{"id":1061,"date":"2023-03-29T15:42:03","date_gmt":"2023-03-29T15:42:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=1061"},"modified":"2024-10-18T20:51:02","modified_gmt":"2024-10-18T20:51:02","slug":"decimals-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/decimals-fresh-take\/","title":{"raw":"Decimals: Fresh Take","rendered":"Decimals: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Write and name decimals<\/li>\r\n\t<li>Turn a decimal into a fraction<\/li>\r\n\t<li>Place decimals on a number line and order them<\/li>\r\n\t<li>Solve equations using decimals<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Write and Name Decimals<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Decimals<\/strong> represent fractions or parts of a whole, based on powers of ten, using a point known as a decimal point.<\/p>\r\n<p>Writing and naming decimals involve expressing numbers that are less than one, or fractions, in decimal notation. A decimal is written with a decimal point, and the place value of each digit after the decimal point indicates its value - tenths, hundredths, thousandths, and so on. For instance, the decimal [latex]0.35[\/latex] is pronounced \"zero point three five\" or \"thirty-five hundredths\". Each digit in a decimal number has a specific place value and is used to represent fractional values.<\/p>\r\n<\/div>\r\n<section class=\"textbox example\">Name each decimal:\r\n\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>[latex]4.3[\/latex]<\/li>\r\n\t<li>[latex]2.45[\/latex]<\/li>\r\n\t<li>[latex]0.009[\/latex]<\/li>\r\n\t<li>[latex]-15.571[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"157184\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"157184\"]<\/p>\r\n<ol>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]4.3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Name the number to the left of the decimal point.<\/td>\r\n<td>four_____<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write \"and\" for the decimal point.<\/td>\r\n<td>four and_____<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Name the number to the right of the decimal point as if it were a whole number.<\/td>\r\n<td>four and three_____<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Name the decimal place of the last digit.<\/td>\r\n<td>four and three tenths<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]2.45[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Name the number to the left of the decimal point.<\/td>\r\n<td>two_____<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write \"and\" for the decimal point.<\/td>\r\n<td>two and_____<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Name the number to the right of the decimal point as if it were a whole number.<\/td>\r\n<td>two and forty-five_____<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Name the decimal place of the last digit.<\/td>\r\n<td>two and forty-five hundredths<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]0.009[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Name the number to the left of the decimal point.<\/td>\r\n<td>Zero is the number to the left of the decimal; it is not included in the name.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Name the number to the right of the decimal point as if it were a whole number.<\/td>\r\n<td>nine_____<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Name the decimal place of the last digit.<\/td>\r\n<td>nine thousandths<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]-15.571[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Name the number to the left of the decimal point.<\/td>\r\n<td>negative fifteen<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write \"and\" for the decimal point.<\/td>\r\n<td>negative fifteen and_____<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Name the number to the right of the decimal point as if it were a whole number.<\/td>\r\n<td>negative fifteen and five hundred seventy-one_____<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Name the decimal place of the last digit.<\/td>\r\n<td>negative fifteen and five hundred seventy-one thousandths<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Write the following numbers as a decimal:\r\n\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>six and seventeen hundredths<\/li>\r\n\t<li>fourteen and thirty-seven hundredths<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"796510\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"796510\"]<\/p>\r\n<ol>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>six and seventeen hundredths<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The word <em>and<\/em> tells us to place a decimal point.<\/td>\r\n<td>___.___<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The word before <em>and<\/em> is the whole number; write it to the left of the decimal point.<\/td>\r\n<td>[latex]6[\/latex]._____<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>The decimal part is seventeen hundredths.<\/p>\r\n<p>Mark two places to the right of the decimal point for hundredths.<\/p>\r\n<\/td>\r\n<td>[latex]6[\/latex]._ _<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the numerals for seventeen in the places marked.<\/td>\r\n<td>[latex]6.17[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>fourteen and thirty-seven hundredths<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Place a decimal point under the word \u2018and\u2019.<\/td>\r\n<td>______. _________<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate the words before \u2018and\u2019 into the whole number and place it to the left of the decimal point.<\/td>\r\n<td>[latex]14[\/latex]. _________<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Mark two places to the right of the decimal point for \"hundredths\".<\/td>\r\n<td>[latex]14[\/latex].__ __<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate the words after \"and\" and write the number to the right of the decimal point.<\/td>\r\n<td>[latex]14.37[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Fourteen and thirty-seven hundredths is written [latex]14.37[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>In the following video, we show more examples of how to write the name of a decimal using a place value chart.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/aLsWWl2-aNE\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Read+and+Write+Decimals.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRead and Write Decimals\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Write a Decimal as a Fraction<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>Writing a decimal as a fraction requires understanding of the place value system. Each digit after the decimal point signifies a specific fractional part: tenths, hundredths, thousandths, etc.<\/p>\r\n<p>To convert a decimal to a fraction, consider the place value of the last digit. For instance, [latex]0.25[\/latex] means twenty-five hundredths and it is represented as the fraction [latex]\\frac{25}{100}[\/latex]. It is important to simplify the fraction to its lowest terms whenever possible, so [latex]\\frac{25}{100}[\/latex] simplifies to [latex]\\frac{1}{4}[\/latex].<\/p>\r\n<\/div>\r\n<section class=\"textbox example\">Write each of the following decimal numbers as a fraction or a mixed number:\r\n\r\n\r\n<ol>\r\n\t<li>[latex]4.09[\/latex]<\/li>\r\n\t<li>[latex]3.7[\/latex]<\/li>\r\n\t<li>[latex]-0.286[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"796511\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"796511\"]<\/p>\r\n<ol>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]4.09[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>There is a [latex]4[\/latex] to the left of the decimal point.<\/p>\r\n<p>Write \"[latex]4[\/latex]\" as the whole number part of the mixed number.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221437\/CNX_BMath_Figure_05_01_014_img-01.png\" alt=\"Decorative Image\" width=\"216\" height=\"57\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Determine the place value of the final digit.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221438\/CNX_BMath_Figure_05_01_014_img-02.png\" alt=\"Decorative Image\" width=\"216\" height=\"41\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Write the fraction.<\/p>\r\n<p>Write [latex]9[\/latex] in the numerator as it is the number to the right of the decimal point.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221439\/CNX_BMath_Figure_05_01_014_img-03.png\" alt=\"Decorative Image\" width=\"216\" height=\"45\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]100[\/latex] in the denominator as the place value of the final digit, [latex]9[\/latex], is hundredth.<\/td>\r\n<td>[latex]4{\\Large\\frac{9}{100}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The fraction is in simplest form.