{"id":8108,"date":"2023-09-20T17:46:41","date_gmt":"2023-09-20T17:46:41","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8108"},"modified":"2025-08-28T03:59:54","modified_gmt":"2025-08-28T03:59:54","slug":"calculations-involving-rational-numbers-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/calculations-involving-rational-numbers-fresh-take\/","title":{"raw":"Calculations Involving Rational Numbers: Fresh Take","rendered":"Calculations Involving Rational Numbers: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Add, subtract, multiple and divide fractions<\/li>\r\n\t<li>Solve real-world problems using arithmetic with rational functions<\/li>\r\n\t<li>Convert between improper fractions and mixed numbers<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Adding and Subtracting Rational Numbers<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> \u00a0 <br \/>\r\n<br \/>\r\nAdding and subtracting rational numbers isn't just about combining numbers; it's about understanding the common ground between them. If the two fractions you are adding have the same denominator or a <strong>common denominator<\/strong>, adding or subtracting the two fractions is pretty straightforward. <br \/>\r\n<br \/>\r\n<strong>Adding\/Subtracting rational numbers with a common denominator<\/strong>: Add or subtract the numerators, and then place that value in the numerator and the common denominator in the denominator.\r\n\r\n<p style=\"text-align: center;\">If [latex]c[\/latex] is a non-zero integer, then [latex]\\frac{a}{c} \\pm \\frac{b}{c}=\\frac{a \\pm b}{c}[\/latex]<\/p>\r\n\r\nAdding and subtracting fractions may seem like a straightforward task, but what happens when the denominators are different? That's where the concept of the Least Common Multiple (LCM) comes into play. <br \/>\r\n<br \/>\r\n<strong>Finding the least common multiple through prime factorization<\/strong>:\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Find the prime factors of each denominator. You can use a factor tree or division method to break down each number into its prime factors.<\/li>\r\n\t<li>List down all the unique prime factors that appear in the prime factorization of each number.<\/li>\r\n\t<li>For each unique prime factor, identify the highest power to which it is raised in any of the given numbers.<\/li>\r\n\t<li>Multiply together the highest powers of all the unique prime factors. The result is the least common multiple (LCM) of the given numbers.<\/li>\r\n<\/ol>\r\n\r\nOnce you have found the least common multiple, adding or subtracting fractions with unlike denominators becomes more straightforward. <br \/>\r\n<br \/>\r\n<strong>Adding\/Subtracting rational numbers with unlike denominators<\/strong>:\r\n\r\n<ol>\r\n\t<li>Find a common denominator using one of the two methods given above.<\/li>\r\n\t<li>Rewrite each fraction using the common denominator.<\/li>\r\n\t<li>Now that the fractions have a common denominator, you can add\/subtract the numerators.<\/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<section class=\"textbox example\">Calculate [latex]\\frac{38}{73}+\\frac{7}{73}[\/latex].[reveal-answer q=\"160936\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160936\"] Since the rational numbers have the same denominator, we perform the addition of the numerators, [latex]38+7[\/latex], and then place the result in the numerator and the common denominator, [latex]73[\/latex], in the denominator.\r\n\r\n<p style=\"text-align: center;\">[latex]\\frac{38}{73}+\\frac{7}{73}=\\frac{38+7}{73}=\\frac{45}{73}[\/latex]<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">Calculate [latex]\\frac{4}{9}+\\frac{7}{12}[\/latex].[reveal-answer q=\"160932\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160932\"]\r\n\r\n<p>The denominators of the fractions are [latex]9[\/latex] and [latex]12[\/latex]. We need to find the LCM of [latex]9[\/latex] and [latex]12[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]LCM(9,12)=36[\/latex]<\/p>\r\n<p>Next, we rewrite each fraction with [latex]36[\/latex] as the common denominator.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{4}{9}=\\frac{4\u00d74}{36}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{7}{12}=\\frac{7\u00d73}{36}[\/latex]<\/p>\r\n<p>Now, we can add the two fractions together.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{4\u00d74}{36}+\\frac{7\u00d73}{36}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{16}{36}+\\frac{21}{36}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{16+21}{36}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{37}{36}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{4}{9}+\\frac{7}{12}=\\frac{37}{36}[\/latex]<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">Calculate [latex]\\frac{21}{40}-\\frac{8}{40}[\/latex] .