{"id":132,"date":"2022-10-06T14:36:40","date_gmt":"2022-10-06T18:36:40","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/math025\/chapter\/equivalent-fractions\/"},"modified":"2025-06-26T23:24:20","modified_gmt":"2025-06-27T03:24:20","slug":"equivalent-fractions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/math025\/chapter\/equivalent-fractions\/","title":{"raw":"Topic A: Equivalent Fractions","rendered":"Topic A: Equivalent Fractions"},"content":{"raw":"<p style=\"text-align: center\">Start from the left side of each drawing and shade in the fraction shown.<\/p>\r\n&nbsp;\r\n<table style=\"width: 50%\" align=\"center\">\r\n<tbody>\r\n<tr>\r\n<td height=\"40\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">This shape is the whole thing<\/p>\r\n&nbsp;\r\n<table style=\"width: 50%\" align=\"center\">\r\n<tbody>\r\n<tr>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">Shade [latex]\\tfrac{1}{2}[\/latex]<\/p>\r\n&nbsp;\r\n<table style=\"width: 50%\" align=\"center\">\r\n<tbody>\r\n<tr>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">Shade [latex]\\tfrac{2}{4}[\/latex]<\/p>\r\n&nbsp;\r\n<table style=\"width: 50%\" align=\"center\">\r\n<tbody>\r\n<tr>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">Shade [latex]\\tfrac{3}{6}[\/latex]<\/p>\r\n&nbsp;\r\n<table style=\"width: 50%\" align=\"center\">\r\n<tbody>\r\n<tr>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">Shade [latex]\\tfrac{4}{8}[\/latex]<\/p>\r\n&nbsp;\r\n<table style=\"width: 50%\" align=\"center\">\r\n<tbody>\r\n<tr>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">Shade [latex]\\tfrac{5}{10}[\/latex]<\/p>\r\nDid you notice that the amount you shaded was the same in each drawing?\r\n\r\nThe fractions that you were asked to shade are <strong>equivalent fractions<\/strong>. Equivalent fractions <strong>are fractions that are equal<\/strong>.\r\n\r\n<strong>Now shade the fractions asked for in these drawings, the same way.<\/strong>\r\n<table style=\"width: 50%\" align=\"center\">\r\n<tbody>\r\n<tr>\r\n<td height=\"40\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">This shape is the whole thing<\/p>\r\n&nbsp;\r\n<table style=\"width: 50%\" align=\"center\">\r\n<tbody>\r\n<tr>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">Shade [latex]\\tfrac{1}{3}[\/latex]<\/p>\r\n&nbsp;\r\n<table style=\"width: 50%\" align=\"center\">\r\n<tbody>\r\n<tr>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">Shade [latex]\\tfrac{2}{6}[\/latex]<\/p>\r\n&nbsp;\r\n<table style=\"width: 50%\" align=\"center\">\r\n<tbody>\r\n<tr>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">Shade [latex]\\tfrac{3}{9}[\/latex]<\/p>\r\n&nbsp;\r\n<table style=\"width: 50%\" align=\"center\">\r\n<tbody>\r\n<tr>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">Shade [latex]\\tfrac{4}{12}[\/latex]<\/p>\r\n&nbsp;\r\n<table style=\"width: 50%\" align=\"center\">\r\n<tbody>\r\n<tr>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<td height=\"40\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p style=\"text-align: center\">Shade [latex]\\tfrac{5}{15}[\/latex]<\/p>\r\n&nbsp;\r\n<p style=\"padding-left: 40px\">These above examples are all equivalent fractions.<\/p>\r\n<p style=\"text-align: center\">[latex]\\dfrac{1}{3}[\/latex] = [latex]\\dfrac{2}{6}[\/latex] = [latex]\\dfrac{3}{9}[\/latex] = [latex]\\dfrac{4}{12}[\/latex] = [latex]\\dfrac{5}{15}[\/latex]<\/p>\r\n<img class=\"alignnone size-full wp-image-385\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-10-28-at-3.33.07-PM.png\" alt=\"Arrow graphic\" width=\"46\" height=\"44\" \/>To work with common fractions, it is often necessary to use an equivalent fraction in place of the fraction that is given. There are several processes to learn which will help you to find equivalent fractions.\r\n<h1>Factors<\/h1>\r\n[pb_glossary id=\"279\"]Factors[\/pb_glossary] are the numbers which are multiplied together to make a [pb_glossary id=\"258\"]product[\/pb_glossary]. An understanding of factors is needed to express fractions in lowest [pb_glossary id=\"2135\"]terms[\/pb_glossary].\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<img class=\"aligncenter wp-image-129 size-medium\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-10-28-at-3.41.31-PM-300x113.png\" alt=\"Labelled illustration. 3 x 4 and label reads Factor. Equals 12 and label reads Product.\" width=\"300\" height=\"113\" \/>\r\n\r\nWe say, \"The factors of 12 are 3 and 4.\"\r\n\r\nDoes 12 have any other factors?\r\n\r\nWhat other numbers can be multiplied together to equal 12?\r\n<ul>\r\n \t<li>[latex]1 \\times 12 = 12[\/latex] or [latex]12 \\times 1 = 12[\/latex]<\/li>\r\n \t<li>[latex]2 \\times 6 = 12[\/latex] or [latex]6 \\times 2 = 12[\/latex]<\/li>\r\n \t<li>[latex]3 \\times 4 = 12[\/latex] or [latex]4 \\times 3 = 12[\/latex]<\/li>\r\n<\/ul>\r\nThe factors of 12 are 1, 2, 3, 4, 6, 12.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example B<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the factors of 10.\r\n<ul>\r\n \t<li>[latex]1 \\times 10 = 10[\/latex]<\/li>\r\n \t<li>[latex]2 \\times 5 = 10[\/latex]<\/li>\r\n<\/ul>\r\nThe factors of 10 are 1, 2, 5, 10.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example C<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the factors of 9.\r\n<ul>\r\n \t<li>[latex]1 \\times 9 = 9[\/latex]<\/li>\r\n \t<li>[latex]3 \\times 3 = 9[\/latex]<\/li>\r\n<\/ul>\r\nThe factors of 9 are 1, 3, 9.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind all the factors\r\n<ol type=\"a\">\r\n \t<li>The factors of 16: [latex]1 \\times 16 = 16; 2 \\times 8 = 16; 4 \\times 4 = 16[\/latex]\u00a0 The factors of 16 are 1, 2, 4, 8, 16.<\/li>\r\n \t<li>The factors of 4: [latex]1 \\times 4 = 4; 2 \\times 2 = 4[\/latex] The factors of 4 are 1, 2, 4.<\/li>\r\n \t<li>The factors of 8:<span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>The factors of 20:<span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>The factors of 5:<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>The factors of 15:<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>The factors of 21:<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>The factors of 6:<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>The factors of 25:<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 1<\/strong>\r\n<ol class=\"threecolumn\" start=\"3\" type=\"a\">\r\n \t<li>1, 2, 4, 8<\/li>\r\n \t<li>1, 2, 4, 5, 10, 20<\/li>\r\n \t<li>1, 5<\/li>\r\n \t<li>1, 3, 5, 15<\/li>\r\n \t<li>1, 3, 7, 21<\/li>\r\n \t<li>1, 2, 3, 6<\/li>\r\n \t<li>1, 5, 25<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nSome numbers <strong>only have two factors, 1 and the number itself.<\/strong> These numbers are called [pb_glossary id=\"257\"]prime numbers[\/pb_glossary]. Look at the chart for some prime numbers.\r\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 0%;height: 306px\" border=\"0\"><caption>Table for Prime numbers and factors<\/caption>\r\n<tbody>\r\n<tr style=\"height: 18px\">\r\n<th style=\"width: 46.9438%;height: 30px;text-align: center\">Prime Numbers<\/th>\r\n<th style=\"width: 52.8117%;height: 18px;text-align: center\">Factors<\/th>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\">1<\/td>\r\n<td style=\"width: 52.8117%;height: 18px\">1,1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px;text-align: left\">2<\/td>\r\n<td style=\"width: 52.8117%;height: 18px\">1,2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\">3<\/td>\r\n<td style=\"width: 52.8117%;height: 18px\">1,3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\">5<\/td>\r\n<td style=\"width: 52.8117%;height: 18px\">1,5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\">7<\/td>\r\n<td style=\"width: 52.8117%;height: 18px\">1,7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\">11<\/td>\r\n<td style=\"width: 52.8117%;height: 18px\">1,11<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\">13<\/td>\r\n<td style=\"width: 52.8117%;height: 18px\">1,13<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\">17<\/td>\r\n<td style=\"width: 52.8117%;height: 18px\">1,17<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\">19<\/td>\r\n<td style=\"width: 52.8117%;height: 18px\">1,19<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\">23<\/td>\r\n<td style=\"width: 52.8117%;height: 18px\">1,23<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\">29<\/td>\r\n<td style=\"width: 52.8117%;height: 18px\">1,29<\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\"><\/td>\r\n<td style=\"width: 52.8117%;height: 18px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\"><\/td>\r\n<td style=\"width: 52.8117%;height: 18px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\"><\/td>\r\n<td style=\"width: 52.8117%;height: 18px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\"><\/td>\r\n<td style=\"width: 52.8117%;height: 18px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 46.9438%;height: 18px\"><\/td>\r\n<td style=\"width: 52.8117%;height: 18px\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<strong>Add other prime numbers to the chart as you find them.<\/strong>\r\n\r\n<strong>Reminder<\/strong>: Prime numbers only have two, prime factors.\r\n<h1>Finding Common Factors<\/h1>\r\nA <strong>common factor<\/strong> is a number used to reduce the numerator and denominator.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example D<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat are the common factors for [latex]\\tfrac{4}{6}[\/latex]?\r\n<ul>\r\n \t<li>Find the factors of 4 and 6.\r\n<ul>\r\n \t<li>The factors of 4 are 1, 2, 4.<\/li>\r\n \t<li>The factors of 6 are 1, 2, 3, 6.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nWhat factors do 4 and 6 have in common?\r\n<ul>\r\n \t<li>4: 1, 2, 4<\/li>\r\n \t<li>6: 1, 2, 3, 6<\/li>\r\n<\/ul>\r\nThe common factors of 4 and 6 are<strong> 1 and 2<\/strong>\r\n\r\nFor the above equation, the factors are 1 and 2; however, 1 is not used as a <strong>common factor<\/strong>. This is because 1 is a factor of all <strong>whole numbers<\/strong>.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example E<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat are the common factors for [latex]\\tfrac{6}{15}[\/latex]?\r\n<ul>\r\n \t<li>Find the factors of 6 and 15.\r\n<ul>\r\n \t<li>The factors of 6 are 1, 2, 3,6<\/li>\r\n \t<li>The factors of 15 are 1, 3,5,15.