{"id":1097,"date":"2023-03-30T16:05:39","date_gmt":"2023-03-30T16:05:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1097"},"modified":"2024-10-18T20:51:00","modified_gmt":"2024-10-18T20:51:00","slug":"decimals-learn-it-4","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/decimals-learn-it-4\/","title":{"raw":"Decimals: Learn It 4","rendered":"Decimals: Learn It 4"},"content":{"raw":"<h2>Determine Whether a Decimal is a Solution of an Equation<\/h2>\r\n<p>Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family\u2019s budget, or planning for the future of your business, you\u2019ll be solving equations with decimals.<\/p>\r\n<p>Now that we\u2019ve worked with decimals, we are ready to find solutions to equations involving decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal. We\u2019ll list these steps here again for easy reference.<\/p>\r\n<section class=\"textbox questionHelp\">\r\n<p><strong>How To: Determine Whether a Number Is a Solution to an Equation<\/strong><\/p>\r\n<ol id=\"eip-id1168469438337\" class=\"stepwise\">\r\n\t<li>Substitute the number for the variable in the equation.<\/li>\r\n\t<li>Simplify the expressions on both sides of the equation.<\/li>\r\n\t<li>Determine whether the resulting equation is true.<br \/>\r\n<ul id=\"fs-id2482544\">\r\n\t<li>If so, the number is a solution.<\/li>\r\n\t<li>If not, the number is not a solution.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3696[\/ohm2_question]<\/section>\r\n<h2>Solve Equations with Decimals<\/h2>\r\n<p>In previous modules, we solved equations using the properties of equality. We will use these same properties to solve equations with decimals. Remember, when you add, subtract, multiply, or divide the same quantity from both sides of an equation, you still have equality.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>properties of equality<\/h3>\r\n<center>\r\n<table id=\"fs-id1752382\" class=\"unnumbered\" summary=\"A table is shown with 2 rows and 2 columns. The first cell says, \">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>\r\n<p><strong>Subtraction Property of Equality<\/strong><\/p>\r\n<p>For any numbers [latex]a,b,\\text{and }c[\/latex]<\/p>\r\n<p>If [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex]<\/p>\r\n<\/td>\r\n<td>\r\n<p><strong>Addition Property of Equality<\/strong><\/p>\r\n<p>For any numbers [latex]a,b,\\text{and }c[\/latex]<\/p>\r\n<p>If [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>\r\n<p><strong>The Division Property of Equality<\/strong><\/p>\r\n<p>For any numbers [latex]a,b,\\text{and }c,\\text{and }c\\ne 0[\/latex]<\/p>\r\n<p>If [latex]a=b[\/latex], then [latex]{\\Large\\frac{a}{c}}={\\Large\\frac{b}{c}}[\/latex]<\/p>\r\n<\/td>\r\n<td>\r\n<p><strong>The Multiplication Property of Equality<\/strong><\/p>\r\n<p>For any numbers [latex]a,b,\\text{and }c[\/latex]<\/p>\r\n<p>If [latex]a=b[\/latex], then [latex]ac=bc[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/center><\/div>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3697[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3698[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3699[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]3700[\/ohm2_question]<\/section>\r\n<h2>Solving Equations By Clearing Decimals<\/h2>\r\n<p>Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money and percent. But, we know, decimals are really another way to represent fractions. For example, [latex]0.3=\\Large\\frac{3}{10}[\/latex] and [latex]0.17=\\Large\\frac{17}{100}[\/latex]. So, when we have an equation with decimals, we can use the same process we used to clear fractions\u2014multiply both sides of the equation by the least common denominator.<\/p>\r\n<section class=\"textbox example\">\r\n<p>Solve the following:<\/p>\r\n<center>[latex]0.8x - 5=7[\/latex]<\/center>\r\n<p><br \/>\r\n[reveal-answer q=\"571663\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"571663\"]<br \/>\r\nThe only decimal in the equation is [latex]0.8[\/latex]. Since [latex]0.8=\\Large\\frac{8}{10}[\/latex], the LCD is [latex]10[\/latex]. We can multiply both sides by [latex]10[\/latex] to clear the decimal.