{"id":325,"date":"2018-04-16T23:52:05","date_gmt":"2018-04-16T23:52:05","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/?post_type=chapter&#038;p=325"},"modified":"2024-04-26T22:02:47","modified_gmt":"2024-04-26T22:02:47","slug":"multiplying-and-dividing-decimals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/chapter\/multiplying-and-dividing-decimals\/","title":{"raw":"Multiplying and Dividing Decimals","rendered":"Multiplying and Dividing Decimals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use multiplication and division when evaluating expressions with decimals&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:13185,&quot;3&quot;:[null,0],&quot;10&quot;:0,&quot;11&quot;:4,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Use multiplication and division when evaluating expressions with decimals<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nMultiplying and dividing decimals can become complex, but we're going to start with some basic multiplication by ten to warm-up.\r\n<h2>Multiply by Powers of [latex]10[\/latex]<\/h2>\r\nIn many fields, especially in the sciences, it is common to multiply decimals by powers of [latex]10[\/latex]. Let\u2019s see what happens when we multiply [latex]1.9436[\/latex] by some powers of [latex]10[\/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\/24221557\/CNX_BMath_Figure_05_02_022_img.png\" alt=\"The top row says 1.9436 times 10, then 1.9436 times 100, then 1.9436 times 1000. Below each is a vertical multiplication problem. These show that 1.9436 times 10 is 19.4360, 1.9436 times 100 is 194.3600, and 1.9436 times 1000 is 1943.6000.\" data-media-type=\"image\/png\" \/>\r\nLook at the results without the final zeros. Do you notice a pattern?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}1.9436\\left(10\\right)\\hfill &amp; =&amp; 19.436\\hfill \\\\ 1.9436\\left(100\\right)\\hfill &amp; =&amp; 194.36\\hfill \\\\ 1.9436\\left(1000\\right)\\hfill &amp; =&amp; 1943.6\\hfill \\end{array}[\/latex]<\/p>\r\nThe number of places that the decimal point moved is the same as the number of zeros in the power of ten. The table below\u00a0summarizes the results.\r\n<table id=\"fs-id3415452\" summary=\"A table is shown with five rows and three columns. The first row, which is the header row, reads 'Multiply by' in the first column, 'Number of zeros' in the second column, and 'Number of places decimal point moves' in the third column. The first column shows 10, 100, 1,000, and 10,000. The second row shows 1, 2, 3, 4. The third row shows '1 place to the right,' '2 places to the right,' '3 places to the right,' and '4 places to the right'.\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-align=\"center\">Multiply by<\/th>\r\n<th data-align=\"center\">Number of zeros<\/th>\r\n<th data-align=\"center\">Number of places decimal point moves<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]10[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]1[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]1[\/latex] place to the right<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]100[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]2[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]2[\/latex] places to the right<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]1,000[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]3[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]3[\/latex] places to the right<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"center\">[latex]10,000[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]4[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]4[\/latex] places to the right<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can use this pattern as a shortcut to multiply by powers of ten instead of multiplying using the vertical format. We can count the zeros in the power of [latex]10[\/latex] and then move the decimal point that same of places to the right.\r\n\r\nSo, for example, to multiply [latex]45.86[\/latex] by [latex]100[\/latex], move the decimal point [latex]2[\/latex] places to the right.\r\n\r\n<img class=\"aligncenter wp-image-2073 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2985\/2018\/04\/24183315\/image.png\" alt=\"45.86 times 100 is shown to equal 4586. There is an arrow from the decimal going over 2 places from after the 5 to after the 6.\" width=\"153\" height=\"50\" \/>\r\nSometimes when we need to move the decimal point, there are not enough decimal places. In that case, we use zeros as placeholders. For example, let\u2019s multiply [latex]2.4[\/latex] by [latex]100[\/latex]. We need to move the decimal point [latex]2[\/latex] places to the right. Since there is only one digit to the right of the decimal point, we must write a [latex]0[\/latex] in the hundredths place.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221605\/CNX_BMath_Figure_05_02_003_img.png\" alt=\"2.4 times 100 is shown to equal 240. There is an arrow from the decimal going over 2 places from after the 2 to after the 0.