<\/td>\r\n<td>So, [latex]4.09=4{\\Large\\frac{9}{100}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Did you notice that the number of zeros in the denominator is the same as the number of decimal places?<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]3.7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>There is a [latex]3[\/latex] to the left of the decimal point.<\/p>\r\n<p>Write \"[latex]3[\/latex]\" as the whole number part of the mixed number.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221445\/CNX_BMath_Figure_05_01_015_img-01.png\" alt=\"Decorative Image\" width=\"118\" height=\"56\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Determine the place value of the final digit.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221446\/CNX_BMath_Figure_05_01_015_img-02.png\" alt=\"Decorative Image\" width=\"118\" height=\"42\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Write the fraction.<\/p>\r\n<p>Write [latex]7[\/latex] in the numerator as it is the number to the right of the decimal point.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221447\/CNX_BMath_Figure_05_01_015_img-03.png\" alt=\"Decorative Image\" width=\"117\" height=\"44\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]10[\/latex] in the denominator as the place value of the final digit, [latex]7[\/latex], is tenths.<\/td>\r\n<td>[latex]3{\\Large\\frac{7}{10}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The fraction is in simplest form.<\/td>\r\n<td>So, [latex]3.7=3{\\Large\\frac{7}{10}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\u22120.286[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>There is a [latex]0[\/latex] to the left of the decimal point.<\/p>\r\n<p>Write a negative sign before the fraction.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221450\/CNX_BMath_Figure_05_01_016_img-01.png\" alt=\"Decorative Image\" width=\"284\" height=\"53\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Determine the place value of the final digit and write it in the denominator.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221451\/CNX_BMath_Figure_05_01_016_img-02.png\" alt=\"Decorative Image\" width=\"284\" height=\"43\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Write the fraction.<\/p>\r\n<p>Write [latex]286[\/latex] in the numerator as it is the number to the right of the decimal point.<\/p>\r\n<p>Write [latex]1,000[\/latex] in the denominator as the place value of the final digit, [latex]6[\/latex], is thousandths.<\/p>\r\n<\/td>\r\n<td>[latex]-{\\Large\\frac{286}{1000}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We remove a common factor of [latex]2[\/latex] to simplify the fraction.<\/td>\r\n<td>[latex]-{\\Large\\frac{143}{500}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>The following video shows more examples of writing decimals as fractions:<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0yYQLZcTEXc\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+1_+Convert+a+Decimal+to+a+Fraction.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Convert a Decimal to a Fraction\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Rounding Decimals<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>Rounding decimals involves approximating a decimal to the nearest whole number, tenth, hundredth, or other decimal place value. The process is similar to rounding whole numbers. If the digit right after the place value you're rounding to is [latex]5[\/latex] or greater, you round up the last digit kept. If it's less than [latex]5[\/latex], you keep the digit as it is.<\/p>\r\n<p>For example, if rounding [latex]3.78[\/latex] to the nearest tenth, you look at the hundredths place ([latex]8[\/latex]). Since [latex]8[\/latex] is greater than [latex]5[\/latex], you round up, and [latex]3.78[\/latex] becomes [latex]3.8[\/latex].<\/p>\r\n<\/div>\r\n<section class=\"textbox example\">Round [latex]18.379[\/latex] to the nearest:\r\n\r\n\r\n<ol>\r\n\t<li>tenth<\/li>\r\n\t<li>whole number<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"414931\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"414931\"]<\/p>\r\n<ol>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]18.379[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Locate the tenths place and mark it with an arrow.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221520\/CNX_BMath_Figure_05_01_022_img-02.png\" alt=\"Decorative Image\" width=\"133\" height=\"75\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Underline the digit to the right of the tenths digit.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221521\/CNX_BMath_Figure_05_01_022_img-03.png\" alt=\"Decorative Image\" width=\"133\" height=\"77\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Because [latex]7[\/latex] is greater than or equal to [latex]5[\/latex], add [latex]1[\/latex] to the [latex]3[\/latex].<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221523\/CNX_BMath_Figure_05_01_022_img-04.png\" alt=\"Decorative Image\" width=\"133\" height=\"67\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite the number, deleting all digits to the right of the tenths place.<\/td>\r\n<td>[latex]18.4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>So, [latex]18.379[\/latex] rounded to the nearest tenth is [latex]18.4[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]18.379[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Locate the ones place and mark it with an arrow.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221526\/CNX_BMath_Figure_05_01_023_img-02.png\" alt=\"Decorative Image\" width=\"156\" height=\"78\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Underline the digit to the right of the ones place.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221527\/CNX_BMath_Figure_05_01_023_img-03.png\" alt=\"Decorative Image\" width=\"156\" height=\"76\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since [latex]3[\/latex] is not greater than or equal to [latex]5[\/latex], do not add [latex]1[\/latex] to the [latex]8[\/latex].<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221528\/CNX_BMath_Figure_05_01_023_img-04.png\" alt=\"Decorative Image\" width=\"156\" height=\"67\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite the number, deleting all digits to the right of the ones place.<\/td>\r\n<td>[latex]18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>So [latex]18.379[\/latex] rounded to the nearest whole number is [latex]18[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Watch the following video to see an example of how to round a number to several different place values.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qu4Y9DGqXlk\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Examples_+Rounding+Decimals.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExamples: Rounding Decimals\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Locating and Ordering Decimals With a Number Line<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>Locating and ordering decimals on a number line involves understanding the value of decimals and their relative positions. First, identify the whole numbers that the decimal falls between. Then, partition the space between these whole numbers into tenths, hundredths, or thousandths, as needed. Plot the decimals on the line at the corresponding position.<\/p>\r\n<p><br \/>\r\nFor example, [latex]0.5[\/latex] would fall halfway between [latex]0[\/latex] and [latex]1[\/latex]. When ordering decimals, start from the smallest (left on the number line) and move to the largest (right on the number line).<\/p>\r\n<\/div>\r\n<section class=\"textbox example\">Locate [latex]-0.74[\/latex] on a number line.<br \/>\r\n[reveal-answer q=\"669401\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"669401\"]<br \/>\r\nThe decimal [latex]-0.74[\/latex] is equivalent to [latex]-{\\Large\\frac{74}{100}}[\/latex], so it is located between [latex]0[\/latex] and [latex]-1[\/latex]. On a number line, mark off and label the multiples of [latex]-0.10[\/latex] in the interval between [latex]0[\/latex] and [latex]-1[\/latex] ( [latex]-0.10[\/latex] , [latex]-0.20[\/latex] , etc.) and mark [latex]-0.74[\/latex] between [latex]-0.70[\/latex] and [latex]-0.80[\/latex], a little closer to [latex]-0.70[\/latex] .<center><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221503\/CNX_BMath_Figure_05_01_007_img.png\" alt=\"A number line with negative 1.00, negative 0.90, negative 0.80, negative 0.70, negative 0.60, negative 0.50, negative 0.40, negative 0.30, negative 0.20, negative 0.10, and 0.00 labeled. There is a red dot between negative 0.80 and negative 0.70 labeled as negative 0.74.\" width=\"900\" height=\"63\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n<p>In the next video, we show more examples of how to locate a decimal on the number line.