{"id":3914,"date":"2020-02-11T03:28:03","date_gmt":"2020-02-11T03:28:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3914"},"modified":"2020-04-27T16:13:04","modified_gmt":"2020-04-27T16:13:04","slug":"rounding-decimals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/mathforliberalartscorequisite\/chapter\/rounding-decimals\/","title":{"raw":"Rounding Decimals","rendered":"Rounding Decimals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Round a decimal to a given place value<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the United States, gasoline prices are usually written with the decimal part as thousandths of a dollar. For example, a gas station might post the price of unleaded gas at [latex]\\$3.279[\/latex] per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay [latex]$3.28[\/latex] , because the final price would be rounded to the nearest cent. In a lesson about whole numbers, we learned that we can round numbers to get an approximate value when the exact value is not needed.\r\n\r\nSuppose we wanted to round [latex]$2.72[\/latex] to the nearest dollar. Is it closer to [latex]$2[\/latex] or to [latex]$3[\/latex]? What if we wanted to round [latex]$2.72[\/latex] to the nearest ten cents; is it closer to [latex]$2.70[\/latex] or to [latex]$2.80[\/latex]? The number lines in the image below can help us answer those questions.\r\n\r\n(a) We see that [latex]2.72[\/latex] is closer to [latex]3[\/latex] than to [latex]2[\/latex]. So, [latex]2.72[\/latex] rounded to the nearest whole number is [latex]3[\/latex].\r\n\r\n(b) We see that [latex]2.72[\/latex] is closer to [latex]2.70[\/latex] than [latex]2.80[\/latex]. So we say that [latex]2.72[\/latex] rounded to the nearest tenth is [latex]2.7[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221512\/CNX_BMath_Figure_05_01_012.png\" alt=\"In part a, a number line is shown with 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9 and 3. There is a dot between 2.7 and 2.8 labeled as 2.72. In part b, a number line is shown with 2.70, 2.71, 2.72, 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, and 2.80. There is a dot at 2.72. \" \/>\r\nCan we round decimals without number lines? Yes! We use a method based on the one we used to round whole numbers.\r\n<div class=\"textbox shaded\">\r\n<h3>Round a decimal.<\/h3>\r\n<ol class=\"stepwise\">\r\n \t<li>Locate the given place value and mark it with an arrow.<\/li>\r\n \t<li>Underline the digit to the right of the given place value.<\/li>\r\n \t<li>Is this digit greater than or equal to [latex]5?[\/latex]\r\n<ul>\r\n \t<li>Yes - add [latex]1[\/latex] to the digit in the given place value.<\/li>\r\n \t<li>No - do not change the digit in the given place value<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Rewrite the number, removing all digits to the right of the given place value.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n\r\n<span style=\"color: #3366ff;\">Tip:\u00a0 If the digit to the right of your given place value is 5 or above, give your given place value a shove.\u00a0 If the digit to the right of your given place value is 4 or less, let your given place value rest.<\/span>\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nRound [latex]18.379[\/latex] to the nearest hundredth.\r\n\r\nSolution\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 hundredths place and mark it with an arrow.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221515\/CNX_BMath_Figure_05_01_021_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Underline the digit to the right of the [latex]7[\/latex].<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221516\/CNX_BMath_Figure_05_01_021_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Because [latex]9[\/latex] is greater than or equal to [latex]5[\/latex], add [latex]1[\/latex] to the [latex]7[\/latex].<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221517\/CNX_BMath_Figure_05_01_021_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Rewrite the number, deleting all digits to the right of the hundredths place.<\/td>\r\n<td>[latex]18.38[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]18.38[\/latex] is [latex]18.379[\/latex] rounded to the nearest hundredth.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146242[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nRound [latex]18.379[\/latex] to the nearest\r\n<ol>\r\n \t<li>tenth<\/li>\r\n \t<li>whole number\r\n[reveal-answer q=\"414963\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"414963\"]<\/li>\r\n<\/ol>\r\nSolution\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>1. Round [latex]18.379[\/latex] to the nearest tenth.<\/td>\r\n<\/tr>\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 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=\".\" \/><\/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 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=\".\" \/><\/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 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=\".\" \/><\/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<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>2. Round [latex]18.379[\/latex] to the nearest whole number.<\/td>\r\n<\/tr>\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 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=\".\" \/><\/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 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=\".\" \/><\/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 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=\".\" \/><\/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<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146244[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video to see an example of how to round a number to several different place values.\r\n\r\nhttps:\/\/youtu.be\/qu4Y9DGqXlk","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Round a decimal to a given place value<\/li>\n<\/ul>\n<\/div>\n<p>In the United States, gasoline prices are usually written with the decimal part as thousandths of a dollar. For example, a gas station might post the price of unleaded gas at [latex]\\$3.279[\/latex] per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay [latex]$3.28[\/latex] , because the final price would be rounded to the nearest cent. In a lesson about whole numbers, we learned that we can round numbers to get an approximate value when the exact value is not needed.