{\\Large\\frac{5}{10}}={\\Large\\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]{\\Large\\frac{1}{12}}[\/latex] tiles to exactly cover the [latex]{\\Large\\frac{1}{2}}[\/latex] tile, so [latex]{\\Large\\frac{6}{12}}={\\Large\\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]{\\Large\\frac{1}{20}}[\/latex]. How many of them would it take to equal [latex]{\\Large\\frac{1}{2}}[\/latex]? Are you thinking ten tiles? If you are, you\u2019re right, because [latex]{\\Large\\frac{10}{20}}={\\Large\\frac{1}{2}}[\/latex].\r\n\r\nWe have shown that [latex]{\\Large\\frac{1}{2},\\frac{2}{4},\\frac{3}{6},\\frac{4}{8},\\frac{5}{10},\\frac{6}{12}}[\/latex], and [latex]{\\Large\\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<h2>Find Equivalent Fractions<\/h2>\r\nWe used fraction tiles to show that there are many fractions equivalent to [latex]{\\Large\\frac{1}{2}}[\/latex]. For example, [latex]{\\Large\\frac{2}{4},\\frac{3}{6}}[\/latex], and [latex]{\\Large\\frac{4}{8}}[\/latex] are all equivalent to [latex]{\\Large\\frac{1}{2}}[\/latex]. When we lined up the fraction tiles, it took four of the [latex]{\\Large\\frac{1}{8}}[\/latex] tiles to make the same length as a [latex]{\\Large\\frac{1}{2}}[\/latex] tile. This showed that [latex]{\\Large\\frac{4}{8}}={\\Large\\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]{\\Large\\frac{1}{2}}[\/latex] shaded. Image (b) shows a second pizza of the same size, cut into eight pieces with [latex]{\\Large\\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]{\\Large\\frac{1}{2}}[\/latex] is equivalent to [latex]{\\Large\\frac{4}{8}}[\/latex].\r\n\r\nHow can we use mathematics to change [latex]{\\Large\\frac{1}{2}}[\/latex] into [latex]{\\Large\\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]{\\Large\\frac{1\\cdot\\color{blue}{4}}{2\\cdot\\color{blue}{4}}}={\\Large\\frac{4}{8}}[\/latex]<\/p>\r\nThese models lead to the Equivalent Fractions Property, 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]{\\Large\\frac{a}{b}}={\\Large\\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]{\\Large\\frac{1\\cdot\\color{blue}{3}}{2\\cdot\\color{blue}{3}}}={\\Large\\frac{3}{6}}[\/latex] so [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{3}{6}}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]{\\Large\\frac{1\\cdot\\color{blue}{2}}{2\\cdot\\color{blue}{2}}}={\\Large\\frac{2}{4}}[\/latex] so [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{2}{4}}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]{\\Large\\frac{1\\cdot\\color{blue}{10}}{2\\cdot\\color{blue}{10}}}={\\Large\\frac{10}{20}}[\/latex] so [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{10}{20}}[\/latex]<\/p>\r\nSo, we say that [latex]{\\Large\\frac{1}{2},\\frac{2}{4},\\frac{3}{6}}[\/latex], and [latex]{\\Large\\frac{10}{20}}[\/latex] are equivalent fractions.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind three fractions equivalent to [latex]{\\Large\\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]{\\Large\\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]{\\Large\\frac{2\\cdot\\color{blue}{2}}{5\\cdot\\color{blue}{2}}}={\\Large\\frac{4}{10}}[\/latex] &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;[latex]{\\Large\\frac{2\\cdot\\color{blue}{3}}{5\\cdot\\color{blue}{3}}}={\\Large\\frac{6}{15}}[\/latex] &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;[latex]{\\Large\\frac{2\\cdot\\color{blue}{5}}{5\\cdot\\color{blue}{5}}}={\\Large\\frac{10}{25}}[\/latex]\r\n\r\nSo, [latex]{\\Large\\frac{4}{10},\\frac{6}{15}}[\/latex], and [latex]{\\Large\\frac{10}{25}}[\/latex] are equivalent to [latex]{\\Large\\frac{2}{5}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\nFind three fractions equivalent to [latex]{\\Large\\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]{\\Large\\frac{6}{10},\\frac{9}{15}},\\text{and }{\\Large\\frac{12}{20}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nFind three fractions equivalent to [latex]{\\Large\\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]{\\Large\\frac{8}{10},\\frac{12}{15}},\\text{and }{\\Large\\frac{16}{20}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\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]{\\Large\\frac{2}{7}}[\/latex].