[reveal-answer q=\"160933\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160933\"] Since the rational numbers have the same denominator, we perform the subtraction of the numerators, [latex]21-8[\/latex], and then place the result in the numerator and the common denominator, [latex]40[\/latex], in the denominator.\r\n\r\n<p style=\"text-align: center;\">[latex]\\frac{21}{40}-\\frac{8}{40}=\\frac{21-8}{40}=\\frac{13}{40}[\/latex]<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">Calculate [latex]\\frac{10}{99}-\\frac{17}{30}[\/latex].[reveal-answer q=\"160934\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160934\"]\r\n\r\n<p>The denominators of the fractions are [latex]99[\/latex] and [latex]30[\/latex]. We need to find the LCM of [latex]99[\/latex] and [latex]30[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]LCM(99,30)=990[\/latex]<\/p>\r\n<p>Next, we rewrite each fraction with [latex]990[\/latex] as the common denominator.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{10}{99}=\\frac{10\u00d710}{990}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{17}{30}=\\frac{17\u00d733}{990}[\/latex]<\/p>\r\n<p>Now, we can add the two fractions together.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{10\u00d710}{990}-\\frac{17\u00d733}{990}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{100}{990}-\\frac{561}{990}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{100-561}{990}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{-461}{990}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{10}{99}-\\frac{17}{30}=\\frac{-461}{990}[\/latex]<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>The following video goes into more detail of adding fractions with common denominators.<\/p>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328520&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=MZmENadGcK0&amp;video_target=tpm-plugin-gsgekztj-MZmENadGcK0\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Adding+Fractions+with+Common+Denominators+(Step+by+Step)+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAdding Fractions with Common Denominators (Step by Step) | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>The following video goes into more detail of subtracting fractions with common denominators.<\/p>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328521&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=VTCOHFJOAA8&amp;video_target=tpm-plugin-02ze54gb-VTCOHFJOAA8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Subtracting+Fractions+with+Common+Denominators+(Step+by+Step)+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSubtracting Fractions with Common Denominators (Step by Step) | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>Watch the following video to learn why we need a common denominator when adding and subtracting fractions.<\/p>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328522&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=YxAXC8Q16ok&amp;video_target=tpm-plugin-84tqya8n-YxAXC8Q16ok\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Why+Do+We+Need+a+Common+Denominator+When+Adding+and+Subtracting+Fractions%3F+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhy Do We Need a Common Denominator When Adding and Subtracting Fractions? | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>The following video explains finding the least common denominator.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/pZEmFSP3Z0I?si=Luf4ZwR9hFrTX5Vm\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Math+Antics+-+Common+Denominator+LCD.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMath Antics - Common Denominator LCD\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>The following video takes another look at prime factorization.<\/p>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=11328523&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=XBnUWjo3TgM&amp;video_target=tpm-plugin-zz7tw4yk-XBnUWjo3TgM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Prime+Factorization+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPrime Factorization | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>In the following video you will see examples of how to add and subtract fractions with different denominators.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ry3KQ9F76oo?si=HYzGH5nBGLP48-Iu\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Add+and+Subtract+Fractions+with+Unlike+Denominators+(Basic).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Add and Subtract Fractions with Unlike Denominators (Basic)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Multiplying Rational Numbers<\/h2>\r\n<p>A model may help you understand multiplication of rational numbers. When you multiply a fraction by a fraction, you are finding a \u201cfraction of a fraction.\u201d Suppose you have [latex]\\frac{3}{4}[\/latex]\u00a0of a candy bar and you want to find [latex]\\frac{1}{2}[\/latex]\u00a0of the [latex]\\frac{3}{4}[\/latex]:<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"208\"]<img id=\"Picture 24\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170708\/image109.gif\" alt=\"3 out of four boxes are shaded. This is 3\/4.