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nWhat factors do 6 and 15 have in common?\r\n<ul>\r\n \t<li>6: 1, 2, 3,6<\/li>\r\n \t<li>15: 1, 3,5,15<\/li>\r\n<\/ul>\r\nThe common factor of 6 and 15 is 3\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example F<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the common factors of [latex]\\tfrac{16}{24}[\/latex]?\r\n<ul>\r\n \t<li>Find the factors of 16 are 1, 2, 4, 8, 16<\/li>\r\n \t<li>The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24<\/li>\r\n \t<li>The common factors of 16 and 24 are: 2, 4, 8<\/li>\r\n \t<li>8 is called the <strong>greatest common factor (GCF)<\/strong> of 16 and 24 because it is the largest of all the common factors<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the <strong>Common Factors for each set of numbers.<\/strong> Then identify the <strong>Greatest Common Factor (GCF)<\/strong>.\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>10, 15\r\n<ul>\r\n \t<li>Factors of 10: 1, 2, 5, 10<\/li>\r\n \t<li>Factors of 15: 1, 3, 5, 15<\/li>\r\n \t<li>Common factors: 5<\/li>\r\n \t<li>Greatest common factor: 5<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>4, 16\r\n<ul>\r\n \t<li>Factors of 4: 1, 2, 4<\/li>\r\n \t<li>Factors of 16: 1, 2, 4, 8, 16<\/li>\r\n \t<li>Common factors: 2, 4<\/li>\r\n \t<li>Greatest common factor: 4<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>9,12\r\n<ul>\r\n \t<li>Factors of 9:<\/li>\r\n \t<li>Factors of 12:<\/li>\r\n \t<li>Common factors:<\/li>\r\n \t<li>Greatest common factor:<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>20, 30\r\n<ul>\r\n \t<li>Factors of 20:<\/li>\r\n \t<li>Factors of 30:<\/li>\r\n \t<li>Common factors:<\/li>\r\n \t<li>Greatest common factor:<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>18, 12\r\n<ul>\r\n \t<li>Factors of 18:<\/li>\r\n \t<li>Factors of 12:<\/li>\r\n \t<li>Common factors:<\/li>\r\n \t<li>Greatest common factor:<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>24, 32\r\n<ul>\r\n \t<li>Factors of 24:<\/li>\r\n \t<li>Factors of 32:<\/li>\r\n \t<li>Common factors:<\/li>\r\n \t<li>Greatest common factor:<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>8, 12\r\n<ul>\r\n \t<li>Factors of 8:<\/li>\r\n \t<li>Factors of 12:<\/li>\r\n \t<li>Common factors:<\/li>\r\n \t<li>Greatest common factor:<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>6, 9\r\n<ul>\r\n \t<li>Factors of 6:<\/li>\r\n \t<li>Factors of 9:<\/li>\r\n \t<li>Common factors:<\/li>\r\n \t<li>Greatest common factor:<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>9,15\r\n<ul>\r\n \t<li>Factors of 9:<\/li>\r\n \t<li>Factors of 15:<\/li>\r\n \t<li>Common factors:<\/li>\r\n \t<li>Greatest common factor:<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 2<\/strong>\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>10, 15\r\n<ul>\r\n \t<li>Factors of 10: 1, 2, 5, 10<\/li>\r\n \t<li>Factors of 15: 1, 3, 5, 15<\/li>\r\n \t<li>Common factors: 5<\/li>\r\n \t<li>Greatest common factor: 5<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>4, 16\r\n<ul>\r\n \t<li>Factors of 4: 1, 2, 4<\/li>\r\n \t<li>Factors of 16: 1, 2, 4, 8, 16<\/li>\r\n \t<li>Common factors: 2, 4<\/li>\r\n \t<li>Greatest common factor: 4<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>9,12\r\n<ul>\r\n \t<li>Factors of 9: 1, 3, 9<\/li>\r\n \t<li>Factors of 12: 1, 2, 3, 4, 6, 12<\/li>\r\n \t<li>Common factors: 3<\/li>\r\n \t<li>Greatest common factor: 3<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>20, 30\r\n<ul>\r\n \t<li>Factors of 20: 1, 2, 4, 5, 10, 20<\/li>\r\n \t<li>Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30<\/li>\r\n \t<li>Common factors: 2, 5, 10<\/li>\r\n \t<li>Greatest common factor: 10<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>18, 12\r\n<ul>\r\n \t<li>Factors of 18: 1, 2, 3, 6, 9, 18<\/li>\r\n \t<li>Factors of 12:, 1, 2, 3, 4, 6, 12<\/li>\r\n \t<li>Common factors: 2, 3, 6<\/li>\r\n \t<li>Greatest common factor: 6<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>24, 32\r\n<ul>\r\n \t<li>Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24<\/li>\r\n \t<li>Factors of 32:, 1, 2, 4, 8, 16, 32<\/li>\r\n \t<li>Common factors: 2, 4, 8<\/li>\r\n \t<li>Greatest common factor: 8<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>8, 12\r\n<ul>\r\n \t<li>Factors of 8: 1, 2, 4, 8<\/li>\r\n \t<li>Factors of 12: 1, 2, 3, 4, 6, 12<\/li>\r\n \t<li>Common factors: 2, 4<\/li>\r\n \t<li>Greatest common factor: 4<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>6, 9\r\n<ul>\r\n \t<li>Factors of 6: 1, 2, 3, 6<\/li>\r\n \t<li>Factors of 9: 1, 3, 9<\/li>\r\n \t<li>Common factors: 3<\/li>\r\n \t<li>Greatest common factor: 3<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>9, 15\r\n<ul>\r\n \t<li>Factors of 9: 1, 3, 9<\/li>\r\n \t<li>Factors of 15: 1, 3, 5, 15<\/li>\r\n \t<li>Common factors: 3<\/li>\r\n \t<li>Greatest common factor: 3<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<ul>\r\n \t<li style=\"list-style-type: none\">\r\n<ul>\r\n \t<li style=\"list-style-type: none\">\r\n<ul>\r\n \t<li style=\"list-style-type: none\"><\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h1>Expressing Fractions in Lower Terms<\/h1>\r\nExpress means <strong>to say it or write it.<\/strong>\r\n\r\n<strong>Lower terms<\/strong> means to express equivalent fractions with smaller (lower) denominators.\r\n\r\nLook back to page 67. The equivalent fraction in lowest terms is [latex]\\tfrac{1}{2}[\/latex] .\r\n\r\nThe words [pb_glossary id=\"267\"]simplify[\/pb_glossary] and [pb_glossary id=\"264\"]reduce[\/pb_glossary] are another way to say \u201c<strong>express fractions in lower (or lowest) terms<\/strong>.\u201d\r\n\r\n<strong>To express a fraction in lowest terms, do this:<\/strong>\r\n\r\n<strong>Step 1:\u00a0\u00a0<\/strong> Find the greatest common factor (GCF) of the numerator and denominator.\r\n<ul>\r\n \t<li>[latex]\\dfrac{4}{12}[\/latex]\u00a0 \u00a0The factors of 4 are 1, 2, 4<\/li>\r\n \t<li>The factors of 12 are 1, 2, 3, 4, 6, 12<\/li>\r\n<\/ul>\r\n<strong>The GCF is 4.<\/strong>\r\n\r\n<strong>Step 2:<\/strong>\u00a0 \u00a0 \u00a0 \u00a0Divide the numerator and the denominator by the greatest common factor.\r\n<ul>\r\n \t<li>[latex]\\dfrac{4}{12}[\/latex] [latex]\\dfrac{\\div 4}{\\div 4}[\/latex]\u00a0\u00a0 =\u00a0 [latex]\\dfrac{1}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{12}[\/latex] = [latex]\\dfrac{1}{3}[\/latex]<\/li>\r\n<\/ul>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example G<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[latex]\\dfrac{6}{9}[\/latex]\r\n\r\nThe factors of 6 are 1, 2, 3 , 6.\r\nThe factors of 9 are 1, 3 , 9.\r\n\r\n<strong>The GCF is 3.<\/strong>\r\n<ul>\r\n \t<li>[latex]\\dfrac{6}{9}\\dfrac{\\div 3}{\\div 3} = \\dfrac{2}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{6}{9} = \\dfrac{2}{3}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example H<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<div class=\"textbox__content\">\r\n\r\n[latex]\\dfrac{15}{24}[\/latex]\r\n\r\nThe factors of 15 are 1, 3,5,15\r\nThe factors of 24 are 1, 3, 4, 6, 8, 24.\r\n\r\n<strong>The GCF is 3.<\/strong>\r\n<ul>\r\n \t<li>[latex]\\dfrac{15}{24} \\dfrac{\\div 3}{\\div 3} =\\dfrac{5}{8}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{15}{24} = \\dfrac{5}{8}[\/latex]<\/li>\r\n<\/ul>\r\n<strong>There are several reasons lower terms are used:<\/strong>\r\n<ul>\r\n \t<li>The math is usually easier with lower numbers.<\/li>\r\n \t<li>Is it easier to think of [latex]\\tfrac{1}{2}[\/latex] an apple or [latex]\\tfrac{15}{30}[\/latex] of an apple? ([latex]\\tfrac{1}{2}[\/latex] = [latex]\\tfrac{15}{30}[\/latex] )<\/li>\r\n \t<li>Do you want to think about [latex]\\tfrac{155}{620}[\/latex] of your pay cheque or [latex]\\tfrac{1}{4}[\/latex] of your pay cheque? ( [latex]\\tfrac{1}{4}[\/latex] = [latex]\\tfrac{155}{620}[\/latex] )<\/li>\r\n \t<li>Always express fractions in lowest terms!<\/li>\r\n<\/ul>\r\n<strong>Dividing<\/strong> both the numerator and denominator <strong>by the GCF<\/strong> will give an equivalent fraction in lower terms.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nExpress each fraction in <strong>lowest terms.<\/strong> (The directions could also say, \"Simplify each fraction,\" or \"Reduce these fractions\").\r\n<ol type=\"a\">\r\n \t<li>[latex]\\dfrac{2}{4}\\dfrac{\\div 2}{\\div 2}= \\dfrac{1}{2}[\/latex],\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{3}{9}\\dfrac{ \\div \\ \\ \\ }{ \\div \\ \\ \\ }[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>[latex]\\dfrac{2}{12}\\dfrac{\\div2}{\\div2}=\\dfrac{1}{6}[\/latex],\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{3}{15}\\dfrac{ \\div \\ \\ \\ }{ \\div \\ \\ \\ }[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>[latex]\\dfrac{5}{10}\\dfrac{ \\div \\ \\ }{ \\div \\ \\ \\ }[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{4}{24}\\dfrac{ \\div \\ \\ \\ }{ \\div \\ \\ \\ }[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>[latex]\\dfrac{10}{25}\\dfrac{ \\div \\ \\ }{ \\div\\ \\ \\ }[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{9}{12}\\dfrac{ \\div \\ \\ \\ }{ \\div \\ \\ \\ }[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\r\n<p style=\"margin-top: 1em\"><strong>Make sure that you write in the GCF you are dividing with. Do not skip this step until you are absolutely sure you can do it correctly in your head each time.<\/strong><\/p>\r\n(Good mathematicians know when to skip steps and when not to... sometimes easy steps are never skipped by good mathematicians).