\" data-media-type=\"image\/png\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Multiply a decimal by a power of [latex]10[\/latex]<\/h3>\r\n<ol id=\"eip-id1168467251192\" class=\"stepwise\" data-number-style=\"arabic\">\r\n \t<li>Move the decimal point to the right the same number of places as the number of zeros in the power of [latex]10[\/latex].<\/li>\r\n \t<li>Write zeros at the end of the number as placeholders if needed.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the following video we show more examples of how to multiply a decimal by 10, 100, and 1000.\r\n\r\nhttps:\/\/youtu.be\/JFAwf01nPG8\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply [latex]5.63[\/latex] by factors of\r\n\r\n1. [latex]10[\/latex]\r\n\r\n2. [latex]100[\/latex]\r\n\r\n3. [latex]1000[\/latex]\r\n[reveal-answer q=\"296737\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"296737\"]\r\n\r\nSolution\r\nBy looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.\r\n<table id=\"eip-id1168466081784\" class=\"unnumbered unstyled\" summary=\"The top line says 5.63 times 100. The next line says, 'There are 2 zeros in 100, so move the decimal point 2 places to the right.' The image shows 5.63 with an arrow from the decimal point to after a line that has been added after the 3. The last line shows 563.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]56.3\\left(10\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>There is [latex]1[\/latex] zero in [latex]10[\/latex], so move the decimal point [latex]1[\/latex] place to the right.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221607\/CNX_BMath_Figure_05_02_023a_img-01.png\" alt=\"5.63. Move the decimal point 1 place to the right.\" width=\"77\" height=\"56\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]56.3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466121817\" class=\"unnumbered unstyled\" summary=\"The top line says 5.63 times 100. The next line says, \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5.63\\left(100\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>There are [latex]2[\/latex] zeros in [latex]100[\/latex], so move the decimal point [latex]2[\/latex] places to the right.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221608\/CNX_BMath_Figure_05_02_023b_img-01.png\" alt=\"5.63. Move the decimal point 2 places to the right.\" width=\"88\" height=\"60\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]563[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469496249\" class=\"unnumbered unstyled\" summary=\"The top line says 5.63 times 1000. The next line says, \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5.63\\left(1000\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>There are [latex]3[\/latex] zeros in [latex]1000[\/latex], so move the decimal point [latex]3[\/latex] places to the right.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221609\/CNX_BMath_Figure_05_02_023c_img-01.png\" alt=\"5.63. Move the decimal point 3 places to the right.\" width=\"91\" height=\"56\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A zero must be added at the end.<\/td>\r\n<td>[latex]5,630[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY IT<\/h3>\r\n[ohm_question]146599[\/ohm_question]\r\n\r\n<\/div>\r\n<h2 data-type=\"title\">Multiplying Decimals<\/h2>\r\n<p data-type=\"title\">Multiplying decimals is very much like multiplying whole numbers\u2014we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions.<\/p>\r\n<p data-type=\"title\">Do you remember how to multiply fractions? To multiply fractions, you multiply the numerators and then multiply the denominators.<\/p>\r\n<p data-type=\"title\">So let\u2019s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side below. Look for a pattern.<\/p>\r\n\r\n<table id=\"fs-id1620547\" summary=\"A table is shown with 3 columns and 5 rows. The first row is a header row. The first column is not labeled. The other columns are labeled \">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th><\/th>\r\n<th data-align=\"center\">A<\/th>\r\n<th data-align=\"center\">B<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><\/td>\r\n<td data-align=\"center\">[latex]\\left(0.3\\right)\\left(0.7\\right)[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]\\left(0.2\\right)\\left(0.46\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">Convert to fractions.<\/td>\r\n<td data-align=\"center\">[latex]\\left({\\Large\\frac{3}{10}}\\right)\\left({\\Large\\frac{7}{10}}\\right)[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]\\left({\\Large\\frac{2}{10}}\\right)\\left({\\Large\\frac{46}{100}}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">Multiply.<\/td>\r\n<td data-align=\"center\">[latex]{\\Large\\frac{21}{100}}[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]{\\Large\\frac{92}{1000}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">Convert back to decimals.