\" width=\"208\" height=\"65\" \/> Figure 1. A candy bar with 3\/4 shaded[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>By dividing each fourth in half, you can divide the candy bar into eighths.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"208\"]<img id=\"Picture 25\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170709\/image110.gif\" alt=\"Six of 8 boxes are shaded. This is 6\/8.\" width=\"208\" height=\"62\" \/> Figure 2. The candy bar parts were divided in halves, creating 6\/8[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Then, choose half of those to get [latex]\\frac{3}{8}[\/latex].<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"208\"]<img id=\"Picture 27\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170711\/image112.gif\" alt=\"Six of 8 boxes are shaded, and of those six, three of them are shaded purple. The 3 purple boxes represent 3\/8.\" width=\"208\" height=\"54\" \/> Figure 3. The fraction was divided in half, leaving 3\/8[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.<\/p>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> \u00a0\r\n\r\n<p><strong>Multiplying rational numbers<\/strong>:<\/p>\r\n<center>[latex] \\frac{a}{b}\\times \\frac{c}{d}=\\frac{a\\times c}{b\\times d}=\\frac{\\text{product of the numerators}}{\\text{product of the denominators}}[\/latex]<\/center><\/div>\r\n<section class=\"textbox example\">Calculate [latex]\\frac{45}{88} \\times \\frac{28}{75}[\/latex].[reveal-answer q=\"260936\"]Show Solution[\/reveal-answer] [hidden-answer a=\"260936\"]\r\n\r\n<p>Multiply the numerators and place that in the numerator, and then multiply the denominators and place that in the denominator.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\frac{45}{88} \\times \\frac{28}{75}=\\frac{45 \\times 28}{88 \\times 75}=\\frac{1260}{6600}[\/latex]<\/p>\r\n<p>Once we have that result, reduce to lowest terms, which gives [latex]\\frac{1260}{6600}=\\frac{60 \\times 21}{60 \\times 110}=\\frac{\\cancel{60}\u00d721}{\\cancel{60}\u00d7110}=\\frac{21}{110}[\/latex].<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>Watch the following video for more examples of multiplying rational numbers.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/Rxz7OUzNyV0?si=KEQ5_iDfew4KK4xX\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+2_+Multiply+Fractions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Multiply Fractions\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Dividing Rational Numbers<\/h2>\r\n<p>Just like with multiplying rational numbers, a model may help you understand dividing rational numbers. Suppose you have a pizza that is already cut into 4 slices. How many [latex]\\frac{1}{2}[\/latex] slices are there?<\/p>\r\n<table style=\"width: 22.5227%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 49.1803%;\"><center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170724\/image146.gif\" alt=\"A pizza divided into four equal pieces. There are four slices.\" width=\"180\" height=\"179\" \/><\/center><\/td>\r\n<td style=\"width: 49.1803%;\"><center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170725\/image147.gif\" alt=\"A pizza divided into four equal slices. Each slice is then divided in half. There are now 8 slices. \" width=\"180\" height=\"179\" \/><\/center><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p><strong><span style=\"font-size: 10pt;\">Figure 4. The slices are divided into halves, turning quarters into eighths<\/span><\/strong><\/p>\r\n<p>There are [latex]8[\/latex] slices. You can see that dividing [latex]4[\/latex] by [latex] \\frac{1}{2}[\/latex] gives the same result as multiplying [latex]4[\/latex] by [latex]2[\/latex]. What would happen if you needed to divide each slice into thirds?<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"180\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170726\/image148.gif\" alt=\"A pizza divided into four equal slice. Each slice is divided into thirds. There are now 12 slices.\" width=\"180\" height=\"179\" \/> Figure 5. The slices are divided into thirds, turning eighths into twelfths\u00a0[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>You would have [latex]12[\/latex] slices, which is the same as multiplying [latex]4[\/latex] by [latex]3[\/latex].<\/p>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> \u00a0 <br \/>\r\n<br \/>\r\n<strong>Dividing with rational numbers<\/strong>\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\n<section class=\"textbox example\">Divide [latex] \\frac{2}{3}\\div \\frac{1}{6}[\/latex].[reveal-answer q=\"360936\"]Show Solution[\/reveal-answer] [hidden-answer a=\"360936\"]<strong>KEEP<\/strong> [latex] \\frac{2}{3}[\/latex] <strong>CHANGE<\/strong>\u00a0 [latex] \\div [\/latex] to \u00a0[latex] \\times [\/latex] <strong>FLIP\u00a0<\/strong> [latex]\\frac{1}{6}[\/latex]\r\n\r\n<p style=\"text-align: center;\">[latex] \\frac{2}{3} \\times \\frac{6}{1}[\/latex]<\/p>\r\n\r\nMultiply numerators and multiply denominators.