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/F3LAKsOBdNA\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Example_+Identify+Decimals+on+the+Number+Line.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Identify Decimals on the Number Line\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Order Decimals<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>Ordering decimals involves arranging them from least to greatest or vice versa based on their numerical value. Begin by comparing the digits in the tenths place of each decimal. If the tenths place digits are the same, move to the hundredths place, then the thousandths place, and so on, until you find a difference. The decimal with the smaller digit in the first place where the decimals differ is the smaller decimal. If all places are the same, then the decimals are equal.<\/p>\r\n<\/div>\r\n<p>In the following video lesson we show how to order decimals using inequality notation by comparing place values, and by using fractions.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/fjO3fnt3ABA\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Decimal+Notation_+Ordering+Decimals.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDecimal Notation: Ordering Decimals\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Order the following decimals using [latex]&lt;\\text{ or }\\text{&gt;}[\/latex]:\r\n\r\n\r\n<ol>\r\n\t<li>[latex]0.64[\/latex] ____ [latex]0.6[\/latex]<\/li>\r\n\t<li>[latex]0.83[\/latex] ____ [latex]0.803[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"895615\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"895615\"]<\/p>\r\n<ol>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 494.5px;\">\u00a0<\/td>\r\n<td style=\"width: 341.5px;\">[latex]0.64[\/latex] ____ [latex]0.6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 494.5px;\">Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of [latex]0.6[\/latex].<\/td>\r\n<td style=\"width: 341.5px;\">[latex]0.64[\/latex] ____ [latex]0.6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 494.5px;\">Compare the numbers to the right of the decimal point as if they were whole numbers.<\/td>\r\n<td style=\"width: 341.5px;\">[latex]64&gt;60[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 494.5px;\">Order the numbers using the appropriate inequality sign.<\/td>\r\n<td style=\"width: 341.5px;\">\r\n<p>[latex]0.64&gt;0.60[\/latex]<\/p>\r\n<p>[latex]0.64&gt;0.6[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 500.45px;\">\u00a0<\/td>\r\n<td style=\"width: 334.55px;\">[latex]0.83[\/latex] ____ [latex]0.803[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 500.45px;\">Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of [latex]0.83[\/latex].<\/td>\r\n<td style=\"width: 334.55px;\">[latex]0.83[\/latex] ____ [latex]0.803[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 500.45px;\">Compare the numbers to the right of the decimal point as if they were whole numbers.<\/td>\r\n<td style=\"width: 334.55px;\">[latex]830&gt;803[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 500.45px;\">Order the numbers using the appropriate inequality sign.<\/td>\r\n<td style=\"width: 334.55px;\">\r\n<p>[latex]0.830&gt;0.803[\/latex]<\/p>\r\n<p>[latex]0.83&gt;0.803[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Use [latex]&lt;\\text{or}&amp;gt[\/latex] to order: [latex]-0.1[\/latex] ____ [latex]- 0.8[\/latex]<br \/>\r\n[reveal-answer q=\"12939\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"12939\"]\r\n\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 461.067px;\">\u00a0<\/td>\r\n<td style=\"width: 283.933px;\">[latex]-0.1[\/latex] ____ [latex]- 0.8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 461.067px;\">Write the numbers one under the other, lining up the decimal points.<\/td>\r\n<td style=\"width: 283.933px;\">\r\n<p>[latex]-0.1[\/latex]<\/p>\r\n<p>[latex]-0.8[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 461.067px;\">They have the same number of digits.<\/td>\r\n<td style=\"width: 283.933px;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 461.067px;\">Since [latex]-1&gt;-8,-1[\/latex] tenth is greater than [latex]-8[\/latex] tenths.<\/td>\r\n<td style=\"width: 283.933px;\">[latex]-0.1&gt;-0.8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Determine Whether a Decimal is a Solution of an Equation<\/h2>\r\n<section class=\"textbox example\">Determine whether each of the following is a solution of [latex]x - 0.7=1.5[\/latex]\r\n\r\n\r\n<ol style=\"list-style-type: decimal-leading-zero;\">\r\n\t<li>[latex]x=1[\/latex]<\/li>\r\n\t<li>[latex]x=-0.8[\/latex]<\/li>\r\n\t<li>[latex]x=2.2[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"895619\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"895619\"]<\/p>\r\n<ol>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]x-0.7=1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{1}[\/latex] for [latex]x[\/latex].<\/td>\r\n<td>[latex]\\color{red}{1} - 0.7\\stackrel{?}{=}1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]0.3\\not=1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since [latex]x=1[\/latex] does not result in a true equation, [latex]1[\/latex] is not a solution to the equation.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]x-0.7=1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{0.8}[\/latex] for [latex]x[\/latex].<\/td>\r\n<td>[latex]\\color{red}{0.8} - 0.7\\stackrel{?}{=}1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]-1.5\\not=1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since [latex]x=-0.8[\/latex] does not result in a true equation, [latex]-0.8[\/latex] is not a solution to the equation.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]x-0.7=1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\color{red}{2.2}[\/latex] for [latex]x[\/latex].<\/td>\r\n<td>[latex]\\color{red}{2.2} - 0.7\\stackrel{?}{=}1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract.<\/td>\r\n<td>[latex]1.5=1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since [latex]x=2.2[\/latex] results in a true equation, [latex]2.2[\/latex] is a solution to the equation.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Solve Equations with Decimals<\/h2>\r\n<section class=\"textbox example\">Solve: [latex]y+2.3=-4.7[\/latex]<br \/>\r\n[reveal-answer q=\"908331\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"908331\"]We will use the Subtraction Property of Equality to isolate the variable.\r\n\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\">\u00a0<\/td>\r\n<td>[latex]y+2.3=-4.7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Subtract [latex]\\color{red}{2.3}[\/latex] from each side, to undo the addition.<\/td>\r\n<td>[latex]y+2.3\\color{red}{- 2.3}=-4.7\\color{red}{- 2.3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]y-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Check:<\/strong><\/td>\r\n<td>[latex]y+2.3=-4.7[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]y=\\color{red}{-7}[\/latex].<\/td>\r\n<td>[latex]\\color{red}{-7}+2.3\\stackrel{?}{=}-4.7[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-4.7=-4.7[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since [latex]y=-7[\/latex] makes [latex]y+2.3=-4.7[\/latex] a true statement, we know we have found a solution to this equation.<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Solve: [latex]a - 4.75=-1.39[\/latex]<br \/>\r\n[reveal-answer q=\"265677\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"265677\"]<br \/>\r\nWe will use the Addition Property of Equality.\r\n\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\">\u00a0<\/td>\r\n<td>[latex]a-4.75=-1.39[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Add [latex]4.75[\/latex] to each side, to undo the subtraction.<\/td>\r\n<td>[latex]a-4.75+\\color{red}{4.75}=-1.39+\\color{red}{4.75}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Simplify.<\/td>\r\n<td>[latex]a=3.36[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Check:<\/strong><\/td>\r\n<td>[latex]a-4.75=-1.39[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]a=\\color{red}{3.36}[\/latex].<\/td>\r\n<td>[latex]\\color{red}{3.36}-4.75\\stackrel{?}{=}-1.39[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]-1.39=-1.39[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since the result is a true statement, [latex]a=3.36[\/latex] is a solution to the equation.<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>In the following video, we show more examples of using the addition and subtraction properties of equality to solve linear equations that contain decimals.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/D8wKGlxf6bM\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Solving+One+Step+Equations+Using+Addition+and+Subtraction+(Decimals).