<\/td>\r\n<td data-align=\"center\">[latex]0.21[\/latex]<\/td>\r\n<td data-align=\"center\">[latex]0.092[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThere is a pattern that we can use. In A, we multiplied two numbers that each had one decimal place, and the product had two decimal places. In B, we multiplied a number with one decimal place by a number with two decimal places, and the product had three decimal places.\r\n\r\nHow many decimal places would you expect for the product of [latex]\\left(0.01\\right)\\left(0.004\\right)?[\/latex] If you said \"five\", you recognized the pattern. When we multiply two numbers with decimals, we count all the decimal places in the factors\u2014in this case two plus three\u2014to get the number of decimal places in the product\u2014in this case five.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221548\/CNX_BMath_Figure_05_02_018_img.png\" alt=\"The top line says 0.01 times 0.004 equals 0.00004. Below the 0.01, it says 2 places. Below the 0.004, it says 3 places. Below the 0.00004, it says 5 places. The bottom line says 1 over 100 times 4 over 1000 equals 4 over 100,000.\" data-media-type=\"image\/png\" \/>\r\nOnce we know how to determine the number of digits after the decimal point, we can multiply decimal numbers without converting them to fractions first. The number of decimal places in the product is the sum of the number of decimal places in the factors. The rules for multiplying positive and negative numbers apply to decimals too.\r\n<div class=\"textbox shaded\">\r\n<h3>Multiply decimal numbers<\/h3>\r\n<ol id=\"eip-id1168469803517\" class=\"stepwise\" data-number-style=\"arabic\">\r\n \t<li>Determine the sign of the product.<\/li>\r\n \t<li>Write the numbers in vertical format, lining up the numbers on the right.<\/li>\r\n \t<li>Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/li>\r\n \t<li>Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.<\/li>\r\n \t<li>Write the product with the appropriate sign.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(3.9\\right)\\left(4.075\\right)[\/latex]\r\n\r\nSolution\r\n<table id=\"eip-id1168466119810\" class=\"unnumbered unstyled\" summary=\"The top line says 3.9 times 4.075. The next line says, \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\left(3.9\\right)\\left(4.075\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Determine the sign of the product. The signs are the same.<\/td>\r\n<td>The product will be positive.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the numbers in vertical format, lining up the numbers on the right.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221549\/CNX_BMath_Figure_05_02_019_img-01.png\" alt=\"Vertical multiplication for 4.075 times 3.9.\" width=\"149\" height=\"55\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221550\/CNX_BMath_Figure_05_02_019_img-02.png\" alt=\"4.075 times 3.9 equals 36675 plus 122250 which equals 158925.\" width=\"149\" height=\"109\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Place the decimal point. Add the number of decimal places in the factors [latex]\\left(1+3\\right)[\/latex]. Place the decimal point 4 places from the right.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221550\/CNX_BMath_Figure_05_02_019_img-03.png\" alt=\"In our calculation, 4.075 has 3 decimal places, and 3.9 has 1 decimal place. Considering 3 plus 1 equals 4, our solution has 4 decimal places, such that the decimal comes after the first 5 in 158925.\" width=\"149\" height=\"120\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The product is positive.<\/td>\r\n<td>[latex]\\left(3.9\\right)\\left(4.075\\right)=15.8925[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146596[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show another example of how to multiply two decimals.\r\n\r\nhttps:\/\/youtu.be\/55OtS_Dil1Y\r\n<h2>Dividing Decimals<\/h2>\r\nJust as with multiplication, division of decimals is very much like dividing whole numbers \u2014 we 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.\" data-media-type=\"image\/png\" \/>\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.\" data-media-type=\"image\/png\" \/>\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\" data-number-style=\"arabic\">\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, \" data-label=\"\">\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 class=\"alignnone\" 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=\"Long division for 0.12 divided by 3. Our decimal point is located directly above the decimal point in 0.12\" width=\"123\" height=\"40\" data-media-type=\"image\/png\" \/><\/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 class=\"alignnone\" 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=\"Consider how many time 3 goes into 12. 3 goes into 12 4 times, so 4 becomes our hundredths digit in the quotient. 12 minus 12 is 0, so we have a remainder of 0. In total 0.12 divided by 3 equals 0.04.