<\/li>\r\n \t<li>[latex]\\dfrac{3}{30}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{6}{10}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>[latex]\\dfrac{9}{24}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{18}{27}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/li>\r\n \t<li>[latex]\\dfrac{4}{16}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{3}{12}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/li>\r\n \t<li>[latex]\\dfrac{15}{24}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{15}{25}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/li>\r\n \t<li>[latex]\\dfrac{2}{32}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{6}{20}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox__content\"><strong>Answers to Exercise 3\r\n<\/strong>\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{1}{2}[\/latex], [latex]\\dfrac{1}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{6}[\/latex], [latex]\\dfrac{1}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{2}[\/latex], [latex]\\dfrac{1}{6}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{2}{5}[\/latex], [latex]\\dfrac{3}{4}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{10}[\/latex], [latex]\\dfrac{3}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{8}[\/latex], [latex]\\dfrac{2}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{4}[\/latex], [latex]\\dfrac{1}{4}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{5}{8}[\/latex], [latex]\\dfrac{3}{5}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{16}[\/latex], [latex]\\dfrac{3}{10}[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>Expressing Fractions in Higher Terms<\/h1>\r\nHigher Terms are needed when you add and subtract fractions with different denominators.\r\n\r\nYou have learned that <strong>dividing<\/strong> both the numerator and denominator of a fraction by a common factor <strong>gives an equivalent fraction in lower terms<\/strong>. You know that dividing and multiplying are opposite operations, so this next rule will match the one you just learned for reducing:\r\n\r\n<strong>Multiplying both<\/strong> the <strong>numerator and denominator<\/strong> of a fraction by the same number (a <strong>common factor)<\/strong> will give an <strong>equivalent fraction<\/strong> in <strong>higher terms<\/strong>.\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example I<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[latex]\\dfrac{3}{5}\\left(\\dfrac{\\times 2}{\\times 2}\\right)=\\dfrac{6}{10}[\/latex]\r\n\r\n[latex]\\dfrac{3}{5}=\\dfrac{6}{10}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example J<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[latex]\\dfrac{1}{2}\\left(\\dfrac{\\times 8}{\\times 8}\\right)=\\dfrac{8}{16}[\/latex]\r\n\r\n[latex]\\dfrac{1}{2}= \\dfrac{8}{16}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example K<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[latex]\\dfrac{2}{3}\\left(\\dfrac{\\times 3}{\\times 3}\\right)=\\dfrac{6}{9}[\/latex]\r\n\r\n[latex]\\dfrac{2}{3} = \\dfrac{6}{9}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<h1>Are the Fractions Equivalent?<\/h1>\r\nIf the denominators are the same, you can easily judge if the fractions are equivalent by comparing the numerators.\r\n\r\nCompare [latex]\\tfrac{4}{5}[\/latex] and [latex]\\tfrac{3}{5}[\/latex]: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 [latex]\\tfrac{4}{5}[\/latex] \u2260 [latex]\\tfrac{3}{5}[\/latex] (\u2260 means \u2018not equal\u2019)\r\n\r\nCompare [latex]\\tfrac{12}{20}[\/latex] and [latex]\\tfrac{12}{20}[\/latex]: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0[latex]\\tfrac{12}{20} = \\tfrac{12}{20}[\/latex]\r\n\r\nIf the denominators are different, you <strong>might be able to rewrite one or more of the fractions so they have the same denominator.<\/strong>\r\n\r\nCompare [latex]\\tfrac{4}{5}[\/latex] and [latex]\\tfrac{6}{10}[\/latex]: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0[latex]\\tfrac{6}{10}\\tfrac{\\div 2}{\\div 2}[\/latex]=[latex]\\tfrac{3}{5}[\/latex] \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0So:[latex]\\tfrac{4}{5}\\neq \\tfrac{3}{5}[\/latex]\r\n\r\nCompare [latex]\\tfrac{12}{16}[\/latex] and [latex]\\tfrac{5}{8}[\/latex]: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 [latex]\\tfrac{5}{8} \\tfrac{\\times 2}{\\times 2}[\/latex] =[latex]\\tfrac{10}{16}[\/latex] \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0So:[latex]\\tfrac{12}{16}\\neq \\tfrac{10}{16}[\/latex]\r\n\r\nor you could do this: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]\\tfrac{12}{16}[\/latex][latex]\\tfrac{\\div 2}{\\div 2}= \\tfrac{6}{8}[\/latex] \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0So:[latex]\\tfrac{6}{8}\\neq\\tfrac{5}{8}[\/latex]\r\n\r\nA quick method is to <strong>[pb_glossary id=\"284\"]cross multiply[\/pb_glossary]<\/strong>:\r\n<ol>\r\n \t<li>multiply the numerator of one fraction by the denominator of the second fraction<\/li>\r\n \t<li>multiply the numerator of the <strong>other<\/strong> fraction by the denominator of the <strong>first<\/strong> fraction These are called the <em>cross-products<\/em>.\r\nIf the cross products are the same, then the fraction is equivalent.<\/li>\r\n<\/ol>\r\nLook at the examples:\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example L<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nCompare [latex]\\dfrac{4}{7}[\/latex] and [latex]\\dfrac{5}{9}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox__content\">\r\n\r\n<img class=\"wp-image-130 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-11-03-at-10.40.37-AM.png\" alt=\"A picture of two fractions and arrows between them. First fraction is numerator 4 and denominator 7. Second fraction is numberator 5 and denominator 9. Arrows direct attention from the first fraction numerator 4 to second fraction denominator 9. And from first fraction denominator 7 to second fraction numerator 5..\" width=\"212\" height=\"78\" \/>\r\n<ul>\r\n \t<li>Multiply the numerator 4 by the denominator 9\r\n<ul>\r\n \t<li>[latex]4 \\times 9 = 36[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Multiply the denominator 7 by the numerator 5\r\n<ul>\r\n \t<li>[latex]7 \\times 5 = 35[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>The products 36 and 35 <strong>are not <\/strong>the same.<\/li>\r\n<\/ul>\r\nTherefore [latex]\\dfrac{4}{7}\\neq \\dfrac{5}{9}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example M<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nCompare [latex]\\dfrac{2}{3}[\/latex] and [latex]\\dfrac{12}{18}[\/latex]\r\n\r\n<img class=\"aligncenter wp-image-131 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-11-03-at-2.13.14-PM.png\" alt=\"A picture of two fractions and arrows between them. First fraction is numerator 2 and denominator 3. Second fraction is numberator 12 and denominator 18. Arrows direct attention from the first fraction numerator 2 to second fraction denominator 18. And from first fraction denominator 3 to second fraction numerator 12.\" width=\"235\" height=\"96\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>[latex]2 \\times 18 = 36[\/latex]<\/li>\r\n \t<li>[latex]3 \\times 12 = 36[\/latex]<\/li>\r\n \t<li>The products 36 and 36 <strong>are<\/strong> the same<\/li>\r\n<\/ul>\r\nTherefore [latex]\\dfrac{2}{3} = \\dfrac{12}{18}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example N<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nCompare [latex]\\dfrac{24}{40}[\/latex] and [latex]\\dfrac{4}{10}[\/latex]\r\n<ul>\r\n \t<li>[latex]24 \\times 10 = 240[\/latex]<\/li>\r\n \t<li>[latex]40 \\times 4 = 160[\/latex]<\/li>\r\n \t<li>The products 240 and 160 <strong>are not<\/strong> the same.<\/li>\r\n<\/ul>\r\nTherefore [latex]\\dfrac{24}{40} \\neq \\dfrac{4}{10}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nState if each pair is <strong>equivalent (=)<\/strong> or <strong>not equivalent<\/strong> (\u2260). Use whichever method you wish to find the answer.\r\n<ol class=\"threecolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{5}{6}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0 <\/span>[latex]\\dfrac{30}{60}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{12}{24}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u00a0[latex]\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{6}{7}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0[latex]\\dfrac{7}{8}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{2}{3}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0[latex]\\dfrac{12}{18}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{3}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0[latex]\\dfrac{24}{72}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{4}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0[latex]\\dfrac{15}{20}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{12}{14}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0[latex]\\dfrac{6}{7}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{4}{10}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> [latex]\\dfrac{20}{50}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{5}{10}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> [latex]\\dfrac{7}{14}[\/latex]<\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise 4<\/strong>\r\n<ol class=\"threecolumn\" type=\"a\">\r\n \t<li>\u2260<\/li>\r\n \t<li>=<\/li>\r\n \t<li>\u2260<\/li>\r\n \t<li>=<\/li>\r\n \t<li>=<\/li>\r\n \t<li>=<\/li>\r\n \t<li>=<\/li>\r\n \t<li>=<\/li>\r\n \t<li>=<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h1>Rounding Common Fractions to Whole Numbers<\/h1>\r\nWhen rounding to a whole number, if a fraction is less than [latex]\\dfrac{1}{2}[\/latex] \u00a0do not change the whole number:\r\n\r\nExamples:\r\n<ul class=\"twocolumn\">\r\n \t<li>[latex]2\\dfrac{3}{7}\u2248 2[\/latex]<\/li>\r\n \t<li>[latex]23\\dfrac{1}{3} \u2248 23[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{1}{4} \u2248 0[\/latex]<\/li>\r\n \t<li>[latex]5\\dfrac{3}{8}\u2248 5[\/latex]<\/li>\r\n<\/ul>\r\nIf the fraction is [latex]\\dfrac{1}{2}[\/latex] or more, consider the fraction as <strong>another one<\/strong> which must be added to the whole number:\r\n\r\nExamples:\r\n<ul class=\"twocolumn\">\r\n \t<li>[latex]2\\dfrac{1}{2}[\/latex] \u2248 3<\/li>\r\n \t<li>[latex]15\\dfrac{4}{5}[\/latex] \u2248 16<\/li>\r\n \t<li>[latex]6\\dfrac{7}{8}[\/latex] \u2248 7<\/li>\r\n \t<li>[latex]\\dfrac{3}{4}[\/latex] \u2248 1<\/li>\r\n<\/ul>\r\nIf you are not sure if a fraction is more or less than [latex]\\tfrac{1}{2}[\/latex] , you can compare it to [latex]\\tfrac{1}{2}[\/latex], by making equivalent fractions with a common denominator.\r\n<p style=\"text-align: center\">Reminder: greater &gt; smaller<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example O<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nRound [latex]\\tfrac{2}{3}[\/latex] to a whole number.\r\n\r\nIs [latex]\\dfrac{2}{3}[\/latex] &gt;\u00a0 [latex]\\dfrac{1}{2}\\text{?