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving One Step Equations Using Addition and Subtraction (Decimals)\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Solve: [latex]-4.8=0.8n[\/latex]<br \/>\r\n[reveal-answer q=\"249570\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"249570\"]<br \/>\r\nWe will use the Division Property of Equality.<br \/>\r\nUse the Properties of Equality to find a value for [latex]n[\/latex].\r\n\r\n\r\n<table>\r\n<tbody>\r\n<tr style=\"height: 15.46875px;\">\r\n<td style=\"height: 15.46875px;\" colspan=\"2\">\u00a0<\/td>\r\n<td style=\"height: 15.46875px;\">[latex]-4.8=0.8n[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 43px;\">\r\n<td style=\"height: 43px;\" colspan=\"2\">We must divide both sides by [latex]0.8[\/latex] to isolate [latex]n[\/latex].<\/td>\r\n<td style=\"height: 43px;\">[latex]{\\Large\\frac{-4.8}{\\color{red}{0.8}}}={\\Large\\frac{0.8n}{\\color{red}{0.8}}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 25px;\">\r\n<td style=\"height: 25px;\" colspan=\"2\">Simplify.<\/td>\r\n<td style=\"height: 25px;\">[latex]-6=n[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"height: 22px;\"><strong>Check:<\/strong><\/td>\r\n<td style=\"height: 22px;\">[latex]-4.8=0.8n[\/latex]<\/td>\r\n<td style=\"height: 22px;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"height: 28px;\">Substitute [latex]n=\\color{red}{-6}[\/latex].<\/td>\r\n<td style=\"height: 28px;\">[latex]-4.8\\stackrel{?}{=}0.8(\\color{red}{-6})[\/latex]<\/td>\r\n<td style=\"height: 28px;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px;\">\u00a0<\/td>\r\n<td style=\"height: 24px;\">[latex]-4.8=-4.8[\/latex]<\/td>\r\n<td style=\"height: 24px;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Since [latex]n=-6[\/latex] makes [latex]-4.8=0.8n[\/latex] a true statement, we know we have a solution.<\/td>\r\n<td>&nbsp;<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>Watch the next video to see how to solve another equation with decimals that requires division.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/paV-sXbRkHA\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Solve+a+One+Step+Equation+With+Decimals+by+Dividing.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a One Step Equation With Decimals by Dividing\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Solve: [latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex]<br \/>\r\n[reveal-answer q=\"571663\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"571663\"]<br \/>\r\nWe will use the Multiplication Property of Equality.\r\n\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td colspan=\"2\">\u00a0<\/td>\r\n<td>[latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Here, <em>p<\/em> is divided by [latex]\u22121.8[\/latex]. We must multiply by [latex]\u22121.8[\/latex] to isolate [latex]p[\/latex]<\/td>\r\n<td>[latex]\\color{red}{-1.8}({\\Large\\frac{p}{-1.8}})=\\color{red}{-1.8}(-6.5)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td colspan=\"2\">Multiply.<\/td>\r\n<td>[latex]p=11.7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Check:<\/strong><\/td>\r\n<td>[latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]{\\Large\\frac{\\color{red}{11.7}}{-1.8}}\\stackrel{?}{=}-6.5[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0Substitute [latex]p=\\color{red}{11.7}[\/latex].<\/td>\r\n<td>[latex]-6.5=-6.5[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A solution to [latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex] is [latex]p=11.7[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>The following video shows an example of how to solve an equation with decimals that requires multiplication.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/q2s9GaEwSi8\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Solve+a+One+Step+Equation+With+Decimals+by+Multiplying.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a One Step Equation With Decimals by Multiplying\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Solving Equations By Clearing Decimals<\/h2>\r\n<section class=\"textbox example\">Solve: [latex]0.06x+0.02=0.25x - 1.5[\/latex]\r\n\r\n\r\n<p class=\"p1\">[reveal-answer q=\"190834\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"190834\"]<\/p>\r\n<p>Look at the decimals and think of the equivalent fractions.<br \/>\r\n[latex]0.06=\\Large\\frac{6}{100}\\normalsize ,0.02=\\Large\\frac{2}{100}\\normalsize ,0.25=\\Large\\frac{25}{100}\\normalsize ,1.5=1\\Large\\frac{5}{10}[\/latex]<\/p>\r\n<p>Notice, the LCD is [latex]100[\/latex].<br \/>\r\nBy multiplying by the LCD we will clear the decimals.<\/p>\r\n<table>\r\n<tbody>\r\n<tr style=\"height: 23px;\">\r\n<td style=\"height: 23px;\">\u00a0<\/td>\r\n<td style=\"height: 23px;\">[latex]0.06x+0.02=0.25x-1.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px;\">Multiply both sides by [latex]100[\/latex].<\/td>\r\n<td style=\"height: 24px;\">[latex]\\color{red}{100}(0.06x+0.02)=\\color{red}{100}(0.25x-1.5)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 32px;\">\r\n<td style=\"height: 32px;\">Distribute.<\/td>\r\n<td style=\"height: 32px;\">[latex]100(0.06x)+100(0.02)=100(0.25x)-100(1.5)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px;\">Multiply, and now no more decimals.<\/td>\r\n<td style=\"height: 14px;\">[latex]6x+2=25x-150[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 23px;\">\r\n<td style=\"height: 23px;\">Collect the variables to the right.<\/td>\r\n<td style=\"height: 23px;\">[latex]6x\\color{red}{-6x}+2=25x\\color{red}{-6x}-150[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px;\">Simplify.<\/td>\r\n<td style=\"height: 14px;\">[latex]2=19x-150[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 25px;\">\r\n<td style=\"height: 25px;\">Collect the constants to the left.<\/td>\r\n<td style=\"height: 25px;\">[latex]2\\color{red}{+150}=19x-150\\color{red}{+150}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px;\">Simplify.<\/td>\r\n<td style=\"height: 14px;\">[latex]152=19x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 43px;\">\r\n<td style=\"height: 43px;\">Divide by [latex]19[\/latex].<\/td>\r\n<td style=\"height: 43px;\">[latex]\\Large\\frac{152}{\\color{red}{19}}\\normalsize =\\Large\\frac{19x}{\\color{red}{19}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px;\">Simplify.<\/td>\r\n<td style=\"height: 14px;\">[latex]8=x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px;\">Check: Let [latex]x=8[\/latex].<\/td>\r\n<td style=\"height: 14px;\">\u00a0<\/td>\r\n<\/tr>\r\n<tr style=\"height: 87.6426px;\">\r\n<td style=\"height: 87.6426px;\">\r\n<p>[latex]0.06(\\color{red}{8})+0.02=0.25(\\color{red}{8})-1.5[\/latex][latex]0.48+0.02=2.00-1.5[\/latex]<\/p>\r\n<p>[latex]0.50=0.50\\quad\\checkmark[\/latex]<\/p>\r\n<\/td>\r\n<td style=\"height: 87.6426px;\">\u00a0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>The next example uses an equation that is typical of the ones we will see in the money applications. Notice that we will distribute the decimal first before we clear all decimals in the equation.<\/p>\r\n<section class=\"textbox example\">Solve: [latex]0.25x+0.05\\left(x+3\\right)=2.85[\/latex]\r\n\r\n\r\n<p class=\"p1\">[reveal-answer q=\"777666\"]Show Solution[\/reveal-answer]<\/p>\r\n<p class=\"p1\">[hidden-answer a=\"777666\"]<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]0.25x+0.05(x+3)=2.85[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute first.<\/td>\r\n<td>[latex]0.25x+0.05x+0.15=2.85[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]0.30x+0.15=2.85[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>To clear decimals, multiply by [latex]100[\/latex].<\/td>\r\n<td>[latex]\\color{red}{100}(0.30x+0.15)=\\color{red}{100}(2.85)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]30x+15=285[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract [latex]15[\/latex] from both sides.<\/td>\r\n<td>[latex]30x+15\\color{red}{-15}=285\\color{red}{-15}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]30x=270[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide by [latex]30[\/latex].<\/td>\r\n<td>[latex]\\Large\\frac{30x}{\\color{red}{30}}\\normalsize =\\Large\\frac{270}{\\color{red}{30}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x=9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check: Let [latex]x=9[\/latex].<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>[latex]0.25x+0.05(x+3)=2.85[\/latex][latex]0.25(\\color{red}{9})+0.05(\\color{red}{9}+3)\\stackrel{\\text{?}}{=}2.85[\/latex]<\/p>\r\n<p>[latex]2.25+0.05(12)\\stackrel{\\text{?}}{=}2.85[\/latex]<\/p>\r\n<p>[latex]2.85=2.85\\quad\\checkmark[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>In the following video we present another example of how to solve an equation that contains decimals and variable terms on both sides of the equal sign.