<\/p>\r\n<table id=\"eip-id1168467123212\" class=\"unnumbered unstyled\" summary=\"The first line says 0.8x minus 5 equals 7. The next line says, \">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">\u00a0<\/td>\r\n<td style=\"height: 15px;\">[latex]0.8x-5=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Multiply both sides by the LCD.<\/td>\r\n<td style=\"height: 15px;\">[latex]\\color{red}{10}(0.8x-5)=\\color{red}{10}(7)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px;\">Distribute.<\/td>\r\n<td style=\"height: 24px;\">[latex]10(0.8x)-10(5)=10(7)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 23px;\">\r\n<td style=\"height: 23px;\">Multiply, and notice, no more decimals!<\/td>\r\n<td style=\"height: 23px;\">[latex]8x-50=70[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 20px;\">\r\n<td style=\"height: 20px;\">Add 50 to get all constants to the right.<\/td>\r\n<td style=\"height: 20px;\">[latex]8x-50\\color{red}{+50}=70\\color{red}{+50}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 20.75px;\">\r\n<td style=\"height: 20.75px;\">Simplify.<\/td>\r\n<td style=\"height: 20.75px;\">[latex]8x=120[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 42px;\">\r\n<td style=\"height: 42px;\">Divide both sides by [latex]8[\/latex].<\/td>\r\n<td style=\"height: 42px;\">[latex]\\Large\\frac{8x}{\\color{red}{8}}\\normalsize =\\Large\\frac{120}{\\color{red}{8}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 19px;\">\r\n<td style=\"height: 19px;\">Simplify.<\/td>\r\n<td style=\"height: 19px;\">[latex]x=15[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">Check: Let [latex]x=15[\/latex].<\/td>\r\n<td style=\"height: 15px;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 103px;\">\r\n<td style=\"height: 103px;\">\r\n<p>[latex]0.8(\\color{red}{15})-5\\stackrel{\\text{?}}{=}7[\/latex][latex]12-5\\stackrel{\\text{?}}{=}7[\/latex]<\/p>\r\n<p>[latex]7=7\\quad\\checkmark[\/latex]<\/p>\r\n<\/td>\r\n<td style=\"height: 103px;\">\u00a0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]6749[\/ohm2_question]<\/section>","rendered":"<h2>Determine Whether a Decimal is a Solution of an Equation<\/h2>\n<p>Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family\u2019s budget, or planning for the future of your business, you\u2019ll be solving equations with decimals.<\/p>\n<p>Now that we\u2019ve worked with decimals, we are ready to find solutions to equations involving decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal. We\u2019ll list these steps here again for easy reference.<\/p>\n<section class=\"textbox questionHelp\">\n<p><strong>How To: Determine Whether a Number Is a Solution to an Equation<\/strong><\/p>\n<ol id=\"eip-id1168469438337\" class=\"stepwise\">\n<li>Substitute the number for the variable in the equation.<\/li>\n<li>Simplify the expressions on both sides of the equation.<\/li>\n<li>Determine whether the resulting equation is true.\n<ul id=\"fs-id2482544\">\n<li>If so, the number is a solution.<\/li>\n<li>If not, the number is not a solution.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3696\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3696&theme=lumen&iframe_resize_id=ohm3696&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Solve Equations with Decimals<\/h2>\n<p>In previous modules, we solved equations using the properties of equality. We will use these same properties to solve equations with decimals. Remember, when you add, subtract, multiply, or divide the same quantity from both sides of an equation, you still have equality.<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>properties of equality<\/h3>\n<div style=\"text-align: center;\">\n<table id=\"fs-id1752382\" class=\"unnumbered\" summary=\"A table is shown with 2 rows and 2 columns. The first cell says,\">\n<tbody>\n<tr valign=\"top\">\n<td>\n<p><strong>Subtraction Property of Equality<\/strong><\/p>\n<p>For any numbers [latex]a,b,\\text{and }c[\/latex]<\/p>\n<p>If [latex]a=b[\/latex], then [latex]a-c=b-c[\/latex]<\/p>\n<\/td>\n<td>\n<p><strong>Addition Property of Equality<\/strong><\/p>\n<p>For any numbers [latex]a,b,\\text{and }c[\/latex]<\/p>\n<p>If [latex]a=b[\/latex], then [latex]a+c=b+c[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>\n<p><strong>The Division Property of Equality<\/strong><\/p>\n<p>For any numbers [latex]a,b,\\text{and }c,\\text{and }c\\ne 0[\/latex]<\/p>\n<p>If [latex]a=b[\/latex], then [latex]{\\Large\\frac{a}{c}}={\\Large\\frac{b}{c}}[\/latex]<\/p>\n<\/td>\n<td>\n<p><strong>The