}\\longrightarrow\\dfrac{2}{3}= \\dfrac{4}{6}\\text{ and }\\dfrac{1}{2}= \\dfrac{3}{6}[\/latex]\r\n\r\nYES! [latex]\\dfrac{2}{3}[\/latex] &gt; [latex]\\dfrac{1}{2}\\text{so}\\dfrac{2}{3}\u2248 1[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example P<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nRound [latex]2\\tfrac{4}{7}[\/latex] to a whole number.\r\n\r\nIs\u00a0 [latex] \\dfrac{4}{7}[\/latex] &gt; [latex]\\dfrac{1}{2}\\text{?}\\longrightarrow\\dfrac{4}{7}= \\dfrac{8}{14}\\text{ and }\\dfrac{1}{2} = \\dfrac{7}{14}[\/latex]\r\n\r\nYES! [latex]\\tfrac{4}{7}[\/latex] &gt; [latex]\\tfrac{1}{2}[\/latex] so [latex]2\\tfrac{4}{7}[\/latex] \u2248 [latex]3[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Exercise 5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nRound to the nearest whole number\r\n<ol class=\"threecolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{4}{5}\u2248 1[\/latex]<\/li>\r\n \t<li>[latex]2\\dfrac{1}{3}\u2248 2[\/latex]<\/li>\r\n \t<li>[latex]18\\dfrac{1}{2} \u2248 [\/latex]\u00a0\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>[latex]3\\dfrac{7}{8} \u2248 [\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\r\n \t<li>[latex]9\\dfrac{9}{10} \u2248 [\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\r\n \t<li>[latex]\\dfrac{1}{8} \u2248 [\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\r\n \t<li>[latex]4\\dfrac{1}{6} \u2248 [\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\r\n \t<li>[latex]12\\dfrac{7}{9} \u2248 [\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\r\n \t<li>[latex]6\\dfrac{3}{5} \u2248 [\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\r\n \t<li>[latex]20\\dfrac{3}{7} \u2248 [\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\r\n \t<li>[latex]\\dfrac{13}{15} \u2248 [\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\r\n \t<li>[latex]99\\dfrac{2}{3} \u2248 [\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/li>\r\n<\/ol>\r\n<strong>Answers to Exercise Five<\/strong>\r\n<ol class=\"threecolumn\" start=\"3\" type=\"a\">\r\n \t<li>19<\/li>\r\n \t<li>4<\/li>\r\n \t<li>10<\/li>\r\n \t<li>0<\/li>\r\n \t<li>4<\/li>\r\n \t<li>13<\/li>\r\n \t<li>7<\/li>\r\n \t<li>20<\/li>\r\n \t<li>1<\/li>\r\n \t<li>100<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n<h1>Topic A:\u00a0 Self-Test<\/h1>\r\n<strong>Mark\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \/25\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Aim\u00a0\u00a0\u00a0\u00a0\u00a0 20\/25<\/strong>\r\n<ol type=\"A\">\r\n \t<li>Define\u00a0 (3 marks)\r\n<ol>\r\n \t<li style=\"list-style-type: none\">\r\n<ol type=\"a\">\r\n \t<li>equivalent <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>prime number <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n \t<li>greatest common factor (GCF) <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Complete the chart (5 marks)\r\n<table class=\"grid\" style=\"border-collapse: collapse;width: 100%;height: 198px\" border=\"0\">\r\n<thead>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 16.6666%\"><\/td>\r\n<th style=\"width: 16.6666%;height: 18px\" scope=\"col\">Factors<\/th>\r\n<th style=\"width: 33.3333%;height: 18px\" scope=\"col\">Common Factors<\/th>\r\n<th style=\"width: 33.3333%;height: 18px\" scope=\"col\">Greatest Common Factor<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 36px\">\r\n<th style=\"width: 16.6666%\" scope=\"row\">12\r\n18<\/th>\r\n<td style=\"width: 16.6666%;height: 36px\"><\/td>\r\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\r\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 36px\">\r\n<th style=\"width: 16.6666%\" scope=\"row\">15\r\n30<\/th>\r\n<td style=\"width: 16.6666%;height: 36px\"><\/td>\r\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\r\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 36px\">\r\n<th style=\"width: 16.6666%\" scope=\"row\">7\r\n28<\/th>\r\n<td style=\"width: 16.6666%;height: 36px\"><\/td>\r\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\r\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 36px\">\r\n<th style=\"width: 16.6666%\" scope=\"row\">6\r\n16<\/th>\r\n<td style=\"width: 16.6666%;height: 36px\"><\/td>\r\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\r\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 36px\">\r\n<th style=\"width: 16.6666%\" scope=\"row\">18\r\n27<\/th>\r\n<td style=\"width: 16.6666%;height: 36px\"><\/td>\r\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\r\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li>Express in lowest terms. (6 marks)\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{10}{15}[\/latex]= <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>.<\/li>\r\n \t<li>[latex]\\dfrac{14}{16}[\/latex]= <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>.<\/li>\r\n \t<li>[latex]\\dfrac{8}{12}[\/latex]= <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>State if each pair of fractions is equivalent (=) or not equivalent ( ). (6 marks)\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{5}{9}[\/latex] <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> [latex]\\dfrac{15}{27}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{3}{7}[\/latex] <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> [latex]\\dfrac{15}{35}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Round to the nearest whole number (5 marks)\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>4[latex]\\dfrac{5}{8}[\/latex] \u2248 \u00a0\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 <\/span><\/li>\r\n \t<li>19[latex]\\dfrac{4}{10}[\/latex] \u2248 \u00a0\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 <\/span><\/li>\r\n \t<li>[latex]\\dfrac{1}{2}[\/latex] \u2248 \u00a0\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 <\/span><\/li>\r\n \t<li>6[latex]\\dfrac{3}{4}[\/latex] \u2248 \u00a0\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 <\/span><\/li>\r\n \t<li>[latex]\\dfrac{1}{3}[\/latex] \u2248 \u00a0\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 <\/span><\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<h1>Answers to Topic A Self-Test<\/h1>\r\n<ol type=\"A\">\r\n \t<li>Check your definitions in the glossary<\/li>\r\n \t<li>Complete the chart\r\n<table class=\"grid\" style=\"border-collapse: collapse;width: 100%;height: 198px\" border=\"0\">\r\n<thead>\r\n<tr style=\"height: 18px\">\r\n<td style=\"width: 16.6666%\"><\/td>\r\n<th style=\"width: 30.1259%;height: 18px\" scope=\"col\">Factors<\/th>\r\n<th style=\"width: 19.874%;height: 18px\" scope=\"col\">Common Factors<\/th>\r\n<th style=\"width: 33.3333%;height: 18px\" scope=\"col\">Greatest Common Factor<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 36px\">\r\n<th style=\"width: 16.6666%\" scope=\"row\">12\r\n18<\/th>\r\n<td style=\"width: 30.1259%;height: 36px\">\u2026 of 12 are 1, 2, 3, 4, 6, 12\r\n\u2026 of 18 are 1, 2, 3, 6, 9, 18<\/td>\r\n<td style=\"width: 19.874%;height: 36px;text-align: center\">2, 3, 6<\/td>\r\n<td style=\"width: 33.3333%;height: 36px\">6<\/td>\r\n<\/tr>\r\n<tr style=\"height: 36px\">\r\n<th style=\"width: 16.6666%\" scope=\"row\">15\r\n30<\/th>\r\n<td style=\"width: 30.1259%;height: 36px\">\u2026 of 15 are 1, 3, 5, 15\r\n\r\n\u2026 of 30 are 1, 2, 3, 5, 6, 10, 15, 30<\/td>\r\n<td style=\"width: 19.874%;height: 36px;text-align: center\">3, 5, 15<\/td>\r\n<td style=\"width: 33.3333%;height: 36px\">15<\/td>\r\n<\/tr>\r\n<tr style=\"height: 36px\">\r\n<th style=\"width: 16.6666%\" scope=\"row\">7\r\n28<\/th>\r\n<td style=\"width: 30.1259%;height: 36px\">\u2026 of 7 are 1, 7\r\n\u2026 of 28 are 1, 2, 4, 7, 14, 28<\/td>\r\n<td style=\"width: 19.874%;height: 36px;text-align: center\">7<\/td>\r\n<td style=\"width: 33.3333%;height: 36px\">7<\/td>\r\n<\/tr>\r\n<tr style=\"height: 36px\">\r\n<th style=\"width: 16.6666%\" scope=\"row\">6\r\n16<\/th>\r\n<td style=\"width: 30.1259%;height: 36px\">\u2026 of 6 are 1, 2, 3, 6\r\n\u2026 of 16 are 1, 2, 4, 8, 16<\/td>\r\n<td style=\"width: 19.874%;height: 36px;text-align: center\">2<\/td>\r\n<td style=\"width: 33.3333%;height: 36px\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 36px\">\r\n<th style=\"width: 16.6666%\" scope=\"row\">18\r\n27<\/th>\r\n<td style=\"width: 30.1259%;height: 36px\">\u2026 of 18 are 1, 2, 3, 6, 9, 18\r\n\u2026 of 27 are 1, 3, 9, 27<\/td>\r\n<td style=\"width: 19.874%;height: 36px;text-align: center\">3, 9<\/td>\r\n<td style=\"width: 33.3333%;height: 36px\">9<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li><strong>Express in lowest terms.<\/strong>\r\n<ol class=\"threecolumn\" type=\"a\">\r\n \t<li>[latex]\\dfrac{2}{3}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{7}{8}[\/latex]<\/li>\r\n \t<li>[latex]\\dfrac{2}{3}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong>State if each pair of fractions is equivalent (=) or not equivalent ( ).<\/strong>\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li style=\"list-style-type: none\">\r\n<ol class=\"twocolumn\" type=\"a\">\r\n \t<li>=<\/li>\r\n \t<li>=<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong>Round to the nearest whole number.<\/strong>\r\n<ol class=\"threecolumn\" type=\"a\">\r\n \t<li>5<\/li>\r\n \t<li>19<\/li>\r\n \t<li>1<\/li>\r\n \t<li>7<\/li>\r\n \t<li>0<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>","rendered":"<p style=\"text-align: center\">Start from the left side of each drawing and shade in the fraction shown.<\/p>\n<p>&nbsp;<\/p>\n<table style=\"width: 50%; margin: auto;\">\n<tbody>\n<tr>\n<td style=\"height: 40px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">This shape is the whole thing<\/p>\n<p>&nbsp;<\/p>\n<table style=\"width: 50%; margin: auto;\">\n<tbody>\n<tr>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">Shade [latex]\\tfrac{1}{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<table style=\"width: 50%; margin: auto;\">\n<tbody>\n<tr>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">Shade [latex]\\tfrac{2}{4}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<table style=\"width: 50%; margin: auto;\">\n<tbody>\n<tr>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">Shade [latex]\\tfrac{3}{6}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<table style=\"width: 50%; margin: auto;\">\n<tbody>\n<tr>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">Shade [latex]\\tfrac{4}{8}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<table style=\"width: 50%; margin: auto;\">\n<tbody>\n<tr>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">Shade [latex]\\tfrac{5}{10}[\/latex]<\/p>\n<p>Did you notice that the amount you shaded was the same in each drawing?