{"id":3921,"date":"2020-02-11T03:36:41","date_gmt":"2020-02-11T03:36:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3921"},"modified":"2021-02-05T23:57:00","modified_gmt":"2021-02-05T23:57:00","slug":"dividing-decimals","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/dividing-decimals\/","title":{"raw":"Dividing Decimals","rendered":"Dividing Decimals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Divide a decimal by a whole number<\/li>\r\n \t<li>Divide a decimal by another decimal<\/li>\r\n \t<li>Divide a whole number by a decimal<\/li>\r\n<\/ul>\r\n<\/div>\r\nJust as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.\r\n\r\nTo understand decimal division, let\u2019s consider the multiplication problem\r\n<p style=\"text-align: center\">[latex]\\left(0.2\\right)\\left(4\\right)=0.8[\/latex]<\/p>\r\nRemember, a multiplication problem can be rephrased as a division problem. So we can write\r\n<p style=\"text-align: center\">[latex]0.8\\div 4=0.2[\/latex]<\/p>\r\nWe can think of this as \"If we divide 8 tenths into four groups, how many are in each group?\" The number line below\u00a0shows that there are four groups of two-tenths in eight-tenths. So [latex]0.8\\div 4=0.2[\/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\/24221612\/CNX_BMath_Figure_05_02_001.png\" alt=\"A number line is shown with 0, 0.2, 0.4, 0.6, 0.8, and 1. There are braces showing a distance of 0.2 between each adjacent set of 2 numbers.\" \/>\r\nUsing long division notation, we would write\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221614\/CNX_BMath_Figure_05_02_004_img.png\" alt=\"A division problem is shown. 0.8 is on the inside of the division sign, 4 is on the outside. Above the division sign is 0.2.\" \/>\r\nNotice that the decimal point in the quotient is directly above the decimal point in the dividend.\r\n\r\nTo divide a decimal by a whole number, we place the decimal point in the quotient above the decimal point in the dividend and then divide as usual. Sometimes we need to use extra zeros at the end of the dividend to keep dividing until there is no remainder.\r\n<div class=\"textbox shaded\">\r\n<h3>Divide a decimal by a whole number.<\/h3>\r\n<ol id=\"eip-id1168468531099\" class=\"stepwise\">\r\n \t<li>Write as long division, placing the decimal point in the quotient above the decimal point in the dividend.<\/li>\r\n \t<li>Divide as usual.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide: [latex]0.12\\div 3[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168469450973\" class=\"unnumbered unstyled\" summary=\"The top line says 0.12 divided by 3. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]0.12\\div 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write as long division, placing the decimal point in the quotient above the decimal point in the dividend.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221614\/CNX_BMath_Figure_05_02_024_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide as usual. Since [latex]3[\/latex] does not go into [latex]0[\/latex] or [latex]1[\/latex] we use zeros as placeholders.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221615\/CNX_BMath_Figure_05_02_024_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]0.12\\div 3=0.04[\/latex]<\/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]146600[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video to see another example of how to divide a decimal by a whole number.\r\n\r\nhttps:\/\/youtu.be\/LkbSiL6uvtU\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nIn everyday life, we divide whole numbers into decimals\u2014money\u2014to find the price of one item. For example, suppose a case of [latex]24[\/latex] water bottles cost [latex]$3.99[\/latex]. To find the price per water bottle, we would divide [latex]$3.99[\/latex] by [latex]24[\/latex], and round the answer to the nearest cent (hundredth).\r\n\r\nDivide: [latex]$3.99\\div 24[\/latex]\r\n[reveal-answer q=\"929706\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"929706\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469496456\" class=\"unnumbered unstyled\" summary=\"The top line says &gt;.99 divided by 24. The first step says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]$3.99\\div 24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Place the decimal point in the quotient above the decimal point in the dividend.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221616\/CNX_BMath_Figure_05_02_025_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide as usual. When do we stop? Since this division involves money, we round it to the nearest cent (hundredth). To do this, we must carry the division to the thousandths place.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221618\/CNX_BMath_Figure_05_02_025_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Round to the nearest cent.<\/td>\r\n<td>[latex]$0.166\\approx $0.17[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]$3.99\\div 24\\approx $0.17[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis means the price per bottle is [latex]17[\/latex] cents.\r\n\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]145993[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Divide a Decimal by Another Decimal<\/h2>\r\nSo far, we have divided a decimal by a whole number. What happens when we divide a decimal by another decimal? Let\u2019s look at the same multiplication problem we looked at earlier, but in a different way.