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/pZWTJvua-P8\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Solve+a+Linear+Equation+With+Decimals+and+Variables+on+Both+Sides.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a Linear Equation With Decimals and Variables on Both Sides\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Write and name decimals<\/li>\n<li>Turn a decimal into a fraction<\/li>\n<li>Place decimals on a number line and order them<\/li>\n<li>Solve equations using decimals<\/li>\n<\/ul>\n<\/section>\n<h2>Write and Name Decimals<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Decimals<\/strong> represent fractions or parts of a whole, based on powers of ten, using a point known as a decimal point.<\/p>\n<p>Writing and naming decimals involve expressing numbers that are less than one, or fractions, in decimal notation. A decimal is written with a decimal point, and the place value of each digit after the decimal point indicates its value &#8211; tenths, hundredths, thousandths, and so on. For instance, the decimal [latex]0.35[\/latex] is pronounced &#8220;zero point three five&#8221; or &#8220;thirty-five hundredths&#8221;. Each digit in a decimal number has a specific place value and is used to represent fractional values.<\/p>\n<\/div>\n<section class=\"textbox example\">Name each decimal:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]4.3[\/latex]<\/li>\n<li>[latex]2.45[\/latex]<\/li>\n<li>[latex]0.009[\/latex]<\/li>\n<li>[latex]-15.571[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q157184\">Show Solution<\/button><\/p>\n<div id=\"q157184\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td>[latex]4.3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Name the number to the left of the decimal point.<\/td>\n<td>four_____<\/td>\n<\/tr>\n<tr>\n<td>Write &#8220;and&#8221; for the decimal point.<\/td>\n<td>four and_____<\/td>\n<\/tr>\n<tr>\n<td>Name the number to the right of the decimal point as if it were a whole number.<\/td>\n<td>four and three_____<\/td>\n<\/tr>\n<tr>\n<td>Name the decimal place of the last digit.<\/td>\n<td>four and three tenths<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>[latex]2.45[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Name the number to the left of the decimal point.<\/td>\n<td>two_____<\/td>\n<\/tr>\n<tr>\n<td>Write &#8220;and&#8221; for the decimal point.<\/td>\n<td>two and_____<\/td>\n<\/tr>\n<tr>\n<td>Name the number to the right of the decimal point as if it were a whole number.<\/td>\n<td>two and forty-five_____<\/td>\n<\/tr>\n<tr>\n<td>Name the decimal place of the last digit.<\/td>\n<td>two and forty-five hundredths<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>[latex]0.009[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Name the number to the left of the decimal point.<\/td>\n<td>Zero is the number to the left of the decimal; it is not included in the name.<\/td>\n<\/tr>\n<tr>\n<td>Name the number to the right of the decimal point as if it were a whole number.<\/td>\n<td>nine_____<\/td>\n<\/tr>\n<tr>\n<td>Name the decimal place of the last digit.<\/td>\n<td>nine thousandths<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>[latex]-15.571[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Name the number to the left of the decimal point.<\/td>\n<td>negative fifteen<\/td>\n<\/tr>\n<tr>\n<td>Write &#8220;and&#8221; for the decimal point.<\/td>\n<td>negative fifteen and_____<\/td>\n<\/tr>\n<tr>\n<td>Name the number to the right of the decimal point as if it were a whole number.<\/td>\n<td>negative fifteen and five hundred seventy-one_____<\/td>\n<\/tr>\n<tr>\n<td>Name the decimal place of the last digit.<\/td>\n<td>negative fifteen and five hundred seventy-one thousandths<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Write the following numbers as a decimal:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>six and seventeen hundredths<\/li>\n<li>fourteen and thirty-seven hundredths<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q796510\">Show Solution<\/button><\/p>\n<div id=\"q796510\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td>six and seventeen hundredths<\/td>\n<\/tr>\n<tr>\n<td>The word <em>and<\/em> tells us to place a decimal point.<\/td>\n<td>___.___<\/td>\n<\/tr>\n<tr>\n<td>The word before <em>and<\/em> is the whole number; write it to the left of the decimal point.<\/td>\n<td>[latex]6[\/latex]._____<\/td>\n<\/tr>\n<tr>\n<td>\n<p>The decimal part is seventeen hundredths.<\/p>\n<p>Mark two places to the right of the decimal point for hundredths.<\/p>\n<\/td>\n<td>[latex]6[\/latex]._ _<\/td>\n<\/tr>\n<tr>\n<td>Write the numerals for seventeen in the places marked.<\/td>\n<td>[latex]6.17[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>fourteen and thirty-seven hundredths<\/td>\n<\/tr>\n<tr>\n<td>Place a decimal point under the word \u2018and\u2019.<\/td>\n<td>______. _________<\/td>\n<\/tr>\n<tr>\n<td>Translate the words before \u2018and\u2019 into the whole number and place it to the left of the decimal point.<\/td>\n<td>[latex]14[\/latex]. _________<\/td>\n<\/tr>\n<tr>\n<td>Mark two places to the right of the decimal point for &#8220;hundredths&#8221;.<\/td>\n<td>[latex]14[\/latex].__ __<\/td>\n<\/tr>\n<tr>\n<td>Translate the words after &#8220;and&#8221; and write the number to the right of the decimal point.<\/td>\n<td>[latex]14.37[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Fourteen and thirty-seven hundredths is written [latex]14.37[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>In the following video, we show more examples of how to write the name of a decimal using a place value chart.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/aLsWWl2-aNE\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Read+and+Write+Decimals.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cRead and Write Decimals\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Write a Decimal as a Fraction<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Writing a decimal as a fraction requires understanding of the place value system. Each digit after the decimal point signifies a specific fractional part: tenths, hundredths, thousandths, etc.<\/p>\n<p>To convert a decimal to a fraction, consider the place value of the last digit. For instance, [latex]0.25[\/latex] means twenty-five hundredths and it is represented as the fraction [latex]\\frac{25}{100}[\/latex]. It is important to simplify the fraction to its lowest terms whenever possible, so [latex]\\frac{25}{100}[\/latex] simplifies to [latex]\\frac{1}{4}[\/latex].<\/p>\n<\/div>\n<section class=\"textbox example\">Write each of the following decimal numbers as a fraction or a mixed number:<\/p>\n<ol>\n<li>[latex]4.09[\/latex]<\/li>\n<li>[latex]3.7[\/latex]<\/li>\n<li>[latex]-0.286[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q796511\">Show Solution<\/button><\/p>\n<div id=\"q796511\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td>[latex]4.09[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\n<p>There is a [latex]4[\/latex] to the left of the decimal point.<\/p>\n<p>Write &#8220;[latex]4[\/latex]&#8221; as the whole number part of the mixed number.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221437\/CNX_BMath_Figure_05_01_014_img-01.png\" alt=\"Decorative Image\" width=\"216\" height=\"57\" \/><\/td>\n<\/tr>\n<tr>\n<td>Determine the place value of the final digit.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221438\/CNX_BMath_Figure_05_01_014_img-02.png\" alt=\"Decorative Image\" width=\"216\" height=\"41\" \/><\/td>\n<\/tr>\n<tr>\n<td>\n<p>Write the fraction.<\/p>\n<p>Write [latex]9[\/latex] in the numerator as it is the number to the right of the decimal point.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221439\/CNX_BMath_Figure_05_01_014_img-03.png\" alt=\"Decorative Image\" width=\"216\" height=\"45\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write [latex]100[\/latex] in the denominator as the place value of the final digit, [latex]9[\/latex], is hundredth.<\/td>\n<td>[latex]4{\\Large\\frac{9}{100}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The fraction is in simplest form.<\/td>\n<td>So, [latex]4.09=4{\\Large\\frac{9}{100}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Did you notice that the number of zeros in the denominator is the same as the number of decimal places?<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>[latex]3.7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\n<p>There is a [latex]3[\/latex] to the left of the decimal point.<\/p>\n<p>Write &#8220;[latex]3[\/latex]&#8221; as the whole number part of the mixed number.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221445\/CNX_BMath_Figure_05_01_015_img-01.png\" alt=\"Decorative Image\" width=\"118\" height=\"56\" \/><\/td>\n<\/tr>\n<tr>\n<td>Determine the place value of the final digit.