<\/p>\n<p>The fractions that you were asked to shade are <strong>equivalent fractions<\/strong>. Equivalent fractions <strong>are fractions that are equal<\/strong>.<\/p>\n<p><strong>Now shade the fractions asked for in these drawings, the same way.<\/strong><\/p>\n<table style=\"width: 50%; margin: auto;\">\n<tbody>\n<tr>\n<td style=\"height: 40px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">This shape is the whole thing<\/p>\n<p>&nbsp;<\/p>\n<table style=\"width: 50%; margin: auto;\">\n<tbody>\n<tr>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">Shade [latex]\\tfrac{1}{3}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<table style=\"width: 50%; margin: auto;\">\n<tbody>\n<tr>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">Shade [latex]\\tfrac{2}{6}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<table style=\"width: 50%; margin: auto;\">\n<tbody>\n<tr>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">Shade [latex]\\tfrac{3}{9}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<table style=\"width: 50%; margin: auto;\">\n<tbody>\n<tr>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">Shade [latex]\\tfrac{4}{12}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<table style=\"width: 50%; margin: auto;\">\n<tbody>\n<tr>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<td style=\"height: 40px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: center\">Shade [latex]\\tfrac{5}{15}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p style=\"padding-left: 40px\">These above examples are all equivalent fractions.<\/p>\n<p style=\"text-align: center\">[latex]\\dfrac{1}{3}[\/latex] = [latex]\\dfrac{2}{6}[\/latex] = [latex]\\dfrac{3}{9}[\/latex] = [latex]\\dfrac{4}{12}[\/latex] = [latex]\\dfrac{5}{15}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-385\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-10-28-at-3.33.07-PM.png\" alt=\"Arrow graphic\" width=\"46\" height=\"44\" \/>To work with common fractions, it is often necessary to use an equivalent fraction in place of the fraction that is given. There are several processes to learn which will help you to find equivalent fractions.<\/p>\n<h1>Factors<\/h1>\n<p><a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_132_279\">Factors<\/a> are the numbers which are multiplied together to make a <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_132_258\">product<\/a>. An understanding of factors is needed to express fractions in lowest terms.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-129 size-medium\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-10-28-at-3.41.31-PM-300x113.png\" alt=\"Labelled illustration. 3 x 4 and label reads Factor. Equals 12 and label reads Product.\" width=\"300\" height=\"113\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-10-28-at-3.41.31-PM-300x113.png 300w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-10-28-at-3.41.31-PM-65x25.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-10-28-at-3.41.31-PM-225x85.png 225w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-10-28-at-3.41.31-PM-350x132.png 350w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-10-28-at-3.41.31-PM.png 352w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>We say, &#8220;The factors of 12 are 3 and 4.&#8221;<\/p>\n<p>Does 12 have any other factors?<\/p>\n<p>What other numbers can be multiplied together to equal 12?<\/p>\n<ul>\n<li>[latex]1 \\times 12 = 12[\/latex] or [latex]12 \\times 1 = 12[\/latex]<\/li>\n<li>[latex]2 \\times 6 = 12[\/latex] or [latex]6 \\times 2 = 12[\/latex]<\/li>\n<li>[latex]3 \\times 4 = 12[\/latex] or [latex]4 \\times 3 = 12[\/latex]<\/li>\n<\/ul>\n<p>The factors of 12 are 1, 2, 3, 4, 6, 12.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example B<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the factors of 10.<\/p>\n<ul>\n<li>[latex]1 \\times 10 = 10[\/latex]<\/li>\n<li>[latex]2 \\times 5 = 10[\/latex]<\/li>\n<\/ul>\n<p>The factors of 10 are 1, 2, 5, 10.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example C<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the factors of 9.<\/p>\n<ul>\n<li>[latex]1 \\times 9 = 9[\/latex]<\/li>\n<li>[latex]3 \\times 3 = 9[\/latex]<\/li>\n<\/ul>\n<p>The factors of 9 are 1, 3, 9.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find all the factors<\/p>\n<ol type=\"a\">\n<li>The factors of 16: [latex]1 \\times 16 = 16; 2 \\times 8 = 16; 4 \\times 4 = 16[\/latex]\u00a0 The factors of 16 are 1, 2, 4, 8, 16.<\/li>\n<li>The factors of 4: [latex]1 \\times 4 = 4; 2 \\times 2 = 4[\/latex] The factors of 4 are 1, 2, 4.<\/li>\n<li>The factors of 8:<span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>The factors of 20:<span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>The factors of 5:<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>The factors of 15:<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>The factors of 21:<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>The factors of 6:<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>The factors of 25:<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<\/ol>\n<p><strong>Answers to Exercise 1<\/strong><\/p>\n<ol class=\"threecolumn\" start=\"3\" type=\"a\">\n<li>1, 2, 4, 8<\/li>\n<li>1, 2, 4, 5, 10, 20<\/li>\n<li>1, 5<\/li>\n<li>1, 3, 5, 15<\/li>\n<li>1, 3, 7, 21<\/li>\n<li>1, 2, 3, 6<\/li>\n<li>1, 5, 25<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>Some numbers <strong>only have two factors, 1 and the number itself.<\/strong> These numbers are called <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_132_257\">prime numbers<\/a>. Look at the chart for some prime numbers.<\/p>\n<table class=\"grid aligncenter\" style=\"border-collapse: collapse;width: 0%;height: 306px\">\n<caption>Table for Prime numbers and factors<\/caption>\n<tbody>\n<tr style=\"height: 18px\">\n<th style=\"width: 46.9438%;height: 30px;text-align: center\">Prime Numbers<\/th>\n<th style=\"width: 52.8117%;height: 18px;text-align: center\">Factors<\/th>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\">1<\/td>\n<td style=\"width: 52.8117%;height: 18px\">1,1<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px;text-align: left\">2<\/td>\n<td style=\"width: 52.8117%;height: 18px\">1,2<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\">3<\/td>\n<td style=\"width: 52.8117%;height: 18px\">1,3<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\">5<\/td>\n<td style=\"width: 52.8117%;height: 18px\">1,5<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\">7<\/td>\n<td style=\"width: 52.8117%;height: 18px\">1,7<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\">11<\/td>\n<td style=\"width: 52.8117%;height: 18px\">1,11<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\">13<\/td>\n<td style=\"width: 52.8117%;height: 18px\">1,13<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\">17<\/td>\n<td style=\"width: 52.8117%;height: 18px\">1,17<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\">19<\/td>\n<td style=\"width: 52.8117%;height: 18px\">1,19<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\">23<\/td>\n<td style=\"width: 52.8117%;height: 18px\">1,23<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\">29<\/td>\n<td style=\"width: 52.8117%;height: 18px\">1,29<\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\"><\/td>\n<td style=\"width: 52.8117%;height: 18px\"><\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\"><\/td>\n<td style=\"width: 52.8117%;height: 18px\"><\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\"><\/td>\n<td style=\"width: 52.8117%;height: 18px\"><\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\"><\/td>\n<td style=\"width: 52.8117%;height: 18px\"><\/td>\n<\/tr>\n<tr style=\"height: 18px\">\n<td style=\"width: 46.9438%;height: 18px\"><\/td>\n<td style=\"width: 52.8117%;height: 18px\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Add other prime numbers to the chart as you find them.<\/strong><\/p>\n<p><strong>Reminder<\/strong>: Prime numbers only have two, prime factors.<\/p>\n<h1>Finding Common Factors<\/h1>\n<p>A <strong>common factor<\/strong> is a number used to reduce the numerator and denominator.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example D<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What are the common factors for [latex]\\tfrac{4}{6}[\/latex]?<\/p>\n<ul>\n<li>Find the factors of 4 and 6.\n<ul>\n<li>The factors of 4 are 1, 2, 4.<\/li>\n<li>The factors of 6 are 1, 2, 3, 6.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>What factors do 4 and 6 have in common?<\/p>\n<ul>\n<li>4: 1, 2, 4<\/li>\n<li>6: 1, 2, 3, 6<\/li>\n<\/ul>\n<p>The common factors of 4 and 6 are<strong> 1 and 2<\/strong><\/p>\n<p>For the above equation, the factors are 1 and 2; however, 1 is not used as a <strong>common factor<\/strong>. This is because 1 is a factor of all <strong>whole numbers<\/strong>.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example E<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What are the common factors for [latex]\\tfrac{6}{15}[\/latex]?<\/p>\n<ul>\n<li>Find the factors of 6 and 15.\n<ul>\n<li>The factors of 6 are 1, 2, 3,6<\/li>\n<li>The factors of 15 are 1, 3,5,15.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>What factors do 6 and 15 have in common?<\/p>\n<ul>\n<li>6: 1, 2, 3,6<\/li>\n<li>15: 1, 3,5,15<\/li>\n<\/ul>\n<p>The common factor of 6 and 15 is 3<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example F<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the common factors of [latex]\\tfrac{16}{24}[\/latex]?<\/p>\n<ul>\n<li>Find the factors of 16 are 1, 2, 4, 8, 16<\/li>\n<li>The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24<\/li>\n<li>The common factors of 16 and 24 are: 2, 4, 8<\/li>\n<li>8 is called the <strong>greatest common factor (GCF)<\/strong> of 16 and 24 because it is the largest of all the common factors<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the <strong>Common Factors for each set of numbers.<\/strong> Then identify the <strong>Greatest Common Factor (GCF)<\/strong>.