Multiplication Property of Equality<\/strong><\/p>\n<p>For any numbers [latex]a,b,\\text{and }c[\/latex]<\/p>\n<p>If [latex]a=b[\/latex], then [latex]ac=bc[\/latex]<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3697\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3697&theme=lumen&iframe_resize_id=ohm3697&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3698\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3698&theme=lumen&iframe_resize_id=ohm3698&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3699\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3699&theme=lumen&iframe_resize_id=ohm3699&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm3700\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=3700&theme=lumen&iframe_resize_id=ohm3700&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2>Solving Equations By Clearing Decimals<\/h2>\n<p>Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money and percent. But, we know, decimals are really another way to represent fractions. For example, [latex]0.3=\\Large\\frac{3}{10}[\/latex] and [latex]0.17=\\Large\\frac{17}{100}[\/latex]. So, when we have an equation with decimals, we can use the same process we used to clear fractions\u2014multiply both sides of the equation by the least common denominator.<\/p>\n<section class=\"textbox example\">\n<p>Solve the following:<\/p>\n<div style=\"text-align: center;\">[latex]0.8x - 5=7[\/latex]<\/div>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q571663\">Show Solution<\/button><\/p>\n<div id=\"q571663\" class=\"hidden-answer\" style=\"display: none\">\nThe only decimal in the equation is [latex]0.8[\/latex]. Since [latex]0.8=\\Large\\frac{8}{10}[\/latex], the LCD is [latex]10[\/latex]. We can multiply both sides by [latex]10[\/latex] to clear the decimal.<\/p>\n<table id=\"eip-id1168467123212\" class=\"unnumbered unstyled\" summary=\"The first line says 0.8x minus 5 equals 7. The next line says,\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">\u00a0<\/td>\n<td style=\"height: 15px;\">[latex]0.8x-5=7[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Multiply both sides by the LCD.<\/td>\n<td style=\"height: 15px;\">[latex]\\color{red}{10}(0.8x-5)=\\color{red}{10}(7)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px;\">Distribute.<\/td>\n<td style=\"height: 24px;\">[latex]10(0.8x)-10(5)=10(7)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"height: 23px;\">Multiply, and notice, no more decimals!<\/td>\n<td style=\"height: 23px;\">[latex]8x-50=70[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 20px;\">\n<td style=\"height: 20px;\">Add 50 to get all constants to the right.<\/td>\n<td style=\"height: 20px;\">[latex]8x-50\\color{red}{+50}=70\\color{red}{+50}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 20.75px;\">\n<td style=\"height: 20.75px;\">Simplify.<\/td>\n<td style=\"height: 20.75px;\">[latex]8x=120[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 42px;\">\n<td style=\"height: 42px;\">Divide both sides by [latex]8[\/latex].<\/td>\n<td style=\"height: 42px;\">[latex]\\Large\\frac{8x}{\\color{red}{8}}\\normalsize =\\Large\\frac{120}{\\color{red}{8}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 19px;\">\n<td style=\"height: 19px;\">Simplify.<\/td>\n<td style=\"height: 19px;\">[latex]x=15[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">Check: Let [latex]x=15[\/latex].<\/td>\n<td style=\"height: 15px;\">\u00a0<\/td>\n<\/tr>\n<tr style=\"height: 103px;\">\n<td style=\"height: 103px;\">\n<p>[latex]0.8(\\color{red}{15})-5\\stackrel{\\text{?}}{=}7[\/latex][latex]12-5\\stackrel{\\text{?}}{=}7[\/latex]<\/p>\n<p>[latex]7=7\\quad\\checkmark[\/latex]\n<\/td>\n<td style=\"height: 103px;\">\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm6749\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=6749&theme=lumen&iframe_resize_id=ohm6749&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"Lynn Marecek & MaryAnne Anthony-Smith\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/prealgebra\/pages\/5-1-decimals\",\"project\":\"5.1 Decimals\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":54,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Prealgebra","author":"Lynn Marecek & MaryAnne 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