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221446\/CNX_BMath_Figure_05_01_015_img-02.png\" alt=\"Decorative Image\" width=\"118\" height=\"42\" \/><\/td>\n<\/tr>\n<tr>\n<td>\n<p>Write the fraction.<\/p>\n<p>Write [latex]7[\/latex] in the numerator as it is the number to the right of the decimal point.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221447\/CNX_BMath_Figure_05_01_015_img-03.png\" alt=\"Decorative Image\" width=\"117\" height=\"44\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write [latex]10[\/latex] in the denominator as the place value of the final digit, [latex]7[\/latex], is tenths.<\/td>\n<td>[latex]3{\\Large\\frac{7}{10}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The fraction is in simplest form.<\/td>\n<td>So, [latex]3.7=3{\\Large\\frac{7}{10}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>[latex]\u22120.286[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\n<p>There is a [latex]0[\/latex] to the left of the decimal point.<\/p>\n<p>Write a negative sign before the fraction.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221450\/CNX_BMath_Figure_05_01_016_img-01.png\" alt=\"Decorative Image\" width=\"284\" height=\"53\" \/><\/td>\n<\/tr>\n<tr>\n<td>Determine the place value of the final digit and write it in the denominator.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221451\/CNX_BMath_Figure_05_01_016_img-02.png\" alt=\"Decorative Image\" width=\"284\" height=\"43\" \/><\/td>\n<\/tr>\n<tr>\n<td>\n<p>Write the fraction.<\/p>\n<p>Write [latex]286[\/latex] in the numerator as it is the number to the right of the decimal point.<\/p>\n<p>Write [latex]1,000[\/latex] in the denominator as the place value of the final digit, [latex]6[\/latex], is thousandths.<\/p>\n<\/td>\n<td>[latex]-{\\Large\\frac{286}{1000}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>We remove a common factor of [latex]2[\/latex] to simplify the fraction.<\/td>\n<td>[latex]-{\\Large\\frac{143}{500}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>The following video shows more examples of writing decimals as fractions:<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0yYQLZcTEXc\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex+1_+Convert+a+Decimal+to+a+Fraction.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Convert a Decimal to a Fraction\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Rounding Decimals<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Rounding decimals involves approximating a decimal to the nearest whole number, tenth, hundredth, or other decimal place value. The process is similar to rounding whole numbers. If the digit right after the place value you&#8217;re rounding to is [latex]5[\/latex] or greater, you round up the last digit kept. If it&#8217;s less than [latex]5[\/latex], you keep the digit as it is.<\/p>\n<p>For example, if rounding [latex]3.78[\/latex] to the nearest tenth, you look at the hundredths place ([latex]8[\/latex]). Since [latex]8[\/latex] is greater than [latex]5[\/latex], you round up, and [latex]3.78[\/latex] becomes [latex]3.8[\/latex].<\/p>\n<\/div>\n<section class=\"textbox example\">Round [latex]18.379[\/latex] to the nearest:<\/p>\n<ol>\n<li>tenth<\/li>\n<li>whole number<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q414931\">Show Solution<\/button><\/p>\n<div id=\"q414931\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td>[latex]18.379[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Locate the tenths place and mark it with an arrow.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221520\/CNX_BMath_Figure_05_01_022_img-02.png\" alt=\"Decorative Image\" width=\"133\" height=\"75\" \/><\/td>\n<\/tr>\n<tr>\n<td>Underline the digit to the right of the tenths digit.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221521\/CNX_BMath_Figure_05_01_022_img-03.png\" alt=\"Decorative Image\" width=\"133\" height=\"77\" \/><\/td>\n<\/tr>\n<tr>\n<td>Because [latex]7[\/latex] is greater than or equal to [latex]5[\/latex], add [latex]1[\/latex] to the [latex]3[\/latex].<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221523\/CNX_BMath_Figure_05_01_022_img-04.png\" alt=\"Decorative Image\" width=\"133\" height=\"67\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite the number, deleting all digits to the right of the tenths place.<\/td>\n<td>[latex]18.4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>So, [latex]18.379[\/latex] rounded to the nearest tenth is [latex]18.4[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>[latex]18.379[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Locate the ones place and mark it with an arrow.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221526\/CNX_BMath_Figure_05_01_023_img-02.png\" alt=\"Decorative Image\" width=\"156\" height=\"78\" \/><\/td>\n<\/tr>\n<tr>\n<td>Underline the digit to the right of the ones place.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221527\/CNX_BMath_Figure_05_01_023_img-03.png\" alt=\"Decorative Image\" width=\"156\" height=\"76\" \/><\/td>\n<\/tr>\n<tr>\n<td>Since [latex]3[\/latex] is not greater than or equal to [latex]5[\/latex], do not add [latex]1[\/latex] to the [latex]8[\/latex].<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221528\/CNX_BMath_Figure_05_01_023_img-04.png\" alt=\"Decorative Image\" width=\"156\" height=\"67\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite the number, deleting all digits to the right of the ones place.<\/td>\n<td>[latex]18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>So [latex]18.379[\/latex] rounded to the nearest whole number is [latex]18[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video to see an example of how to round a number to several different place values.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/qu4Y9DGqXlk\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Examples_+Rounding+Decimals.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExamples: Rounding Decimals\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Locating and Ordering Decimals With a Number Line<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Locating and ordering decimals on a number line involves understanding the value of decimals and their relative positions. First, identify the whole numbers that the decimal falls between. Then, partition the space between these whole numbers into tenths, hundredths, or thousandths, as needed. Plot the decimals on the line at the corresponding position.<\/p>\n<p>\nFor example, [latex]0.5[\/latex] would fall halfway between [latex]0[\/latex] and [latex]1[\/latex]. When ordering decimals, start from the smallest (left on the number line) and move to the largest (right on the number line).<\/p>\n<\/div>\n<section class=\"textbox example\">Locate [latex]-0.74[\/latex] on a number line.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q669401\">Show Solution<\/button><\/p>\n<div id=\"q669401\" class=\"hidden-answer\" style=\"display: none\">\nThe decimal [latex]-0.74[\/latex] is equivalent to [latex]-{\\Large\\frac{74}{100}}[\/latex], so it is located between [latex]0[\/latex] and [latex]-1[\/latex]. On a number line, mark off and label the multiples of [latex]-0.10[\/latex] in the interval between [latex]0[\/latex] and [latex]-1[\/latex] ( [latex]-0.10[\/latex] , [latex]-0.20[\/latex] , etc.) and mark [latex]-0.74[\/latex] between [latex]-0.70[\/latex] and [latex]-0.80[\/latex], a little closer to [latex]-0.70[\/latex] .<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221503\/CNX_BMath_Figure_05_01_007_img.png\" alt=\"A number line with negative 1.00, negative 0.90, negative 0.80, negative 0.70, negative 0.60, negative 0.50, negative 0.40, negative 0.30, negative 0.20, negative 0.10, and 0.00 labeled. There is a red dot between negative 0.80 and negative 0.70 labeled as negative 0.74.\" width=\"900\" height=\"63\" \/><\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<p>In the next video, we show more examples of how to locate a decimal on the number line.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/F3LAKsOBdNA\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Example_+Identify+Decimals+on+the+Number+Line.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExample: Identify Decimals on the Number Line\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Order Decimals<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Ordering decimals involves arranging them from least to greatest or vice versa based on their numerical value. Begin by comparing the digits in the tenths place of each decimal. If the tenths place digits are the same, move to the hundredths place, then the thousandths place, and so on, until you find a difference. The decimal with the smaller digit in the first place where the decimals differ is the smaller decimal. If all places are the same, then the decimals are equal.