<\/p>\n<p>Suppose we wanted to round [latex]$2.72[\/latex] to the nearest dollar. Is it closer to [latex]$2[\/latex] or to [latex]$3[\/latex]? What if we wanted to round [latex]$2.72[\/latex] to the nearest ten cents; is it closer to [latex]$2.70[\/latex] or to [latex]$2.80[\/latex]? The number lines in the image below can help us answer those questions.<\/p>\n<p>(a) We see that [latex]2.72[\/latex] is closer to [latex]3[\/latex] than to [latex]2[\/latex]. So, [latex]2.72[\/latex] rounded to the nearest whole number is [latex]3[\/latex].<\/p>\n<p>(b) We see that [latex]2.72[\/latex] is closer to [latex]2.70[\/latex] than [latex]2.80[\/latex]. So we say that [latex]2.72[\/latex] rounded to the nearest tenth is [latex]2.7[\/latex].<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221512\/CNX_BMath_Figure_05_01_012.png\" alt=\"In part a, a number line is shown with 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9 and 3. There is a dot between 2.7 and 2.8 labeled as 2.72. In part b, a number line is shown with 2.70, 2.71, 2.72, 2.73, 2.74, 2.75, 2.76, 2.77, 2.78, 2.79, and 2.80. There is a dot at 2.72.\" \/><br \/>\nCan we round decimals without number lines? Yes! We use a method based on the one we used to round whole numbers.<\/p>\n<div class=\"textbox shaded\">\n<h3>Round a decimal.<\/h3>\n<ol class=\"stepwise\">\n<li>Locate the given place value and mark it with an arrow.<\/li>\n<li>Underline the digit to the right of the given place value.<\/li>\n<li>Is this digit greater than or equal to [latex]5?[\/latex]\n<ul>\n<li>Yes &#8211; add [latex]1[\/latex] to the digit in the given place value.<\/li>\n<li>No &#8211; do not change the digit in the given place value<\/li>\n<\/ul>\n<\/li>\n<li>Rewrite the number, removing all digits to the right of the given place value.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<p><span style=\"color: #3366ff;\">Tip:\u00a0 If the digit to the right of your given place value is 5 or above, give your given place value a shove.\u00a0 If the digit to the right of your given place value is 4 or less, let your given place value rest.<\/span><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Round [latex]18.379[\/latex] to the nearest hundredth.<\/p>\n<p>Solution<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]18.379[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Locate the hundredths place and mark it with an arrow.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221515\/CNX_BMath_Figure_05_01_021_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Underline the digit to the right of the [latex]7[\/latex].<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221516\/CNX_BMath_Figure_05_01_021_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Because [latex]9[\/latex] is greater than or equal to [latex]5[\/latex], add [latex]1[\/latex] to the [latex]7[\/latex].<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221517\/CNX_BMath_Figure_05_01_021_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Rewrite the number, deleting all digits to the right of the hundredths place.<\/td>\n<td>[latex]18.38[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]18.38[\/latex] is [latex]18.379[\/latex] rounded to the nearest hundredth.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146242\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146242&theme=oea&iframe_resize_id=ohm146242&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Round [latex]18.379[\/latex] to the nearest<\/p>\n<ol>\n<li>tenth<\/li>\n<li>whole number\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q414963\">Show Solution<\/span><\/p>\n<div id=\"q414963\" class=\"hidden-answer\" style=\"display: none\"><\/li>\n<\/ol>\n<p>Solution<\/p>\n<table>\n<tbody>\n<tr>\n<td>1. Round [latex]18.379[\/latex] to the nearest tenth.<\/td>\n<\/tr>\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 decoding=\"async\" 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=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Underline the digit to the right of the tenths digit.<\/td>\n<td><img decoding=\"async\" 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=\".\" \/><\/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 decoding=\"async\" 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=\".\" \/><\/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<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<td>2. Round [latex]18.379[\/latex] to the nearest whole number.<\/td>\n<\/tr>\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 decoding=\"async\" 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=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Underline the digit to the right of the ones place.<\/td>\n<td><img decoding=\"async\" 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=\".\" \/><\/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 decoding=\"async\" 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=\".\" \/><\/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<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146244\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146244&theme=oea&iframe_resize_id=ohm146244&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video to see an example of how to round a number to several different place values.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Examples:  Rounding Decimals\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/qu4Y9DGqXlk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3914\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 146239, 146242, 146244. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Examples: Rounding Decimals. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/qu4Y9DGqXlk\">https:\/\/youtu.be\/qu4Y9DGqXlk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":25777,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"cc\",\"description\":\"Examples: Rounding Decimals\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/qu4Y9DGqXlk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 146239, 146242, 146244\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + 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