\r\n\r\n<p style=\"text-align: center;\">[latex]\\frac{2 \\times 6}{3 \\times 1}=\\frac{12}{3}[\/latex]<\/p>\r\n\r\n\u00a0 Simplify.\r\n\r\n<p style=\"text-align: center;\">[latex] \\frac{12}{3}=4[\/latex]<\/p>\r\n<h4>Answer<\/h4>\r\n\r\n[latex] \\frac{2}{3}\\div \\frac{1}{6}=4[\/latex] [\/hidden-answer]<\/section>\r\n<p>Watch the following video for more examples of dividing rational numbers.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/XHQlyQhIeNM?si=tsS5v_dDZ7s_c4pI\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+3_+Divide+Fractions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 3: Divide Fractions\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Solving Problems Involving Rational Numbers<\/h2>\r\n<p>Watch the following for more examples of determining a fraction of a large whole number.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/mENiI0nwX2g?si=CCqCPpfvpD-wWGyA\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Determine+a+Fraction+of+a+Large+Whole+Number.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine a Fraction of a Large Whole Number\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Converting Between Improper Fractions and Mixed Numbers<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> \u00a0\r\n\r\n<p><strong>Convert an improper fraction to a mixed number.<\/strong> \u00a0<\/p>\r\n<ol id=\"eip-241\" class=\"stepwise\">\r\n\t<li>Divide the denominator into the numerator.<\/li>\r\n\t<li>Identify the quotient, remainder, and divisor.<\/li>\r\n\t<li>Write the mixed number as quotient [latex]{\\Large\\frac{\\text{remainder}}{\\text{divisor}}}[\/latex] .<\/li>\r\n<\/ol>\r\n<p><strong>Convert a mixed number to an improper fraction.<\/strong> \u00a0<\/p>\r\n<ol id=\"eip-id1168467427407\" class=\"stepwise\">\r\n\t<li>Multiply the whole number by the denominator.<\/li>\r\n\t<li>Add the numerator to the product found in Step 1.<\/li>\r\n\t<li>Write the final sum over the original denominator.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox recall\">\r\n<p>A <strong>quotient<\/strong> is the result obtained when one number is divided by another; a <strong>remainder<\/strong> is the amount left over after division when the dividend cannot be evenly divided by the divisor; and a <strong>divisor<\/strong> is the number by which another number (the dividend) is divided.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Rewrite [latex]\\frac{95}{26}[\/latex] as a mixed number.<\/p>\r\n\r\n[reveal-answer q=\"160938\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160938\"] When [latex]95[\/latex] is divided by [latex]26[\/latex], the result is [latex]3[\/latex] with a remainder of [latex]17[\/latex]. So, we can rewrite [latex]\\frac{95}{26}[\/latex] as [latex]3\\frac{17}{26}[\/latex]. [\/hidden-answer]<\/section>\r\n<p>Now you can watch worked examples of how to convert an improper fraction to a mixed number in the following video.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/e6uoYVg5Q30?si=0yp8AigQ9jp74hp6\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Examples_+Convert+an+Improper+Fraction+to+a+Mixed+Number.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExamples: Convert an Improper Fraction to a Mixed Number\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Rewrite [latex]9\\frac{5}{14}[\/latex] as an improper fraction.<\/p>\r\n\r\n[reveal-answer q=\"160939\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160939\"]\r\n\r\n<ul>\r\n\t<li><strong>Step 1:<\/strong> Multiply the integer part, [latex]9[\/latex], by the denominator, [latex]14[\/latex], which gives [latex]9\u00d714=126[\/latex].<\/li>\r\n\t<li><strong>Step 2:<\/strong> Add that product to the numerator, which gives [latex]5+126=131[\/latex].<\/li>\r\n\t<li><strong>Step 3:<\/strong> Write the number as the sum, [latex]131[\/latex], divided by the denominator, [latex]14[\/latex], which gives [latex]\\frac{131}{14}[\/latex].<\/li>\r\n<\/ul>\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>In the following video we show more example of how to convert a mixed number to an improper fraction.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/p_YRBcZ4u4g?si=0W-_ozEb_qAoxKzu\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Examples_+Converting+a+Mixed+Number+to+an+Improper+Fraction.