<\/p>\n<\/div>\n<p>In the following video lesson we show how to order decimals using inequality notation by comparing place values, and by using fractions.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/fjO3fnt3ABA\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Decimal+Notation_+Ordering+Decimals.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDecimal Notation: Ordering Decimals\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Order the following decimals using [latex]<\\text{ or }\\text{>}[\/latex]:<\/p>\n<ol>\n<li>[latex]0.64[\/latex] ____ [latex]0.6[\/latex]<\/li>\n<li>[latex]0.83[\/latex] ____ [latex]0.803[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q895615\">Show Solution<\/button><\/p>\n<div id=\"q895615\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td style=\"width: 494.5px;\">\u00a0<\/td>\n<td style=\"width: 341.5px;\">[latex]0.64[\/latex] ____ [latex]0.6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 494.5px;\">Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of [latex]0.6[\/latex].<\/td>\n<td style=\"width: 341.5px;\">[latex]0.64[\/latex] ____ [latex]0.6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 494.5px;\">Compare the numbers to the right of the decimal point as if they were whole numbers.<\/td>\n<td style=\"width: 341.5px;\">[latex]64>60[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 494.5px;\">Order the numbers using the appropriate inequality sign.<\/td>\n<td style=\"width: 341.5px;\">\n[latex]0.64>0.60[\/latex]<br \/>\n[latex]0.64>0.6[\/latex]\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td style=\"width: 500.45px;\">\u00a0<\/td>\n<td style=\"width: 334.55px;\">[latex]0.83[\/latex] ____ [latex]0.803[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 500.45px;\">Check to see if both numbers have the same number of decimal places. They do not, so write one zero at the right of [latex]0.83[\/latex].<\/td>\n<td style=\"width: 334.55px;\">[latex]0.83[\/latex] ____ [latex]0.803[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 500.45px;\">Compare the numbers to the right of the decimal point as if they were whole numbers.<\/td>\n<td style=\"width: 334.55px;\">[latex]830>803[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 500.45px;\">Order the numbers using the appropriate inequality sign.<\/td>\n<td style=\"width: 334.55px;\">\n[latex]0.830>0.803[\/latex]<br \/>\n[latex]0.83>0.803[\/latex]\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Use [latex]<\\text{or}&gt[\/latex] to order: [latex]-0.1[\/latex] ____ [latex]- 0.8[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q12939\">Show Solution<\/button><\/p>\n<div id=\"q12939\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td style=\"width: 461.067px;\">\u00a0<\/td>\n<td style=\"width: 283.933px;\">[latex]-0.1[\/latex] ____ [latex]- 0.8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 461.067px;\">Write the numbers one under the other, lining up the decimal points.<\/td>\n<td style=\"width: 283.933px;\">\n[latex]-0.1[\/latex]<br \/>\n[latex]-0.8[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 461.067px;\">They have the same number of digits.<\/td>\n<td style=\"width: 283.933px;\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 461.067px;\">Since [latex]-1>-8,-1[\/latex] tenth is greater than [latex]-8[\/latex] tenths.<\/td>\n<td style=\"width: 283.933px;\">[latex]-0.1>-0.8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<h2>Determine Whether a Decimal is a Solution of an Equation<\/h2>\n<section class=\"textbox example\">Determine whether each of the following is a solution of [latex]x - 0.7=1.5[\/latex]<\/p>\n<ol style=\"list-style-type: decimal-leading-zero;\">\n<li>[latex]x=1[\/latex]<\/li>\n<li>[latex]x=-0.8[\/latex]<\/li>\n<li>[latex]x=2.2[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q895619\">Show Solution<\/button><\/p>\n<div id=\"q895619\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]x-0.7=1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{1}[\/latex] for [latex]x[\/latex].<\/td>\n<td>[latex]\\color{red}{1} - 0.7\\stackrel{?}{=}1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]0.3\\not=1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Since [latex]x=1[\/latex] does not result in a true equation, [latex]1[\/latex] is not a solution to the equation.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]x-0.7=1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{0.8}[\/latex] for [latex]x[\/latex].<\/td>\n<td>[latex]\\color{red}{0.8} - 0.7\\stackrel{?}{=}1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]-1.5\\not=1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Since [latex]x=-0.8[\/latex] does not result in a true equation, [latex]-0.8[\/latex] is not a solution to the equation.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]x-0.7=1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\color{red}{2.2}[\/latex] for [latex]x[\/latex].<\/td>\n<td>[latex]\\color{red}{2.2} - 0.7\\stackrel{?}{=}1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract.<\/td>\n<td>[latex]1.5=1.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Since [latex]x=2.2[\/latex] results in a true equation, [latex]2.2[\/latex] is a solution to the equation.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>Solve Equations with Decimals<\/h2>\n<section class=\"textbox example\">Solve: [latex]y+2.3=-4.7[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q908331\">Show Solution<\/button><\/p>\n<div id=\"q908331\" class=\"hidden-answer\" style=\"display: none\">We will use the Subtraction Property of Equality to isolate the variable.<\/p>\n<table>\n<tbody>\n<tr>\n<td colspan=\"2\">\u00a0<\/td>\n<td>[latex]y+2.3=-4.7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Subtract [latex]\\color{red}{2.3}[\/latex] from each side, to undo the addition.<\/td>\n<td>[latex]y+2.3\\color{red}{- 2.3}=-4.7\\color{red}{- 2.3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]y-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Check:<\/strong><\/td>\n<td>[latex]y+2.3=-4.7[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]y=\\color{red}{-7}[\/latex].<\/td>\n<td>[latex]\\color{red}{-7}+2.3\\stackrel{?}{=}-4.7[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-4.7=-4.7[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Since [latex]y=-7[\/latex] makes [latex]y+2.3=-4.7[\/latex] a true statement, we know we have found a solution to this equation.<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Solve: [latex]a - 4.75=-1.39[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q265677\">Show Solution<\/button><\/p>\n<div id=\"q265677\" class=\"hidden-answer\" style=\"display: none\">\nWe will use the Addition Property of Equality.<\/p>\n<table>\n<tbody>\n<tr>\n<td colspan=\"2\">\u00a0<\/td>\n<td>[latex]a-4.75=-1.39[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Add [latex]4.75[\/latex] to each side, to undo the subtraction.<\/td>\n<td>[latex]a-4.75+\\color{red}{4.75}=-1.39+\\color{red}{4.75}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Simplify.<\/td>\n<td>[latex]a=3.36[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Check:<\/strong><\/td>\n<td>[latex]a-4.75=-1.39[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]a=\\color{red}{3.36}[\/latex].<\/td>\n<td>[latex]\\color{red}{3.36}-4.75\\stackrel{?}{=}-1.39[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]-1.39=-1.39[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Since the result is a true statement, [latex]a=3.36[\/latex] is a solution to the equation.<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<p>In the following video, we show more examples of using the addition and subtraction properties of equality to solve linear equations that contain decimals.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/D8wKGlxf6bM\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Solving+One+Step+Equations+Using+Addition+and+Subtraction+(Decimals).txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSolving One Step Equations Using Addition and Subtraction (Decimals)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Solve: [latex]-4.8=0.8n[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q249570\">Show Solution<\/button><\/p>\n<div id=\"q249570\" class=\"hidden-answer\" style=\"display: none\">\nWe will use the Division Property of Equality.<br \/>\nUse the Properties of Equality to find a value for [latex]n[\/latex].<\/p>\n<table>\n<tbody>\n<tr style=\"height: 15.46875px;\">\n<td style=\"height: 15.46875px;\" colspan=\"2\">\u00a0<\/td>\n<td style=\"height: 15.46875px;\">[latex]-4.8=0.8n[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 43px;\">\n<td style=\"height: 43px;\" colspan=\"2\">We must divide both sides by [latex]0.8[\/latex] to isolate [latex]n[\/latex].