<\/p>\n<ol class=\"twocolumn\" type=\"a\">\n<li>10, 15\n<ul>\n<li>Factors of 10: 1, 2, 5, 10<\/li>\n<li>Factors of 15: 1, 3, 5, 15<\/li>\n<li>Common factors: 5<\/li>\n<li>Greatest common factor: 5<\/li>\n<\/ul>\n<\/li>\n<li>4, 16\n<ul>\n<li>Factors of 4: 1, 2, 4<\/li>\n<li>Factors of 16: 1, 2, 4, 8, 16<\/li>\n<li>Common factors: 2, 4<\/li>\n<li>Greatest common factor: 4<\/li>\n<\/ul>\n<\/li>\n<li>9,12\n<ul>\n<li>Factors of 9:<\/li>\n<li>Factors of 12:<\/li>\n<li>Common factors:<\/li>\n<li>Greatest common factor:<\/li>\n<\/ul>\n<\/li>\n<li>20, 30\n<ul>\n<li>Factors of 20:<\/li>\n<li>Factors of 30:<\/li>\n<li>Common factors:<\/li>\n<li>Greatest common factor:<\/li>\n<\/ul>\n<\/li>\n<li>18, 12\n<ul>\n<li>Factors of 18:<\/li>\n<li>Factors of 12:<\/li>\n<li>Common factors:<\/li>\n<li>Greatest common factor:<\/li>\n<\/ul>\n<\/li>\n<li>24, 32\n<ul>\n<li>Factors of 24:<\/li>\n<li>Factors of 32:<\/li>\n<li>Common factors:<\/li>\n<li>Greatest common factor:<\/li>\n<\/ul>\n<\/li>\n<li>8, 12\n<ul>\n<li>Factors of 8:<\/li>\n<li>Factors of 12:<\/li>\n<li>Common factors:<\/li>\n<li>Greatest common factor:<\/li>\n<\/ul>\n<\/li>\n<li>6, 9\n<ul>\n<li>Factors of 6:<\/li>\n<li>Factors of 9:<\/li>\n<li>Common factors:<\/li>\n<li>Greatest common factor:<\/li>\n<\/ul>\n<\/li>\n<li>9,15\n<ul>\n<li>Factors of 9:<\/li>\n<li>Factors of 15:<\/li>\n<li>Common factors:<\/li>\n<li>Greatest common factor:<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 2<\/strong><\/p>\n<ol class=\"twocolumn\" type=\"a\">\n<li>10, 15\n<ul>\n<li>Factors of 10: 1, 2, 5, 10<\/li>\n<li>Factors of 15: 1, 3, 5, 15<\/li>\n<li>Common factors: 5<\/li>\n<li>Greatest common factor: 5<\/li>\n<\/ul>\n<\/li>\n<li>4, 16\n<ul>\n<li>Factors of 4: 1, 2, 4<\/li>\n<li>Factors of 16: 1, 2, 4, 8, 16<\/li>\n<li>Common factors: 2, 4<\/li>\n<li>Greatest common factor: 4<\/li>\n<\/ul>\n<\/li>\n<li>9,12\n<ul>\n<li>Factors of 9: 1, 3, 9<\/li>\n<li>Factors of 12: 1, 2, 3, 4, 6, 12<\/li>\n<li>Common factors: 3<\/li>\n<li>Greatest common factor: 3<\/li>\n<\/ul>\n<\/li>\n<li>20, 30\n<ul>\n<li>Factors of 20: 1, 2, 4, 5, 10, 20<\/li>\n<li>Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30<\/li>\n<li>Common factors: 2, 5, 10<\/li>\n<li>Greatest common factor: 10<\/li>\n<\/ul>\n<\/li>\n<li>18, 12\n<ul>\n<li>Factors of 18: 1, 2, 3, 6, 9, 18<\/li>\n<li>Factors of 12:, 1, 2, 3, 4, 6, 12<\/li>\n<li>Common factors: 2, 3, 6<\/li>\n<li>Greatest common factor: 6<\/li>\n<\/ul>\n<\/li>\n<li>24, 32\n<ul>\n<li>Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24<\/li>\n<li>Factors of 32:, 1, 2, 4, 8, 16, 32<\/li>\n<li>Common factors: 2, 4, 8<\/li>\n<li>Greatest common factor: 8<\/li>\n<\/ul>\n<\/li>\n<li>8, 12\n<ul>\n<li>Factors of 8: 1, 2, 4, 8<\/li>\n<li>Factors of 12: 1, 2, 3, 4, 6, 12<\/li>\n<li>Common factors: 2, 4<\/li>\n<li>Greatest common factor: 4<\/li>\n<\/ul>\n<\/li>\n<li>6, 9\n<ul>\n<li>Factors of 6: 1, 2, 3, 6<\/li>\n<li>Factors of 9: 1, 3, 9<\/li>\n<li>Common factors: 3<\/li>\n<li>Greatest common factor: 3<\/li>\n<\/ul>\n<\/li>\n<li>9, 15\n<ul>\n<li>Factors of 9: 1, 3, 9<\/li>\n<li>Factors of 15: 1, 3, 5, 15<\/li>\n<li>Common factors: 3<\/li>\n<li>Greatest common factor: 3<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<ul>\n<li style=\"list-style-type: none\">\n<ul>\n<li style=\"list-style-type: none\">\n<ul>\n<li style=\"list-style-type: none\"><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h1>Expressing Fractions in Lower Terms<\/h1>\n<p>Express means <strong>to say it or write it.<\/strong><\/p>\n<p><strong>Lower terms<\/strong> means to express equivalent fractions with smaller (lower) denominators.<\/p>\n<p>Look back to page 67. The equivalent fraction in lowest terms is [latex]\\tfrac{1}{2}[\/latex] .<\/p>\n<p>The words <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_132_267\">simplify<\/a> and <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_132_264\">reduce<\/a> are another way to say \u201c<strong>express fractions in lower (or lowest) terms<\/strong>.\u201d<\/p>\n<p><strong>To express a fraction in lowest terms, do this:<\/strong><\/p>\n<p><strong>Step 1:\u00a0\u00a0<\/strong> Find the greatest common factor (GCF) of the numerator and denominator.<\/p>\n<ul>\n<li>[latex]\\dfrac{4}{12}[\/latex]\u00a0 \u00a0The factors of 4 are 1, 2, 4<\/li>\n<li>The factors of 12 are 1, 2, 3, 4, 6, 12<\/li>\n<\/ul>\n<p><strong>The GCF is 4.<\/strong><\/p>\n<p><strong>Step 2:<\/strong>\u00a0 \u00a0 \u00a0 \u00a0Divide the numerator and the denominator by the greatest common factor.<\/p>\n<ul>\n<li>[latex]\\dfrac{4}{12}[\/latex] [latex]\\dfrac{\\div 4}{\\div 4}[\/latex]\u00a0\u00a0 =\u00a0 [latex]\\dfrac{1}{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{12}[\/latex] = [latex]\\dfrac{1}{3}[\/latex]<\/li>\n<\/ul>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example G<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>[latex]\\dfrac{6}{9}[\/latex]<\/p>\n<p>The factors of 6 are 1, 2, 3 , 6.<br \/>\nThe factors of 9 are 1, 3 , 9.<\/p>\n<p><strong>The GCF is 3.<\/strong><\/p>\n<ul>\n<li>[latex]\\dfrac{6}{9}\\dfrac{\\div 3}{\\div 3} = \\dfrac{2}{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{6}{9} = \\dfrac{2}{3}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example H<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div class=\"textbox__content\">\n<p>[latex]\\dfrac{15}{24}[\/latex]<\/p>\n<p>The factors of 15 are 1, 3,5,15<br \/>\nThe factors of 24 are 1, 3, 4, 6, 8, 24.<\/p>\n<p><strong>The GCF is 3.<\/strong><\/p>\n<ul>\n<li>[latex]\\dfrac{15}{24} \\dfrac{\\div 3}{\\div 3} =\\dfrac{5}{8}[\/latex]<\/li>\n<li>[latex]\\dfrac{15}{24} = \\dfrac{5}{8}[\/latex]<\/li>\n<\/ul>\n<p><strong>There are several reasons lower terms are used:<\/strong><\/p>\n<ul>\n<li>The math is usually easier with lower numbers.<\/li>\n<li>Is it easier to think of [latex]\\tfrac{1}{2}[\/latex] an apple or [latex]\\tfrac{15}{30}[\/latex] of an apple? ([latex]\\tfrac{1}{2}[\/latex] = [latex]\\tfrac{15}{30}[\/latex] )<\/li>\n<li>Do you want to think about [latex]\\tfrac{155}{620}[\/latex] of your pay cheque or [latex]\\tfrac{1}{4}[\/latex] of your pay cheque? ( [latex]\\tfrac{1}{4}[\/latex] = [latex]\\tfrac{155}{620}[\/latex] )<\/li>\n<li>Always express fractions in lowest terms!<\/li>\n<\/ul>\n<p><strong>Dividing<\/strong> both the numerator and denominator <strong>by the GCF<\/strong> will give an equivalent fraction in lower terms.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Express each fraction in <strong>lowest terms.<\/strong> (The directions could also say, &#8220;Simplify each fraction,&#8221; or &#8220;Reduce these fractions&#8221;).<\/p>\n<ol type=\"a\">\n<li>[latex]\\dfrac{2}{4}\\dfrac{\\div 2}{\\div 2}= \\dfrac{1}{2}[\/latex],\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{3}{9}\\dfrac{ \\div \\ \\ \\ }{ \\div \\ \\ \\ }[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>[latex]\\dfrac{2}{12}\\dfrac{\\div2}{\\div2}=\\dfrac{1}{6}[\/latex],\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{3}{15}\\dfrac{ \\div \\ \\ \\ }{ \\div \\ \\ \\ }[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>[latex]\\dfrac{5}{10}\\dfrac{ \\div \\ \\ }{ \\div \\ \\ \\ }[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{4}{24}\\dfrac{ \\div \\ \\ \\ }{ \\div \\ \\ \\ }[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>[latex]\\dfrac{10}{25}\\dfrac{ \\div \\ \\ }{ \\div\\ \\ \\ }[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{9}{12}\\dfrac{ \\div \\ \\ \\ }{ \\div \\ \\ \\ }[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\n<p style=\"margin-top: 1em\"><strong>Make sure that you write in the GCF you are dividing with. Do not skip this step until you are absolutely sure you can do it correctly in your head each time.<\/strong><\/p>\n<p>(Good mathematicians know when to skip steps and when not to&#8230; sometimes easy steps are never skipped by good mathematicians).<\/li>\n<li>[latex]\\dfrac{3}{30}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{6}{10}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>[latex]\\dfrac{9}{24}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{18}{27}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/li>\n<li>[latex]\\dfrac{4}{16}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{3}{12}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/li>\n<li>[latex]\\dfrac{15}{24}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{15}{25}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/li>\n<li>[latex]\\dfrac{2}{32}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0[latex]\\dfrac{6}{20}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox__content\"><strong>Answers to Exercise 3<br \/>\n<\/strong><\/p>\n<ol class=\"twocolumn\" type=\"a\">\n<li>[latex]\\dfrac{1}{2}[\/latex], [latex]\\dfrac{1}{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{6}[\/latex], [latex]\\dfrac{1}{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{2}[\/latex], [latex]\\dfrac{1}{6}[\/latex]<\/li>\n<li>[latex]\\dfrac{2}{5}[\/latex], [latex]\\dfrac{3}{4}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{10}[\/latex], [latex]\\dfrac{3}{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{8}[\/latex], [latex]\\dfrac{2}{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{4}[\/latex], [latex]\\dfrac{1}{4}[\/latex]<\/li>\n<li>[latex]\\dfrac{5}{8}[\/latex], [latex]\\dfrac{3}{5}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{16}[\/latex], [latex]\\dfrac{3}{10}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>Expressing Fractions in Higher Terms<\/h1>\n<p>Higher Terms are needed when you add and subtract fractions with different denominators.<\/p>\n<p>You have learned that <strong>dividing<\/strong> both the numerator and denominator of a fraction by a common factor <strong>gives an equivalent fraction in lower terms<\/strong>. You know that dividing and multiplying are opposite operations, so this next rule will match the one you just learned for reducing:<\/p>\n<p><strong>Multiplying both<\/strong> the <strong>numerator and denominator<\/strong> of a fraction by the same number (a <strong>common factor)<\/strong> will give an <strong>equivalent fraction<\/strong> in <strong>higher terms<\/strong>.