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExamples: Converting a Mixed Number to an Improper Fraction\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Add, subtract, multiple and divide fractions<\/li>\n<li>Solve real-world problems using arithmetic with rational functions<\/li>\n<li>Convert between improper fractions and mixed numbers<\/li>\n<\/ul>\n<\/section>\n<h2>Adding and Subtracting Rational Numbers<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> \u00a0 <\/p>\n<p>Adding and subtracting rational numbers isn&#8217;t just about combining numbers; it&#8217;s about understanding the common ground between them. If the two fractions you are adding have the same denominator or a <strong>common denominator<\/strong>, adding or subtracting the two fractions is pretty straightforward. <\/p>\n<p><strong>Adding\/Subtracting rational numbers with a common denominator<\/strong>: Add or subtract the numerators, and then place that value in the numerator and the common denominator in the denominator.<\/p>\n<p style=\"text-align: center;\">If [latex]c[\/latex] is a non-zero integer, then [latex]\\frac{a}{c} \\pm \\frac{b}{c}=\\frac{a \\pm b}{c}[\/latex]<\/p>\n<p>Adding and subtracting fractions may seem like a straightforward task, but what happens when the denominators are different? That&#8217;s where the concept of the Least Common Multiple (LCM) comes into play. <\/p>\n<p><strong>Finding the least common multiple through prime factorization<\/strong>:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Find the prime factors of each denominator. You can use a factor tree or division method to break down each number into its prime factors.<\/li>\n<li>List down all the unique prime factors that appear in the prime factorization of each number.<\/li>\n<li>For each unique prime factor, identify the highest power to which it is raised in any of the given numbers.<\/li>\n<li>Multiply together the highest powers of all the unique prime factors. The result is the least common multiple (LCM) of the given numbers.<\/li>\n<\/ol>\n<p>Once you have found the least common multiple, adding or subtracting fractions with unlike denominators becomes more straightforward. <\/p>\n<p><strong>Adding\/Subtracting rational numbers with unlike denominators<\/strong>:<\/p>\n<ol>\n<li>Find a common denominator using one of the two methods given above.<\/li>\n<li>Rewrite each fraction using the common denominator.<\/li>\n<li>Now that the fractions have a common denominator, you can add\/subtract the numerators.<\/li>\n<li>Simplify by canceling out all common factors in the numerator and denominator.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Calculate [latex]\\frac{38}{73}+\\frac{7}{73}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q160936\">Show Solution<\/button> <\/p>\n<div id=\"q160936\" class=\"hidden-answer\" style=\"display: none\"> Since the rational numbers have the same denominator, we perform the addition of the numerators, [latex]38+7[\/latex], and then place the result in the numerator and the common denominator, [latex]73[\/latex], in the denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{38}{73}+\\frac{7}{73}=\\frac{38+7}{73}=\\frac{45}{73}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Calculate [latex]\\frac{4}{9}+\\frac{7}{12}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q160932\">Show Solution<\/button> <\/p>\n<div id=\"q160932\" class=\"hidden-answer\" style=\"display: none\">\n<p>The denominators of the fractions are [latex]9[\/latex] and [latex]12[\/latex]. We need to find the LCM of [latex]9[\/latex] and [latex]12[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]LCM(9,12)=36[\/latex]<\/p>\n<p>Next, we rewrite each fraction with [latex]36[\/latex] as the common denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{4}{9}=\\frac{4\u00d74}{36}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{7}{12}=\\frac{7\u00d73}{36}[\/latex]<\/p>\n<p>Now, we can add the two fractions together.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{4\u00d74}{36}+\\frac{7\u00d73}{36}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{16}{36}+\\frac{21}{36}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{16+21}{36}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{37}{36}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{4}{9}+\\frac{7}{12}=\\frac{37}{36}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Calculate [latex]\\frac{21}{40}-\\frac{8}{40}[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q160933\">Show Solution<\/button> <\/p>\n<div id=\"q160933\" class=\"hidden-answer\" style=\"display: none\"> Since the rational numbers have the same denominator, we perform the subtraction of the numerators, [latex]21-8[\/latex], and then place the result in the numerator and the common denominator, [latex]40[\/latex], in the denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{21}{40}-\\frac{8}{40}=\\frac{21-8}{40}=\\frac{13}{40}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Calculate [latex]\\frac{10}{99}-\\frac{17}{30}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q160934\">Show Solution<\/button> <\/p>\n<div id=\"q160934\" class=\"hidden-answer\" style=\"display: none\">\n<p>The denominators of the fractions are [latex]99[\/latex] and [latex]30[\/latex]. We need to find the LCM of [latex]99[\/latex] and [latex]30[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]LCM(99,30)=990[\/latex]<\/p>\n<p>Next, we rewrite each fraction with [latex]990[\/latex] as the common denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{10}{99}=\\frac{10\u00d710}{990}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{17}{30}=\\frac{17\u00d733}{990}[\/latex]<\/p>\n<p>Now, we can add the two fractions together.