<\/td>\n<td style=\"height: 43px;\">[latex]{\\Large\\frac{-4.8}{\\color{red}{0.8}}}={\\Large\\frac{0.8n}{\\color{red}{0.8}}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 25px;\">\n<td style=\"height: 25px;\" colspan=\"2\">Simplify.<\/td>\n<td style=\"height: 25px;\">[latex]-6=n[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"height: 22px;\"><strong>Check:<\/strong><\/td>\n<td style=\"height: 22px;\">[latex]-4.8=0.8n[\/latex]<\/td>\n<td style=\"height: 22px;\">\u00a0<\/td>\n<\/tr>\n<tr style=\"height: 28px;\">\n<td style=\"height: 28px;\">Substitute [latex]n=\\color{red}{-6}[\/latex].<\/td>\n<td style=\"height: 28px;\">[latex]-4.8\\stackrel{?}{=}0.8(\\color{red}{-6})[\/latex]<\/td>\n<td style=\"height: 28px;\">\u00a0<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px;\">\u00a0<\/td>\n<td style=\"height: 24px;\">[latex]-4.8=-4.8[\/latex]<\/td>\n<td style=\"height: 24px;\">\u00a0<\/td>\n<\/tr>\n<tr>\n<td>Since [latex]n=-6[\/latex] makes [latex]-4.8=0.8n[\/latex] a true statement, we know we have a solution.<\/td>\n<td>&nbsp;<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the next video to see how to solve another equation with decimals that requires division.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/paV-sXbRkHA\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Solve+a+One+Step+Equation+With+Decimals+by+Dividing.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a One Step Equation With Decimals by Dividing\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Solve: [latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex]<\/p>\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\">\nWe will use the Multiplication Property of Equality.<\/p>\n<table>\n<tbody>\n<tr>\n<td colspan=\"2\">\u00a0<\/td>\n<td>[latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Here, <em>p<\/em> is divided by [latex]\u22121.8[\/latex]. We must multiply by [latex]\u22121.8[\/latex] to isolate [latex]p[\/latex]<\/td>\n<td>[latex]\\color{red}{-1.8}({\\Large\\frac{p}{-1.8}})=\\color{red}{-1.8}(-6.5)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td colspan=\"2\">Multiply.<\/td>\n<td>[latex]p=11.7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Check:<\/strong><\/td>\n<td>[latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]{\\Large\\frac{\\color{red}{11.7}}{-1.8}}\\stackrel{?}{=}-6.5[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>\u00a0Substitute [latex]p=\\color{red}{11.7}[\/latex].<\/td>\n<td>[latex]-6.5=-6.5[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>A solution to [latex]{\\Large\\frac{p}{-1.8}}=-6.5[\/latex] is [latex]p=11.7[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<p>The following video shows an example of how to solve an equation with decimals that requires multiplication.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/q2s9GaEwSi8\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Solve+a+One+Step+Equation+With+Decimals+by+Multiplying.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a One Step Equation With Decimals by Multiplying\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Solving Equations By Clearing Decimals<\/h2>\n<section class=\"textbox example\">Solve: [latex]0.06x+0.02=0.25x - 1.5[\/latex]<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q190834\">Show Solution<\/button><\/p>\n<p class=\"p1\">\n<div id=\"q190834\" class=\"hidden-answer\" style=\"display: none\">\n<p>Look at the decimals and think of the equivalent fractions.<br \/>\n[latex]0.06=\\Large\\frac{6}{100}\\normalsize ,0.02=\\Large\\frac{2}{100}\\normalsize ,0.25=\\Large\\frac{25}{100}\\normalsize ,1.5=1\\Large\\frac{5}{10}[\/latex]<\/p>\n<p>Notice, the LCD is [latex]100[\/latex].<br \/>\nBy multiplying by the LCD we will clear the decimals.<\/p>\n<table>\n<tbody>\n<tr style=\"height: 23px;\">\n<td style=\"height: 23px;\">\u00a0<\/td>\n<td style=\"height: 23px;\">[latex]0.06x+0.02=0.25x-1.5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px;\">Multiply both sides by [latex]100[\/latex].<\/td>\n<td style=\"height: 24px;\">[latex]\\color{red}{100}(0.06x+0.02)=\\color{red}{100}(0.25x-1.5)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 32px;\">\n<td style=\"height: 32px;\">Distribute.<\/td>\n<td style=\"height: 32px;\">[latex]100(0.06x)+100(0.02)=100(0.25x)-100(1.5)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px;\">Multiply, and now no more decimals.<\/td>\n<td style=\"height: 14px;\">[latex]6x+2=25x-150[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 23px;\">\n<td style=\"height: 23px;\">Collect the variables to the right.<\/td>\n<td style=\"height: 23px;\">[latex]6x\\color{red}{-6x}+2=25x\\color{red}{-6x}-150[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px;\">Simplify.<\/td>\n<td style=\"height: 14px;\">[latex]2=19x-150[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 25px;\">\n<td style=\"height: 25px;\">Collect the constants to the left.<\/td>\n<td style=\"height: 25px;\">[latex]2\\color{red}{+150}=19x-150\\color{red}{+150}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px;\">Simplify.<\/td>\n<td style=\"height: 14px;\">[latex]152=19x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 43px;\">\n<td style=\"height: 43px;\">Divide by [latex]19[\/latex].<\/td>\n<td style=\"height: 43px;\">[latex]\\Large\\frac{152}{\\color{red}{19}}\\normalsize =\\Large\\frac{19x}{\\color{red}{19}}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px;\">Simplify.<\/td>\n<td style=\"height: 14px;\">[latex]8=x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px;\">Check: Let [latex]x=8[\/latex].<\/td>\n<td style=\"height: 14px;\">\u00a0<\/td>\n<\/tr>\n<tr style=\"height: 87.6426px;\">\n<td style=\"height: 87.6426px;\">\n<p>[latex]0.06(\\color{red}{8})+0.02=0.25(\\color{red}{8})-1.5[\/latex][latex]0.48+0.02=2.00-1.5[\/latex]<\/p>\n<p>[latex]0.50=0.50\\quad\\checkmark[\/latex]\n<\/td>\n<td style=\"height: 87.6426px;\">\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<p>The next example uses an equation that is typical of the ones we will see in the money applications. Notice that we will distribute the decimal first before we clear all decimals in the equation.<\/p>\n<section class=\"textbox example\">Solve: [latex]0.25x+0.05\\left(x+3\\right)=2.85[\/latex]<\/p>\n<p class=\"p1\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q777666\">Show Solution<\/button><\/p>\n<p class=\"p1\">\n<div id=\"q777666\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]0.25x+0.05(x+3)=2.85[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute first.<\/td>\n<td>[latex]0.25x+0.05x+0.15=2.85[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]0.30x+0.15=2.85[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>To clear decimals, multiply by [latex]100[\/latex].<\/td>\n<td>[latex]\\color{red}{100}(0.30x+0.15)=\\color{red}{100}(2.85)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]30x+15=285[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract [latex]15[\/latex] from both sides.<\/td>\n<td>[latex]30x+15\\color{red}{-15}=285\\color{red}{-15}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]30x=270[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide by [latex]30[\/latex].<\/td>\n<td>[latex]\\Large\\frac{30x}{\\color{red}{30}}\\normalsize =\\Large\\frac{270}{\\color{red}{30}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x=9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check: Let [latex]x=9[\/latex].<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>\n<p>[latex]0.25x+0.05(x+3)=2.85[\/latex][latex]0.25(\\color{red}{9})+0.05(\\color{red}{9}+3)\\stackrel{\\text{?}}{=}2.85[\/latex]<\/p>\n<p>[latex]2.25+0.05(12)\\stackrel{\\text{?}}{=}2.85[\/latex]<br \/>\n[latex]2.85=2.85\\quad\\checkmark[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<p>In the following video we present another example of how to solve an equation that contains decimals and variable terms on both sides of the equal sign.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/pZWTJvua-P8\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Solve+a+Linear+Equation+With+Decimals+and+Variables+on+Both+Sides.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Solve a Linear Equation With Decimals and Variables on Both Sides\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":54,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1061"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":40,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1061\/revisions"}],"predecessor-version":[{"id":15194,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1061\/revisions\/15194"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/54"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/1061\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=1061"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=1061"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=1061"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=1061"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}