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example I<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>[latex]\\dfrac{3}{5}\\left(\\dfrac{\\times 2}{\\times 2}\\right)=\\dfrac{6}{10}[\/latex]<\/p>\n<p>[latex]\\dfrac{3}{5}=\\dfrac{6}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example J<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>[latex]\\dfrac{1}{2}\\left(\\dfrac{\\times 8}{\\times 8}\\right)=\\dfrac{8}{16}[\/latex]<\/p>\n<p>[latex]\\dfrac{1}{2}= \\dfrac{8}{16}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example K<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>[latex]\\dfrac{2}{3}\\left(\\dfrac{\\times 3}{\\times 3}\\right)=\\dfrac{6}{9}[\/latex]<\/p>\n<p>[latex]\\dfrac{2}{3} = \\dfrac{6}{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<h1>Are the Fractions Equivalent?<\/h1>\n<p>If the denominators are the same, you can easily judge if the fractions are equivalent by comparing the numerators.<\/p>\n<p>Compare [latex]\\tfrac{4}{5}[\/latex] and [latex]\\tfrac{3}{5}[\/latex]: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 [latex]\\tfrac{4}{5}[\/latex] \u2260 [latex]\\tfrac{3}{5}[\/latex] (\u2260 means \u2018not equal\u2019)<\/p>\n<p>Compare [latex]\\tfrac{12}{20}[\/latex] and [latex]\\tfrac{12}{20}[\/latex]: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0[latex]\\tfrac{12}{20} = \\tfrac{12}{20}[\/latex]<\/p>\n<p>If the denominators are different, you <strong>might be able to rewrite one or more of the fractions so they have the same denominator.<\/strong><\/p>\n<p>Compare [latex]\\tfrac{4}{5}[\/latex] and [latex]\\tfrac{6}{10}[\/latex]: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0[latex]\\tfrac{6}{10}\\tfrac{\\div 2}{\\div 2}[\/latex]=[latex]\\tfrac{3}{5}[\/latex] \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0So:[latex]\\tfrac{4}{5}\\neq \\tfrac{3}{5}[\/latex]<\/p>\n<p>Compare [latex]\\tfrac{12}{16}[\/latex] and [latex]\\tfrac{5}{8}[\/latex]: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 [latex]\\tfrac{5}{8} \\tfrac{\\times 2}{\\times 2}[\/latex] =[latex]\\tfrac{10}{16}[\/latex] \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0So:[latex]\\tfrac{12}{16}\\neq \\tfrac{10}{16}[\/latex]<\/p>\n<p>or you could do this: \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 [latex]\\tfrac{12}{16}[\/latex][latex]\\tfrac{\\div 2}{\\div 2}= \\tfrac{6}{8}[\/latex] \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0So:[latex]\\tfrac{6}{8}\\neq\\tfrac{5}{8}[\/latex]<\/p>\n<p>A quick method is to <strong><a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_132_284\">cross multiply<\/a><\/strong>:<\/p>\n<ol>\n<li>multiply the numerator of one fraction by the denominator of the second fraction<\/li>\n<li>multiply the numerator of the <strong>other<\/strong> fraction by the denominator of the <strong>first<\/strong> fraction These are called the <em>cross-products<\/em>.<br \/>\nIf the cross products are the same, then the fraction is equivalent.<\/li>\n<\/ol>\n<p>Look at the examples:<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example L<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Compare [latex]\\dfrac{4}{7}[\/latex] and [latex]\\dfrac{5}{9}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox__content\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-130 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-11-03-at-10.40.37-AM.png\" alt=\"A picture of two fractions and arrows between them. First fraction is numerator 4 and denominator 7. Second fraction is numberator 5 and denominator 9. Arrows direct attention from the first fraction numerator 4 to second fraction denominator 9. And from first fraction denominator 7 to second fraction numerator 5..\" width=\"212\" height=\"78\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-11-03-at-10.40.37-AM.png 212w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-11-03-at-10.40.37-AM-65x24.png 65w\" sizes=\"auto, (max-width: 212px) 100vw, 212px\" \/><\/p>\n<ul>\n<li>Multiply the numerator 4 by the denominator 9\n<ul>\n<li>[latex]4 \\times 9 = 36[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Multiply the denominator 7 by the numerator 5\n<ul>\n<li>[latex]7 \\times 5 = 35[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>The products 36 and 35 <strong>are not <\/strong>the same.<\/li>\n<\/ul>\n<p>Therefore [latex]\\dfrac{4}{7}\\neq \\dfrac{5}{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example M<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Compare [latex]\\dfrac{2}{3}[\/latex] and [latex]\\dfrac{12}{18}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-131 size-full\" src=\"https:\/\/pressbooks.bccampus.ca\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-11-03-at-2.13.14-PM.png\" alt=\"A picture of two fractions and arrows between them. First fraction is numerator 2 and denominator 3. Second fraction is numberator 12 and denominator 18. Arrows direct attention from the first fraction numerator 2 to second fraction denominator 18. And from first fraction denominator 3 to second fraction numerator 12.\" width=\"235\" height=\"96\" srcset=\"https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-11-03-at-2.13.14-PM.png 235w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-11-03-at-2.13.14-PM-65x27.png 65w, https:\/\/pressbooks.bccampus.ca\/math025\/wp-content\/uploads\/sites\/2383\/2025\/01\/Screen-Shot-2022-11-03-at-2.13.14-PM-225x92.png 225w\" sizes=\"auto, (max-width: 235px) 100vw, 235px\" \/><\/p>\n<\/div>\n<div class=\"textbox__content\">\n<ul>\n<li>[latex]2 \\times 18 = 36[\/latex]<\/li>\n<li>[latex]3 \\times 12 = 36[\/latex]<\/li>\n<li>The products 36 and 36 <strong>are<\/strong> the same<\/li>\n<\/ul>\n<p>Therefore [latex]\\dfrac{2}{3} = \\dfrac{12}{18}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example N<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Compare [latex]\\dfrac{24}{40}[\/latex] and [latex]\\dfrac{4}{10}[\/latex]<\/p>\n<ul>\n<li>[latex]24 \\times 10 = 240[\/latex]<\/li>\n<li>[latex]40 \\times 4 = 160[\/latex]<\/li>\n<li>The products 240 and 160 <strong>are not<\/strong> the same.<\/li>\n<\/ul>\n<p>Therefore [latex]\\dfrac{24}{40} \\neq \\dfrac{4}{10}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>State if each pair is <strong>equivalent (=)<\/strong> or <strong>not equivalent<\/strong> (\u2260). Use whichever method you wish to find the answer.<\/p>\n<ol class=\"threecolumn\" type=\"a\">\n<li>[latex]\\dfrac{5}{6}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0 <\/span>[latex]\\dfrac{30}{60}[\/latex]<\/li>\n<li>[latex]\\dfrac{12}{24}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span>\u00a0[latex]\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]\\dfrac{6}{7}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0[latex]\\dfrac{7}{8}[\/latex]<\/li>\n<li>[latex]\\dfrac{2}{3}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0[latex]\\dfrac{12}{18}[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{3}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0[latex]\\dfrac{24}{72}[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{4}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0[latex]\\dfrac{15}{20}[\/latex]<\/li>\n<li>[latex]\\dfrac{12}{14}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>\u00a0[latex]\\dfrac{6}{7}[\/latex]<\/li>\n<li>[latex]\\dfrac{4}{10}[\/latex] =\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> [latex]\\dfrac{20}{50}[\/latex]<\/li>\n<li>[latex]\\dfrac{5}{10}[\/latex] = <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> [latex]\\dfrac{7}{14}[\/latex]<\/li>\n<\/ol>\n<p><strong>Answers to Exercise 4<\/strong><\/p>\n<ol class=\"threecolumn\" type=\"a\">\n<li>\u2260<\/li>\n<li>=<\/li>\n<li>\u2260<\/li>\n<li>=<\/li>\n<li>=<\/li>\n<li>=<\/li>\n<li>=<\/li>\n<li>=<\/li>\n<li>=<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h1>Rounding Common Fractions to Whole Numbers<\/h1>\n<p>When rounding to a whole number, if a fraction is less than [latex]\\dfrac{1}{2}[\/latex] \u00a0do not change the whole number:<\/p>\n<p>Examples:<\/p>\n<ul class=\"twocolumn\">\n<li>[latex]2\\dfrac{3}{7}\u2248 2[\/latex]<\/li>\n<li>[latex]23\\dfrac{1}{3} \u2248 23[\/latex]<\/li>\n<li>[latex]\\dfrac{1}{4} \u2248 0[\/latex]<\/li>\n<li>[latex]5\\dfrac{3}{8}\u2248 5[\/latex]<\/li>\n<\/ul>\n<p>If the fraction is [latex]\\dfrac{1}{2}[\/latex] or more, consider the fraction as <strong>another one<\/strong> which must be added to the whole number:<\/p>\n<p>Examples:<\/p>\n<ul class=\"twocolumn\">\n<li>[latex]2\\dfrac{1}{2}[\/latex] \u2248 3<\/li>\n<li>[latex]15\\dfrac{4}{5}[\/latex] \u2248 16<\/li>\n<li>[latex]6\\dfrac{7}{8}[\/latex] \u2248 7<\/li>\n<li>[latex]\\dfrac{3}{4}[\/latex] \u2248 1<\/li>\n<\/ul>\n<p>If you are not sure if a fraction is more or less than [latex]\\tfrac{1}{2}[\/latex] , you can compare it to [latex]\\tfrac{1}{2}[\/latex], by making equivalent fractions with a common denominator.<\/p>\n<p style=\"text-align: center\">Reminder: greater &gt; smaller<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example O<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Round [latex]\\tfrac{2}{3}[\/latex] to a whole number.<\/p>\n<p>Is [latex]\\dfrac{2}{3}[\/latex] &gt;\u00a0 [latex]\\dfrac{1}{2}\\text{?}\\longrightarrow\\dfrac{2}{3}= \\dfrac{4}{6}\\text{ and }\\dfrac{1}{2}= \\dfrac{3}{6}[\/latex]<\/p>\n<p>YES! [latex]\\dfrac{2}{3}[\/latex] &gt; [latex]\\dfrac{1}{2}\\text{so}\\dfrac{2}{3}\u2248 1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example P<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Round [latex]2\\tfrac{4}{7}[\/latex] to a whole number.<\/p>\n<p>Is\u00a0 [latex]\\dfrac{4}{7}[\/latex] &gt; [latex]\\dfrac{1}{2}\\text{?