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{10\u00d710}{990}-\\frac{17\u00d733}{990}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{100}{990}-\\frac{561}{990}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{100-561}{990}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{-461}{990}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{10}{99}-\\frac{17}{30}=\\frac{-461}{990}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>The following video goes into more detail of adding fractions with common denominators.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328520&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=MZmENadGcK0&amp;video_target=tpm-plugin-gsgekztj-MZmENadGcK0\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Adding+Fractions+with+Common+Denominators+(Step+by+Step)+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAdding Fractions with Common Denominators (Step by Step) | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>The following video goes into more detail of subtracting fractions with common denominators.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328521&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=VTCOHFJOAA8&amp;video_target=tpm-plugin-02ze54gb-VTCOHFJOAA8\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Subtracting+Fractions+with+Common+Denominators+(Step+by+Step)+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSubtracting Fractions with Common Denominators (Step by Step) | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>Watch the following video to learn why we need a common denominator when adding and subtracting fractions.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328522&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=YxAXC8Q16ok&amp;video_target=tpm-plugin-84tqya8n-YxAXC8Q16ok\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Why+Do+We+Need+a+Common+Denominator+When+Adding+and+Subtracting+Fractions%3F+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWhy Do We Need a Common Denominator When Adding and Subtracting Fractions? | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>The following video explains finding the least common denominator.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/pZEmFSP3Z0I?si=Luf4ZwR9hFrTX5Vm\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Math+Antics+-+Common+Denominator+LCD.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cMath Antics &#8211; Common Denominator LCD\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>The following video takes another look at prime factorization.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=11328523&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=XBnUWjo3TgM&amp;video_target=tpm-plugin-zz7tw4yk-XBnUWjo3TgM\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Prime+Factorization+%7C+Math+with+Mr.+J.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cPrime Factorization | Math with Mr. J\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>In the following video you will see examples of how to add and subtract fractions with different denominators.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ry3KQ9F76oo?si=HYzGH5nBGLP48-Iu\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Add+and+Subtract+Fractions+with+Unlike+Denominators+(Basic).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Add and Subtract Fractions with Unlike Denominators (Basic)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Multiplying Rational Numbers<\/h2>\n<p>A model may help you understand multiplication of rational numbers. When you multiply a fraction by a fraction, you are finding a \u201cfraction of a fraction.\u201d Suppose you have [latex]\\frac{3}{4}[\/latex]\u00a0of a candy bar and you want to find [latex]\\frac{1}{2}[\/latex]\u00a0of the [latex]\\frac{3}{4}[\/latex]:<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 208px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170708\/image109.gif\" alt=\"3 out of four boxes are shaded. This is 3\/4.\" width=\"208\" height=\"65\" \/><figcaption class=\"wp-caption-text\">Figure 1. A candy bar with 3\/4 shaded<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>By dividing each fourth in half, you can divide the candy bar into eighths.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 208px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170709\/image110.gif\" alt=\"Six of 8 boxes are shaded. This is 6\/8.