\r\n<p style=\"text-align: center\">[latex]\\left(0.2\\right)\\left(4\\right)=0.8[\/latex]<\/p>\r\nRemember, again, that a multiplication problem can be rephrased as a division problem. This time we ask, \"How many times does [latex]0.2[\/latex] go into [latex]0.8\\text{?\"}[\/latex] Because [latex]\\left(0.2\\right)\\left(4\\right)=0.8[\/latex], we can say that [latex]0.2[\/latex] goes into [latex]0.8[\/latex] four times. This means that [latex]0.8[\/latex] divided by [latex]0.2[\/latex] is [latex]4[\/latex].\r\n<p style=\"text-align: center\">[latex]0.8\\div 0.2=4[\/latex]<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221612\/CNX_BMath_Figure_05_02_001.png\" alt=\"A number line is shown with 0, 0.2, 0.4, 0.6, 0.8, and 1. There are braces showing a distance of 0.2 between each adjacent set of 2 numbers.\" \/>\r\nWe would get the same answer, [latex]4[\/latex], if we divide [latex]8[\/latex] by [latex]2[\/latex], both whole numbers. Why is this so? Let\u2019s think about the division problem as a fraction.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}{\\Large\\frac{0.8}{0.2}}\\\\ \\\\ {\\Large\\frac{\\left(0.8\\right)10}{\\left(0.2\\right)10}}\\\\ \\\\ {\\Large\\frac{8}{2}}\\\\ \\\\ 4\\end{array}[\/latex]<\/p>\r\nWe multiplied the numerator and denominator by [latex]10[\/latex] and ended up just dividing [latex]8[\/latex] by [latex]4[\/latex]. To divide decimals, we multiply both the numerator and denominator by the same power of [latex]10[\/latex] to make the denominator a whole number. Because of the Equivalent Fractions Property, we haven\u2019t changed the value of the fraction. The effect is to move the decimal points in the numerator and denominator the same number of places to the right.\r\n\r\nWe use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.\r\n\r\nIt may help to review the vocabulary for division:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221619\/CNX_BMath_Figure_05_02_026_img.png\" alt=\"a divided by b is shown with a labeled as the dividend and b labeled as the divisor. Then a over b is shown with a labeled as the divided and b labeled as the divisor. Then a is shown inside a division problem with b on the outside with a labeled as the dividend and b labeled as the divisor.\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Divide decimal numbers<\/h3>\r\n<ol id=\"eip-id1168468296206\" class=\"stepwise\">\r\n \t<li>Determine the sign of the quotient.<\/li>\r\n \t<li>Make the divisor a whole number by moving the decimal point all the way to the right. Move the decimal point in the dividend the same number of places to the right, writing zeros as needed.<\/li>\r\n \t<li>Divide. Place the decimal point in the quotient above the decimal point in the dividend.<\/li>\r\n \t<li>Write the quotient with the appropriate sign.<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide: [latex]-2.89\\div \\text{(}3.4\\text{)}[\/latex]\r\n[reveal-answer q=\"15844\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"15844\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468454903\" class=\"unnumbered unstyled\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td>Determine the sign of the quotient.<\/td>\r\n<td>The quotient will be negative.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Make the divisor the whole number by 'moving' the decimal point all the way to the right. 'Move' the decimal point in the dividend the same number of places to the right.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221621\/CNX_BMath_Figure_05_02_027_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide. Place the decimal point in the quotient above the decimal point in the dividend. Add zeros as needed until the remainder is zero.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221622\/CNX_BMath_Figure_05_02_027_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the quotient with the appropriate sign.<\/td>\r\n<td>[latex]-2.89\\div \\left(3.4\\right)=-0.85[\/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]146601[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide: [latex]-25.65\\div \\text{(}-0.06\\text{)}[\/latex]\r\n[reveal-answer q=\"177883\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"177883\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469481183\" class=\"unnumbered unstyled\" summary=\"The first line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-25.65\\div \\left(-0.06\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are the same.<\/td>\r\n<td>The quotient is positive.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Make the divisor a whole number by 'moving' the decimal point all the way to the right.\r\n\r\n'Move' the decimal point in the dividend the same number of places.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221623\/CNX_BMath_Figure_05_02_028_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.\r\n\r\nPlace the decimal point in the quotient above the decimal point in the dividend.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221624\/CNX_BMath_Figure_05_02_028_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the quotient with the appropriate sign.<\/td>\r\n<td>[latex]-25.65\\div \\left(-0.06\\right)=427.5[\/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]146604[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next example we show how to divide two decimals that include negatives.\r\n\r\nhttps:\/\/youtu.be\/o2ad4UzTlj0\r\n\r\nNow we will divide a whole number by a decimal number.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide: [latex]4\\div 0.05[\/latex]\r\n[reveal-answer q=\"616625\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"616625\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468408069\" class=\"unnumbered unstyled\" summary=\"The first line says 4 divided by 0.05. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4\\div 0.05[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The signs are the same.<\/td>\r\n<td>The quotient is positive.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Make the divisor a whole number by 'moving' the decimal point all the way to the right.\r\n\r\nMove the decimal point in the dividend the same number of places, adding zeros as needed.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221625\/CNX_BMath_Figure_05_02_029_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.\r\n\r\nPlace the decimal point in the quotient above the decimal point in the dividend.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221627\/CNX_BMath_Figure_05_02_029_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the quotient with the appropriate sign.<\/td>\r\n<td>[latex]4\\div 0.05=80[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can relate this example to money. How many nickels are there in four dollars? Because [latex]4\\div 0.05=80[\/latex], there are [latex]80[\/latex] nickels in [latex]$4[\/latex].\r\n\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]146607[\/ohm_question]\r\n\r\n<\/div>\r\nThe following example shows how to divide a whole number by a decimal using base ten blocks.\r\n\r\nhttps:\/\/youtu.be\/LmWzhGvDt58","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Divide a decimal by a whole number<\/li>\n<li>Divide a decimal by another decimal<\/li>\n<li>Divide a whole number by a decimal<\/li>\n<\/ul>\n<\/div>\n<p>Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.<\/p>\n<p>To understand decimal division, let\u2019s consider the multiplication problem<\/p>\n<p style=\"text-align: center\">[latex]\\left(0.2\\right)\\left(4\\right)=0.8[\/latex]<\/p>\n<p>Remember, a multiplication problem can be rephrased as a division problem. So we can write<\/p>\n<p style=\"text-align: center\">[latex]0.8\\div 4=0.2[\/latex]<\/p>\n<p>We can think of this as &#8220;If we divide 8 tenths into four groups, how many are in each group?&#8221; The number line below\u00a0shows that there are four groups of two-tenths in eight-tenths. So [latex]0.8\\div 4=0.2[\/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\/24221612\/CNX_BMath_Figure_05_02_001.png\" alt=\"A number line is shown with 0, 0.2, 0.4, 0.6, 0.8, and 1. There are braces showing a distance of 0.2 between each adjacent set of 2 numbers.\" \/><br \/>\nUsing long division notation, we would write<\/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\/24221614\/CNX_BMath_Figure_05_02_004_img.png\" alt=\"A division problem is shown. 0.8 is on the inside of the division sign, 4 is on the outside. Above the division sign is 0.2.\" \/><br \/>\nNotice that the decimal point in the quotient is directly above the decimal point in the dividend.<\/p>\n<p>To divide a decimal by a whole number, we place the decimal point in the quotient above the decimal point in the dividend and then divide as usual. Sometimes we need to use extra zeros at the end of the dividend to keep dividing until there is no remainder.<\/p>\n<div class=\"textbox shaded\">\n<h3>Divide a decimal by a whole number.<\/h3>\n<ol id=\"eip-id1168468531099\" class=\"stepwise\">\n<li>Write as long division, placing the decimal point in the quotient above the decimal point in the dividend.<\/li>\n<li>Divide as usual.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide: [latex]0.12\\div 3[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168469450973\" class=\"unnumbered unstyled\" summary=\"The top line says 0.12 divided by 3. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]0.12\\div 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write as long division, placing the decimal point in the quotient above the decimal point in the dividend.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221614\/CNX_BMath_Figure_05_02_024_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Divide as usual. Since [latex]3[\/latex] does not go into [latex]0[\/latex] or [latex]1[\/latex] we use zeros as placeholders.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221615\/CNX_BMath_Figure_05_02_024_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]0.12\\div 3=0.04[\/latex]<\/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=\"ohm146600\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146600&theme=oea&iframe_resize_id=ohm146600&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the following video to see another example of how to divide a decimal by a whole number.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Example:  Dividing a Decimal by a Whole Number\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/LkbSiL6uvtU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>In everyday life, we divide whole numbers into decimals\u2014money\u2014to find the price of one item. For example, suppose a case of [latex]24[\/latex] water bottles cost [latex]$3.99[\/latex]. To find the price per water bottle, we would divide [latex]$3.99[\/latex] by [latex]24[\/latex], and round the answer to the nearest cent (hundredth).<\/p>\n<p>Divide: [latex]$3.99\\div 24[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q929706\">Show Solution<\/span><\/p>\n<div id=\"q929706\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469496456\" class=\"unnumbered unstyled\" summary=\"The top line says &gt;.99 divided by 24. The first step says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]$3.99\\div 24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Place the decimal point in the quotient above the decimal point in the dividend.