}\\longrightarrow\\dfrac{4}{7}= \\dfrac{8}{14}\\text{ and }\\dfrac{1}{2} = \\dfrac{7}{14}[\/latex]<\/p>\n<p>YES! [latex]\\tfrac{4}{7}[\/latex] &gt; [latex]\\tfrac{1}{2}[\/latex] so [latex]2\\tfrac{4}{7}[\/latex] \u2248 [latex]3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Exercise 5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Round to the nearest whole number<\/p>\n<ol class=\"threecolumn\" type=\"a\">\n<li>[latex]\\dfrac{4}{5}\u2248 1[\/latex]<\/li>\n<li>[latex]2\\dfrac{1}{3}\u2248 2[\/latex]<\/li>\n<li>[latex]18\\dfrac{1}{2} \u2248[\/latex]\u00a0\u00a0<span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>[latex]3\\dfrac{7}{8} \u2248[\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\n<li>[latex]9\\dfrac{9}{10} \u2248[\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\n<li>[latex]\\dfrac{1}{8} \u2248[\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\n<li>[latex]4\\dfrac{1}{6} \u2248[\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\n<li>[latex]12\\dfrac{7}{9} \u2248[\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\n<li>[latex]6\\dfrac{3}{5} \u2248[\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\n<li>[latex]20\\dfrac{3}{7} \u2248[\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\n<li>[latex]\\dfrac{13}{15} \u2248[\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 <\/span><\/li>\n<li>[latex]99\\dfrac{2}{3} \u2248[\/latex]\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/li>\n<\/ol>\n<p><strong>Answers to Exercise Five<\/strong><\/p>\n<ol class=\"threecolumn\" start=\"3\" type=\"a\">\n<li>19<\/li>\n<li>4<\/li>\n<li>10<\/li>\n<li>0<\/li>\n<li>4<\/li>\n<li>13<\/li>\n<li>7<\/li>\n<li>20<\/li>\n<li>1<\/li>\n<li>100<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h1>Topic A:\u00a0 Self-Test<\/h1>\n<p><strong>Mark\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \/25\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Aim\u00a0\u00a0\u00a0\u00a0\u00a0 20\/25<\/strong><\/p>\n<ol type=\"A\">\n<li>Define\u00a0 (3 marks)\n<ol>\n<li style=\"list-style-type: none\">\n<ol type=\"a\">\n<li>equivalent <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>prime number <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<li>greatest common factor (GCF) <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<li>Complete the chart (5 marks)<br \/>\n<table class=\"grid\" style=\"border-collapse: collapse;width: 100%;height: 198px\">\n<thead>\n<tr style=\"height: 18px\">\n<td style=\"width: 16.6666%\"><\/td>\n<th style=\"width: 16.6666%;height: 18px\" scope=\"col\">Factors<\/th>\n<th style=\"width: 33.3333%;height: 18px\" scope=\"col\">Common Factors<\/th>\n<th style=\"width: 33.3333%;height: 18px\" scope=\"col\">Greatest Common Factor<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 36px\">\n<th style=\"width: 16.6666%\" scope=\"row\">12<br \/>\n18<\/th>\n<td style=\"width: 16.6666%;height: 36px\"><\/td>\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\n<\/tr>\n<tr style=\"height: 36px\">\n<th style=\"width: 16.6666%\" scope=\"row\">15<br \/>\n30<\/th>\n<td style=\"width: 16.6666%;height: 36px\"><\/td>\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\n<\/tr>\n<tr style=\"height: 36px\">\n<th style=\"width: 16.6666%\" scope=\"row\">7<br \/>\n28<\/th>\n<td style=\"width: 16.6666%;height: 36px\"><\/td>\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\n<\/tr>\n<tr style=\"height: 36px\">\n<th style=\"width: 16.6666%\" scope=\"row\">6<br \/>\n16<\/th>\n<td style=\"width: 16.6666%;height: 36px\"><\/td>\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\n<\/tr>\n<tr style=\"height: 36px\">\n<th style=\"width: 16.6666%\" scope=\"row\">18<br \/>\n27<\/th>\n<td style=\"width: 16.6666%;height: 36px\"><\/td>\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\n<td style=\"width: 33.3333%;height: 36px\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Express in lowest terms. (6 marks)\n<ol class=\"twocolumn\" type=\"a\">\n<li>[latex]\\dfrac{10}{15}[\/latex]= <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>.<\/li>\n<li>[latex]\\dfrac{14}{16}[\/latex]= <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>.<\/li>\n<li>[latex]\\dfrac{8}{12}[\/latex]= <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>.<\/li>\n<\/ol>\n<\/li>\n<li>State if each pair of fractions is equivalent (=) or not equivalent ( ). (6 marks)\n<ol class=\"twocolumn\" type=\"a\">\n<li>[latex]\\dfrac{5}{9}[\/latex] <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> [latex]\\dfrac{15}{27}[\/latex]<\/li>\n<li>[latex]\\dfrac{3}{7}[\/latex] <span style=\"text-decoration: underline\" aria-label=\"blank;\">\u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span> [latex]\\dfrac{15}{35}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li>Round to the nearest whole number (5 marks)\n<ol class=\"twocolumn\" type=\"a\">\n<li>4[latex]\\dfrac{5}{8}[\/latex] \u2248 \u00a0\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 <\/span><\/li>\n<li>19[latex]\\dfrac{4}{10}[\/latex] \u2248 \u00a0\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 <\/span><\/li>\n<li>[latex]\\dfrac{1}{2}[\/latex] \u2248 \u00a0\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 <\/span><\/li>\n<li>6[latex]\\dfrac{3}{4}[\/latex] \u2248 \u00a0\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 <\/span><\/li>\n<li>[latex]\\dfrac{1}{3}[\/latex] \u2248 \u00a0\u00a0 <span style=\"text-decoration: underline\" aria-label=\"blank;\"> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 <\/span><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<h1>Answers to Topic A Self-Test<\/h1>\n<ol type=\"A\">\n<li>Check your definitions in the glossary<\/li>\n<li>Complete the chart<br \/>\n<table class=\"grid\" style=\"border-collapse: collapse;width: 100%;height: 198px\">\n<thead>\n<tr style=\"height: 18px\">\n<td style=\"width: 16.6666%\"><\/td>\n<th style=\"width: 30.1259%;height: 18px\" scope=\"col\">Factors<\/th>\n<th style=\"width: 19.874%;height: 18px\" scope=\"col\">Common Factors<\/th>\n<th style=\"width: 33.3333%;height: 18px\" scope=\"col\">Greatest Common Factor<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 36px\">\n<th style=\"width: 16.6666%\" scope=\"row\">12<br \/>\n18<\/th>\n<td style=\"width: 30.1259%;height: 36px\">\u2026 of 12 are 1, 2, 3, 4, 6, 12<br \/>\n\u2026 of 18 are 1, 2, 3, 6, 9, 18<\/td>\n<td style=\"width: 19.874%;height: 36px;text-align: center\">2, 3, 6<\/td>\n<td style=\"width: 33.3333%;height: 36px\">6<\/td>\n<\/tr>\n<tr style=\"height: 36px\">\n<th style=\"width: 16.6666%\" scope=\"row\">15<br \/>\n30<\/th>\n<td style=\"width: 30.1259%;height: 36px\">\u2026 of 15 are 1, 3, 5, 15<\/p>\n<p>\u2026 of 30 are 1, 2, 3, 5, 6, 10, 15, 30<\/td>\n<td style=\"width: 19.874%;height: 36px;text-align: center\">3, 5, 15<\/td>\n<td style=\"width: 33.3333%;height: 36px\">15<\/td>\n<\/tr>\n<tr style=\"height: 36px\">\n<th style=\"width: 16.6666%\" scope=\"row\">7<br \/>\n28<\/th>\n<td style=\"width: 30.1259%;height: 36px\">\u2026 of 7 are 1, 7<br \/>\n\u2026 of 28 are 1, 2, 4, 7, 14, 28<\/td>\n<td style=\"width: 19.874%;height: 36px;text-align: center\">7<\/td>\n<td style=\"width: 33.3333%;height: 36px\">7<\/td>\n<\/tr>\n<tr style=\"height: 36px\">\n<th style=\"width: 16.6666%\" scope=\"row\">6<br \/>\n16<\/th>\n<td style=\"width: 30.1259%;height: 36px\">\u2026 of 6 are 1, 2, 3, 6<br \/>\n\u2026 of 16 are 1, 2, 4, 8, 16<\/td>\n<td style=\"width: 19.874%;height: 36px;text-align: center\">2<\/td>\n<td style=\"width: 33.3333%;height: 36px\">2<\/td>\n<\/tr>\n<tr style=\"height: 36px\">\n<th style=\"width: 16.6666%\" scope=\"row\">18<br \/>\n27<\/th>\n<td style=\"width: 30.1259%;height: 36px\">\u2026 of 18 are 1, 2, 3, 6, 9, 18<br \/>\n\u2026 of 27 are 1, 3, 9, 27<\/td>\n<td style=\"width: 19.874%;height: 36px;text-align: center\">3, 9<\/td>\n<td style=\"width: 33.3333%;height: 36px\">9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li><strong>Express in lowest terms.<\/strong>\n<ol class=\"threecolumn\" type=\"a\">\n<li>[latex]\\dfrac{2}{3}[\/latex]<\/li>\n<li>[latex]\\dfrac{7}{8}[\/latex]<\/li>\n<li>[latex]\\dfrac{2}{3}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<li><strong>State if each pair of fractions is equivalent (=) or not equivalent ( ).<\/strong>\n<ol class=\"twocolumn\" type=\"a\">\n<li style=\"list-style-type: none\">\n<ol class=\"twocolumn\" type=\"a\">\n<li>=<\/li>\n<li>=<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<li><strong>Round to the nearest whole number.<\/strong>\n<ol class=\"threecolumn\" type=\"a\">\n<li>5<\/li>\n<li>19<\/li>\n<li>1<\/li>\n<li>7<\/li>\n<li>0<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_132_279\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_132_279\"><div tabindex=\"-1\"><p>The numbers or quantities that are multiplied together to form a given product. 5 \u00d7 2 = 10, so 5 and 2 are factors of 10.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_132_258\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_132_258\"><div tabindex=\"-1\"><p>The result of a multiplying question, the answer.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_132_2135\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_132_2135\"><div tabindex=\"-1\"><\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_132_257\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_132_257\"><div tabindex=\"-1\"><p>A number that can only be divided evenly by itself and 1.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_132_267\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_132_267\"><div tabindex=\"-1\"><p>See <em>reduce<\/em>.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_132_264\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_132_264\"><div tabindex=\"-1\"><p>Write a common fraction in lowest terms. Divide both terms by same factor.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><template id=\"term_132_284\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_132_284\"><div tabindex=\"-1\"><p>In a proportion, multiply the numerator of the first fraction times the denominator of the second fraction. Then multiply the denominator of the first fraction times the numerator of the second fraction. In a true proportion, the products of the cross multiplication are equal.<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":999,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-132","chapter","type-chapter","status-publish","hentry"],"part":116,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/chapters\/132","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/wp\/v2\/users\/999"}],"version-history":[{"count":16,"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/chapters\/132\/revisions"}],"predecessor-version":[{"id":329,"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/chapters\/132\/revisions\/329"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/parts\/116"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/chapters\/132\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/wp\/v2\/media?parent=132"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/pressbooks\/v2\/chapter-type?post=132"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/wp\/v2\/contributor?post=132"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/math025\/wp-json\/wp\/v2\/license?post=132"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}