\" width=\"208\" height=\"62\" \/><figcaption class=\"wp-caption-text\">Figure 2. The candy bar parts were divided in halves, creating 6\/8<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Then, choose half of those to get [latex]\\frac{3}{8}[\/latex].<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 208px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170711\/image112.gif\" alt=\"Six of 8 boxes are shaded, and of those six, three of them are shaded purple. The 3 purple boxes represent 3\/8.\" width=\"208\" height=\"54\" \/><figcaption class=\"wp-caption-text\">Figure 3. The fraction was divided in half, leaving 3\/8<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.<\/p>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> \u00a0<\/p>\n<p><strong>Multiplying rational numbers<\/strong>:<\/p>\n<div style=\"text-align: center;\">[latex]\\frac{a}{b}\\times \\frac{c}{d}=\\frac{a\\times c}{b\\times d}=\\frac{\\text{product of the numerators}}{\\text{product of the denominators}}[\/latex]<\/div>\n<\/div>\n<section class=\"textbox example\">Calculate [latex]\\frac{45}{88} \\times \\frac{28}{75}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q260936\">Show Solution<\/button> <\/p>\n<div id=\"q260936\" class=\"hidden-answer\" style=\"display: none\">\n<p>Multiply the numerators and place that in the numerator, and then multiply the denominators and place that in the denominator.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{45}{88} \\times \\frac{28}{75}=\\frac{45 \\times 28}{88 \\times 75}=\\frac{1260}{6600}[\/latex]<\/p>\n<p>Once we have that result, reduce to lowest terms, which gives [latex]\\frac{1260}{6600}=\\frac{60 \\times 21}{60 \\times 110}=\\frac{\\cancel{60}\u00d721}{\\cancel{60}\u00d7110}=\\frac{21}{110}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for more examples of multiplying rational numbers.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/Rxz7OUzNyV0?si=KEQ5_iDfew4KK4xX\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+2_+Multiply+Fractions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Multiply Fractions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Dividing Rational Numbers<\/h2>\n<p>Just like with multiplying rational numbers, a model may help you understand dividing rational numbers. Suppose you have a pizza that is already cut into 4 slices. How many [latex]\\frac{1}{2}[\/latex] slices are there?<\/p>\n<table style=\"width: 22.5227%;\">\n<tbody>\n<tr>\n<td style=\"width: 49.1803%;\">\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170724\/image146.gif\" alt=\"A pizza divided into four equal pieces. There are four slices.\" width=\"180\" height=\"179\" \/><\/div>\n<\/td>\n<td style=\"width: 49.1803%;\">\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170725\/image147.gif\" alt=\"A pizza divided into four equal slices. Each slice is then divided in half. There are now 8 slices.\" width=\"180\" height=\"179\" \/><\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong><span style=\"font-size: 10pt;\">Figure 4. The slices are divided into halves, turning quarters into eighths<\/span><\/strong><\/p>\n<p>There are [latex]8[\/latex] slices. You can see that dividing [latex]4[\/latex] by [latex]\\frac{1}{2}[\/latex] gives the same result as multiplying [latex]4[\/latex] by [latex]2[\/latex]. What would happen if you needed to divide each slice into thirds?<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 180px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/17170726\/image148.gif\" alt=\"A pizza divided into four equal slice. Each slice is divided into thirds. There are now 12 slices.\" width=\"180\" height=\"179\" \/><figcaption class=\"wp-caption-text\">Figure 5. The slices are divided into thirds, turning eighths into twelfths\u00a0<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>You would have [latex]12[\/latex] slices, which is the same as multiplying [latex]4[\/latex] by [latex]3[\/latex].<\/p>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> \u00a0 <\/p>\n<p><strong>Dividing with rational numbers<\/strong><\/p>\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<section class=\"textbox example\">Divide [latex]\\frac{2}{3}\\div \\frac{1}{6}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q360936\">Show Solution<\/button> <\/p>\n<div id=\"q360936\" class=\"hidden-answer\" style=\"display: none\"><strong>KEEP<\/strong> [latex]\\frac{2}{3}[\/latex] <strong>CHANGE<\/strong>\u00a0 [latex]\\div[\/latex] to \u00a0[latex]\\times[\/latex] <strong>FLIP\u00a0<\/strong> [latex]\\frac{1}{6}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2}{3} \\times \\frac{6}{1}[\/latex]<\/p>\n<p>Multiply numerators and multiply denominators.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{2 \\times 6}{3 \\times 1}=\\frac{12}{3}[\/latex]<\/p>\n<p>\u00a0 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><\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for more examples of dividing rational numbers.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/XHQlyQhIeNM?si=tsS5v_dDZ7s_c4pI\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+3_+Divide+Fractions.