\r\n[reveal-answer q=\"810854\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"810854\"]\r\n\r\nSolution\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]{\\Large\\frac{2}{7}}={\\Large\\frac{2\\cdot\\color{blue}{3}}{7\\cdot\\color{blue}{3}}}={\\Large\\frac{6}{21}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try it<\/h3>\r\n[ohm_question height=\"270\"]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\r\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use fraction tiles or visual aids to create equivalent fractions<\/li>\n<li>Find an equivalent fraction given a fraction<\/li>\n<\/ul>\n<\/div>\n<p>Let\u2019s think about Andy and Bobby and their favorite food again. If Andy eats [latex]{\\Large\\frac{1}{2}}[\/latex] of a pizza and Bobby eats [latex]{\\Large\\frac{2}{4}}[\/latex] of the pizza, have they eaten the same amount of pizza? In other words, does [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{2}{4}}?[\/latex] We can use fraction tiles to find out whether Andy 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&nbsp;and extend it to include eighths, tenths, and twelfths.<\/p>\n<p>Start with a [latex]{\\Large\\frac{1}{2}}[\/latex] tile. How many fourths equal one-half? How many of the [latex]{\\Large\\frac{1}{4}}[\/latex] tiles exactly cover the [latex]{\\Large\\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]{\\Large\\frac{1}{4}}[\/latex] tiles cover the [latex]{\\Large\\frac{1}{2}}[\/latex] tile, we see that [latex]{\\Large\\frac{2}{4}}[\/latex] is the same as [latex]{\\Large\\frac{1}{2}}[\/latex], or [latex]{\\Large\\frac{2}{4}}={\\Large\\frac{1}{2}}[\/latex].<\/p>\n<p>How many of the [latex]{\\Large\\frac{1}{6}}[\/latex] tiles cover the [latex]{\\Large\\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]{\\Large\\frac{1}{6}}[\/latex] tiles cover the [latex]{\\Large\\frac{1}{2}}[\/latex] tile, we see that [latex]{\\Large\\frac{3}{6}}[\/latex] is the same as [latex]{\\Large\\frac{1}{2}}[\/latex].<\/p>\n<p>So, [latex]{\\Large\\frac{3}{6}}={\\Large\\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]{\\Large\\frac{1}{8}}[\/latex]) equal one-half ([latex]{\\Large\\frac{1}{2}}[\/latex])?<\/li>\n<li>How many tenths ([latex]{\\Large\\frac{1}{10}}[\/latex]) equal one-half ([latex]{\\Large\\frac{1}{2}}[\/latex])?<\/li>\n<li>How many twelfths ([latex]{\\Large\\frac{1}{12}}[\/latex]) equal one-half ([latex]{\\Large\\frac{1}{2}}[\/latex])?<\/li>\n<\/ol>\n<p>Solution<br \/>\n1. It takes four [latex]{\\Large\\frac{1}{8}}[\/latex] tiles to exactly cover the [latex]{\\Large\\frac{1}{2}}[\/latex] tile, so [latex]{\\Large\\frac{4}{8}}={\\Large\\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]{\\Large\\frac{1}{10}}[\/latex] tiles to exactly cover the [latex]{\\Large\\frac{1}{2}}[\/latex] tile, so [latex]{\\Large\\frac{5}{10}}={\\Large\\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]{\\Large\\frac{1}{12}}[\/latex] tiles to exactly cover the [latex]{\\Large\\frac{1}{2}}[\/latex] tile, so [latex]{\\Large\\frac{6}{12}}={\\Large\\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]{\\Large\\frac{1}{20}}[\/latex]. How many of them would it take to equal [latex]{\\Large\\frac{1}{2}}[\/latex]? Are you thinking ten tiles? If you are, you\u2019re right, because [latex]{\\Large\\frac{10}{20}}={\\Large\\frac{1}{2}}[\/latex].<\/p>\n<p>We have shown that [latex]{\\Large\\frac{1}{2},\\frac{2}{4},\\frac{3}{6},\\frac{4}{8},\\frac{5}{10},\\frac{6}{12}}[\/latex], and [latex]{\\Large\\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<h2>Find Equivalent Fractions<\/h2>\n<p>We used fraction tiles to show that there are many fractions equivalent to [latex]{\\Large\\frac{1}{2}}[\/latex]. For example, [latex]{\\Large\\frac{2}{4},\\frac{3}{6}}[\/latex], and [latex]{\\Large\\frac{4}{8}}[\/latex] are all equivalent to [latex]{\\Large\\frac{1}{2}}[\/latex]. When we lined up the fraction tiles, it took four of the [latex]{\\Large\\frac{1}{8}}[\/latex] tiles to make the same length as a [latex]{\\Large\\frac{1}{2}}[\/latex] tile. This showed that [latex]{\\Large\\frac{4}{8}}={\\Large\\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]{\\Large\\frac{1}{2}}[\/latex] shaded. Image (b) shows a second pizza of the same size, cut into eight pieces with [latex]{\\Large\\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]{\\Large\\frac{1}{2}}[\/latex] is equivalent to [latex]{\\Large\\frac{4}{8}}[\/latex].