\" width=\"123\" height=\"85\" data-media-type=\"image\/png\" \/><\/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<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\r\nMost commonly, this calculation will be done with money while shopping. Prices of products tend to be presented as a combination of dollars and cents (i.e. [latex]\\$5.75[\/latex], [latex]2.99[\/latex], or [latex]\\$25.60[\/latex]). You can then divide by the number of items or the number of units to determine the cost per item or unit.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSuppose 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 Answer[\/reveal-answer]\r\n[hidden-answer a=\"929706\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469496456\" class=\"unnumbered unstyled\" style=\"height: 255px;\" summary=\"The top line says &gt;.99 divided by 24. The first step says, \" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 676.265625px;\"><\/td>\r\n<td style=\"height: 12px; width: 145.03125px;\">[latex]$3.99\\div 24[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 38px;\">\r\n<td style=\"height: 38px; width: 676.265625px;\">Place the decimal point in the quotient above the decimal point in the dividend.<\/td>\r\n<td style=\"height: 38px; width: 145.03125px;\"><img class=\"alignnone\" 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=\"Long division for 3.99 divided by 24. Our decimal point is located directly above the decimal point in 3.99.\" width=\"153\" height=\"38\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 157px;\">\r\n<td style=\"height: 157px; width: 676.265625px;\">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 style=\"height: 157px; width: 145.03125px;\"><img class=\"alignnone\" 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=\"24 goes into 39 1 time, so 1 becomes our tenths digit in the quotient. 39 minus 24 equals 15. We carry down the 9, so 15 becomes 159. 24 goes into 159 6 times, so 6 becomes the hundredths digit in the quotient. 159 minus 144 equals 15. We carry down the trailing 0 so 15 becomes 150. 24 goes into 150 6 times, so 6 becomes the thousandths digit in the quotient. 150 minus 144 equals 6.\" width=\"153\" height=\"164\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px; width: 676.265625px;\">Round to the nearest cent.<\/td>\r\n<td style=\"height: 24px; width: 145.03125px;\" data-align=\"right\">[latex]$0.166\\approx $0.17[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 24px;\">\r\n<td style=\"height: 24px; width: 676.265625px;\"><\/td>\r\n<td style=\"height: 24px; width: 145.03125px;\" data-align=\"right\">[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<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]145993[\/ohm_question]\r\n\r\n[ohm_question]156951[\/ohm_question]\r\n\r\n<\/div>\r\nNext, we will divide a whole number by a decimal.\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 Answer[\/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, \" data-label=\"\">\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 class=\"alignnone\" 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=\"Long division for 4.00 divided by 0.05. Accounting for the 2 decimal places in 0.05, we shift both 0.05 and 4.00 2 decimal places to the right.\" width=\"197\" height=\"36\" data-media-type=\"image\/png\" \/><\/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 class=\"alignnone\" 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=\"Our equation becomes 400 divided by 5. 5 goes into 40 8 times, so eight becomes the tens place in the quotient. 40 minus 40 equals 0, and we carry down the 0, so 0 stays as 0. 5 does not go into 0, so 0 becomes the ones place in the quotient and we have a remainder of 0. In total, 4.00 divided by 0.05 equals 80.\" width=\"197\" height=\"104\" data-media-type=\"image\/png\" \/><\/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<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 Objectives<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use multiplication and division when evaluating expressions with decimals&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:13185,&quot;3&quot;:[null,0],&quot;10&quot;:0,&quot;11&quot;:4,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Use multiplication and division when evaluating expressions with decimals<\/span><\/li>\n<\/ul>\n<\/div>\n<p>Multiplying and dividing decimals can become complex, but we&#8217;re going to start with some basic multiplication by ten to warm-up.<\/p>\n<h2>Multiply by Powers of [latex]10[\/latex]<\/h2>\n<p>In many fields, especially in the sciences, it is common to multiply decimals by powers of [latex]10[\/latex]. Let\u2019s see what happens when we multiply [latex]1.9436[\/latex] by some powers of [latex]10[\/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\/24221557\/CNX_BMath_Figure_05_02_022_img.png\" alt=\"The top row says 1.9436 times 10, then 1.9436 times 100, then 1.9436 times 1000. Below each is a vertical multiplication problem. These show that 1.9436 times 10 is 19.4360, 1.9436 times 100 is 194.3600, and 1.9436 times 1000 is 1943.6000.