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 3: Divide Fractions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Solving Problems Involving Rational Numbers<\/h2>\n<p>Watch the following for more examples of determining a fraction of a large whole number.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/mENiI0nwX2g?si=CCqCPpfvpD-wWGyA\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Determine+a+Fraction+of+a+Large+Whole+Number.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermine a Fraction of a Large Whole Number\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Converting Between Improper Fractions and Mixed Numbers<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<\/strong> \u00a0<\/p>\n<p><strong>Convert an improper fraction to a mixed number.<\/strong> \u00a0<\/p>\n<ol id=\"eip-241\" class=\"stepwise\">\n<li>Divide the denominator into the numerator.<\/li>\n<li>Identify the quotient, remainder, and divisor.<\/li>\n<li>Write the mixed number as quotient [latex]{\\Large\\frac{\\text{remainder}}{\\text{divisor}}}[\/latex] .<\/li>\n<\/ol>\n<p><strong>Convert a mixed number to an improper fraction.<\/strong> \u00a0<\/p>\n<ol id=\"eip-id1168467427407\" class=\"stepwise\">\n<li>Multiply the whole number by the denominator.<\/li>\n<li>Add the numerator to the product found in Step 1.<\/li>\n<li>Write the final sum over the original denominator.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox recall\">\n<p>A <strong>quotient<\/strong> is the result obtained when one number is divided by another; a <strong>remainder<\/strong> is the amount left over after division when the dividend cannot be evenly divided by the divisor; and a <strong>divisor<\/strong> is the number by which another number (the dividend) is divided.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Rewrite [latex]\\frac{95}{26}[\/latex] as a mixed number.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q160938\">Show Solution<\/button> <\/p>\n<div id=\"q160938\" class=\"hidden-answer\" style=\"display: none\"> When [latex]95[\/latex] is divided by [latex]26[\/latex], the result is [latex]3[\/latex] with a remainder of [latex]17[\/latex]. So, we can rewrite [latex]\\frac{95}{26}[\/latex] as [latex]3\\frac{17}{26}[\/latex]. <\/div>\n<\/div>\n<\/section>\n<p>Now you can watch worked examples of how to convert an improper fraction to a mixed number in the following video.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/e6uoYVg5Q30?si=0yp8AigQ9jp74hp6\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Examples_+Convert+an+Improper+Fraction+to+a+Mixed+Number.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExamples: Convert an Improper Fraction to a Mixed Number\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Rewrite [latex]9\\frac{5}{14}[\/latex] as an improper fraction.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q160939\">Show Solution<\/button> <\/p>\n<div id=\"q160939\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li><strong>Step 1:<\/strong> Multiply the integer part, [latex]9[\/latex], by the denominator, [latex]14[\/latex], which gives [latex]9\u00d714=126[\/latex].<\/li>\n<li><strong>Step 2:<\/strong> Add that product to the numerator, which gives [latex]5+126=131[\/latex].<\/li>\n<li><strong>Step 3:<\/strong> Write the number as the sum, [latex]131[\/latex], divided by the denominator, [latex]14[\/latex], which gives [latex]\\frac{131}{14}[\/latex].<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<p>In the following video we show more example of how to convert a mixed number to an improper fraction.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/p_YRBcZ4u4g?si=0W-_ozEb_qAoxKzu\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Examples_+Converting+a+Mixed+Number+to+an+Improper+Fraction.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExamples: Converting a Mixed Number to an Improper Fraction\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Contemporary Mathematics\",\"author\":\"Donna Kirk\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/3-4-rational-numbers\",\"project\":\"3.4 Rational Numbers\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/contemporary-mathematics\/pages\/1-introduction\"},{\"type\":\"copyrighted_video\",\"description\":\"Adding Fractions with Common Denominators (Step by Step) | Math with Mr. J\",\"author\":\"Math with Mr. J\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/MZmENadGcK0\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Subtracting Fractions with Common Denominators (Step by Step) | Math with Mr. J\",\"author\":\"Math with Mr. J\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/VTCOHFJOAA8\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Why Do We Need a Common Denominator When Adding and Subtracting Fractions? 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