<\/p>\n<p>How can we use mathematics to change [latex]{\\Large\\frac{1}{2}}[\/latex] into [latex]{\\Large\\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]{\\Large\\frac{1\\cdot\\color{blue}{4}}{2\\cdot\\color{blue}{4}}}={\\Large\\frac{4}{8}}[\/latex]<\/p>\n<p>These models lead to the Equivalent Fractions Property, 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]{\\Large\\frac{a}{b}}={\\Large\\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]{\\Large\\frac{1\\cdot\\color{blue}{3}}{2\\cdot\\color{blue}{3}}}={\\Large\\frac{3}{6}}[\/latex] so [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{3}{6}}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]{\\Large\\frac{1\\cdot\\color{blue}{2}}{2\\cdot\\color{blue}{2}}}={\\Large\\frac{2}{4}}[\/latex] so [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{2}{4}}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]{\\Large\\frac{1\\cdot\\color{blue}{10}}{2\\cdot\\color{blue}{10}}}={\\Large\\frac{10}{20}}[\/latex] so [latex]{\\Large\\frac{1}{2}}={\\Large\\frac{10}{20}}[\/latex]<\/p>\n<p>So, we say that [latex]{\\Large\\frac{1}{2},\\frac{2}{4},\\frac{3}{6}}[\/latex], and [latex]{\\Large\\frac{10}{20}}[\/latex] are equivalent fractions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find three fractions equivalent to [latex]{\\Large\\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]{\\Large\\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]{\\Large\\frac{2\\cdot\\color{blue}{2}}{5\\cdot\\color{blue}{2}}}={\\Large\\frac{4}{10}}[\/latex] &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;[latex]{\\Large\\frac{2\\cdot\\color{blue}{3}}{5\\cdot\\color{blue}{3}}}={\\Large\\frac{6}{15}}[\/latex] &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;[latex]{\\Large\\frac{2\\cdot\\color{blue}{5}}{5\\cdot\\color{blue}{5}}}={\\Large\\frac{10}{25}}[\/latex]<\/p>\n<p>So, [latex]{\\Large\\frac{4}{10},\\frac{6}{15}}[\/latex], and [latex]{\\Large\\frac{10}{25}}[\/latex] are equivalent to [latex]{\\Large\\frac{2}{5}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p>Find three fractions equivalent to [latex]{\\Large\\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]{\\Large\\frac{6}{10},\\frac{9}{15}},\\text{and }{\\Large\\frac{12}{20}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Find three fractions equivalent to [latex]{\\Large\\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]{\\Large\\frac{8}{10},\\frac{12}{15}},\\text{and }{\\Large\\frac{16}{20}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/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]{\\Large\\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>Solution<br \/>\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].<\/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]{\\Large\\frac{2}{7}}={\\Large\\frac{2\\cdot\\color{blue}{3}}{7\\cdot\\color{blue}{3}}}={\\Large\\frac{6}{21}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"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\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-4625\">\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>Question ID: 146001, 146005. <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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Determine Equivalent Fractions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/8gJS0kvtGFU\">https:\/\/youtu.be\/8gJS0kvtGFU<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Ex: Determine Equivalent Fractions\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/8gJS0kvtGFU\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID: 146001, 146005\",\"author\":\"Alyson Day\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + 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