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221616\/CNX_BMath_Figure_05_02_025_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Divide as usual. When do we stop? Since this division involves money, we round it to the nearest cent (hundredth). To do this, we must carry the division to the thousandths place.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221618\/CNX_BMath_Figure_05_02_025_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Round to the nearest cent.<\/td>\n<td>[latex]$0.166\\approx $0.17[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]$3.99\\div 24\\approx $0.17[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This means the price per bottle is [latex]17[\/latex] cents.<\/p>\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=\"ohm145993\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=145993&theme=oea&iframe_resize_id=ohm145993&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Divide a Decimal by Another Decimal<\/h2>\n<p>So far, we have divided a decimal by a whole number. What happens when we divide a decimal by another decimal? Let\u2019s look at the same multiplication problem we looked at earlier, but in a different way.<\/p>\n<p style=\"text-align: center\">[latex]\\left(0.2\\right)\\left(4\\right)=0.8[\/latex]<\/p>\n<p>Remember, again, that a multiplication problem can be rephrased as a division problem. This time we ask, &#8220;How many times does [latex]0.2[\/latex] go into [latex]0.8\\text{?\"}[\/latex] Because [latex]\\left(0.2\\right)\\left(4\\right)=0.8[\/latex], we can say that [latex]0.2[\/latex] goes into [latex]0.8[\/latex] four times. This means that [latex]0.8[\/latex] divided by [latex]0.2[\/latex] is [latex]4[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]0.8\\div 0.2=4[\/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\/24221612\/CNX_BMath_Figure_05_02_001.png\" alt=\"A number line is shown with 0, 0.2, 0.4, 0.6, 0.8, and 1. There are braces showing a distance of 0.2 between each adjacent set of 2 numbers.\" \/><br \/>\nWe would get the same answer, [latex]4[\/latex], if we divide [latex]8[\/latex] by [latex]2[\/latex], both whole numbers. Why is this so? Let\u2019s think about the division problem as a fraction.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}{\\Large\\frac{0.8}{0.2}}\\\\ \\\\ {\\Large\\frac{\\left(0.8\\right)10}{\\left(0.2\\right)10}}\\\\ \\\\ {\\Large\\frac{8}{2}}\\\\ \\\\ 4\\end{array}[\/latex]<\/p>\n<p>We multiplied the numerator and denominator by [latex]10[\/latex] and ended up just dividing [latex]8[\/latex] by [latex]4[\/latex]. To divide decimals, we multiply both the numerator and denominator by the same power of [latex]10[\/latex] to make the denominator a whole number. Because of the Equivalent Fractions Property, we haven\u2019t changed the value of the fraction. The effect is to move the decimal points in the numerator and denominator the same number of places to the right.<\/p>\n<p>We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.<\/p>\n<p>It may help to review the vocabulary for division:<\/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\/24221619\/CNX_BMath_Figure_05_02_026_img.png\" alt=\"a divided by b is shown with a labeled as the dividend and b labeled as the divisor. Then a over b is shown with a labeled as the divided and b labeled as the divisor. Then a is shown inside a division problem with b on the outside with a labeled as the dividend and b labeled as the divisor.\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Divide decimal numbers<\/h3>\n<ol id=\"eip-id1168468296206\" class=\"stepwise\">\n<li>Determine the sign of the quotient.<\/li>\n<li>Make the divisor a whole number by moving the decimal point all the way to the right. Move the decimal point in the dividend the same number of places to the right, writing zeros as needed.<\/li>\n<li>Divide. Place the decimal point in the quotient above the decimal point in the dividend.<\/li>\n<li>Write the quotient with the appropriate sign.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide: [latex]-2.89\\div \\text{(}3.4\\text{)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q15844\">Show Solution<\/span><\/p>\n<div id=\"q15844\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468454903\" class=\"unnumbered unstyled\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td>Determine the sign of the quotient.<\/td>\n<td>The quotient will be negative.<\/td>\n<\/tr>\n<tr>\n<td>Make the divisor the whole number by &#8216;moving&#8217; the decimal point all the way to the right. &#8216;Move&#8217; the decimal point in the dividend the same number of places to the right.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221621\/CNX_BMath_Figure_05_02_027_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Divide. Place the decimal point in the quotient above the decimal point in the dividend. Add zeros as needed until the remainder is zero.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221622\/CNX_BMath_Figure_05_02_027_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write the quotient with the appropriate sign.<\/td>\n<td>[latex]-2.89\\div \\left(3.4\\right)=-0.85[\/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=\"ohm146601\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146601&theme=oea&iframe_resize_id=ohm146601&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>Divide: [latex]-25.65\\div \\text{(}-0.06\\text{)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q177883\">Show Solution<\/span><\/p>\n<div id=\"q177883\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469481183\" class=\"unnumbered unstyled\" summary=\"The first line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-25.65\\div \\left(-0.06\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The signs are the same.<\/td>\n<td>The quotient is positive.<\/td>\n<\/tr>\n<tr>\n<td>Make the divisor a whole number by &#8216;moving&#8217; the decimal point all the way to the right.<\/p>\n<p>&#8216;Move&#8217; the decimal point in the dividend the same number of places.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221623\/CNX_BMath_Figure_05_02_028_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/p>\n<p>Place the decimal point in the quotient above the decimal point in the dividend.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221624\/CNX_BMath_Figure_05_02_028_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write the quotient with the appropriate sign.<\/td>\n<td>[latex]-25.65\\div \\left(-0.06\\right)=427.5[\/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=\"ohm146604\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146604&theme=oea&iframe_resize_id=ohm146604&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next example we show how to divide two decimals that include negatives.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Dividing Signed Decimals\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/o2ad4UzTlj0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Now we will divide a whole number by a decimal number.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide: [latex]4\\div 0.05[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q616625\">Show Solution<\/span><\/p>\n<div id=\"q616625\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468408069\" class=\"unnumbered unstyled\" summary=\"The first line says 4 divided by 0.05. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]4\\div 0.05[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The signs are the same.<\/td>\n<td>The quotient is positive.<\/td>\n<\/tr>\n<tr>\n<td>Make the divisor a whole number by &#8216;moving&#8217; the decimal point all the way to the right.<\/p>\n<p>Move the decimal point in the dividend the same number of places, adding zeros as needed.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221625\/CNX_BMath_Figure_05_02_029_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/p>\n<p>Place the decimal point in the quotient above the decimal point in the dividend.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221627\/CNX_BMath_Figure_05_02_029_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Write the quotient with the appropriate sign.<\/td>\n<td>[latex]4\\div 0.05=80[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can relate this example to money. How many nickels are there in four dollars? Because [latex]4\\div 0.05=80[\/latex], there are [latex]80[\/latex] nickels in [latex]$4[\/latex].<\/p>\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=\"ohm146607\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146607&theme=oea&iframe_resize_id=ohm146607&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following example shows how to divide a whole number by a decimal using base ten blocks.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex: Determine the Quotient of a Whole Number and Decimal using Base Ten Blocks\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/LmWzhGvDt58?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-3921\">\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 146607, 146604, 146601,146600. <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>Ex: Dividing a Decimal by a Whole Number. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/LkbSiL6uvtU\">https:\/\/youtu.be\/LkbSiL6uvtU<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Dividing Signed Decimals. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/o2ad4UzTlj0\">https:\/\/youtu.be\/o2ad4UzTlj0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Determine the Quotient of a Whole Number and Decimal using Base Ten Blocks. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/LmWzhGvDt58\">https:\/\/youtu.be\/LmWzhGvDt58<\/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":9,"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\":\"Ex: Dividing a Decimal by a Whole Number\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/LkbSiL6uvtU\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Dividing Signed Decimals\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/o2ad4UzTlj0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Determine the Quotient of a Whole Number and Decimal using Base Ten Blocks\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/LmWzhGvDt58\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 146607, 146604, 146601,146600\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3921","chapter","type-chapter","status-web-only","hentry"],"part":377,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3921","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3921\/revisions"}],"predecessor-version":[{"id":3922,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3921\/revisions\/3922"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/377"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3921\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=3921"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=3921"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=3921"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=3921"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}