\" data-media-type=\"image\/png\" \/><br \/>\nLook at the results without the final zeros. Do you notice a pattern?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}1.9436\\left(10\\right)\\hfill & =& 19.436\\hfill \\\\ 1.9436\\left(100\\right)\\hfill & =& 194.36\\hfill \\\\ 1.9436\\left(1000\\right)\\hfill & =& 1943.6\\hfill \\end{array}[\/latex]<\/p>\n<p>The number of places that the decimal point moved is the same as the number of zeros in the power of ten. The table below\u00a0summarizes the results.<\/p>\n<table id=\"fs-id3415452\" summary=\"A table is shown with five rows and three columns. The first row, which is the header row, reads 'Multiply by' in the first column, 'Number of zeros' in the second column, and 'Number of places decimal point moves' in the third column. The first column shows 10, 100, 1,000, and 10,000. The second row shows 1, 2, 3, 4. The third row shows '1 place to the right,' '2 places to the right,' '3 places to the right,' and '4 places to the right'.\">\n<thead>\n<tr valign=\"top\">\n<th data-align=\"center\">Multiply by<\/th>\n<th data-align=\"center\">Number of zeros<\/th>\n<th data-align=\"center\">Number of places decimal point moves<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]10[\/latex]<\/td>\n<td data-align=\"center\">[latex]1[\/latex]<\/td>\n<td data-align=\"center\">[latex]1[\/latex] place to the right<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]100[\/latex]<\/td>\n<td data-align=\"center\">[latex]2[\/latex]<\/td>\n<td data-align=\"center\">[latex]2[\/latex] places to the right<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]1,000[\/latex]<\/td>\n<td data-align=\"center\">[latex]3[\/latex]<\/td>\n<td data-align=\"center\">[latex]3[\/latex] places to the right<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"center\">[latex]10,000[\/latex]<\/td>\n<td data-align=\"center\">[latex]4[\/latex]<\/td>\n<td data-align=\"center\">[latex]4[\/latex] places to the right<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can use this pattern as a shortcut to multiply by powers of ten instead of multiplying using the vertical format. We can count the zeros in the power of [latex]10[\/latex] and then move the decimal point that same of places to the right.<\/p>\n<p>So, for example, to multiply [latex]45.86[\/latex] by [latex]100[\/latex], move the decimal point [latex]2[\/latex] places to the right.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2073 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2985\/2018\/04\/24183315\/image.png\" alt=\"45.86 times 100 is shown to equal 4586. There is an arrow from the decimal going over 2 places from after the 5 to after the 6.\" width=\"153\" height=\"50\" \/><br \/>\nSometimes when we need to move the decimal point, there are not enough decimal places. In that case, we use zeros as placeholders. For example, let\u2019s multiply [latex]2.4[\/latex] by [latex]100[\/latex]. We need to move the decimal point [latex]2[\/latex] places to the right. Since there is only one digit to the right of the decimal point, we must write a [latex]0[\/latex] in the hundredths place.<\/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\/24221605\/CNX_BMath_Figure_05_02_003_img.png\" alt=\"2.4 times 100 is shown to equal 240. There is an arrow from the decimal going over 2 places from after the 2 to after the 0.\" data-media-type=\"image\/png\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Multiply a decimal by a power of [latex]10[\/latex]<\/h3>\n<ol id=\"eip-id1168467251192\" class=\"stepwise\" data-number-style=\"arabic\">\n<li>Move the decimal point to the right the same number of places as the number of zeros in the power of [latex]10[\/latex].<\/li>\n<li>Write zeros at the end of the number as placeholders if needed.<\/li>\n<\/ol>\n<\/div>\n<p>In the following video we show more examples of how to multiply a decimal by 10, 100, and 1000.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Multiply Decimals by 10, 100, and 1000\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/JFAwf01nPG8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply [latex]5.63[\/latex] by factors of<\/p>\n<p>1. [latex]10[\/latex]<\/p>\n<p>2. [latex]100[\/latex]<\/p>\n<p>3. [latex]1000[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q296737\">Show Answer<\/span><\/p>\n<div id=\"q296737\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nBy looking at the number of zeros in the multiple of ten, we see the number of places we need to move the decimal to the right.<\/p>\n<table id=\"eip-id1168466081784\" class=\"unnumbered unstyled\" summary=\"The top line says 5.63 times 100. The next line says, 'There are 2 zeros in 100, so move the decimal point 2 places to the right.' The image shows 5.63 with an arrow from the decimal point to after a line that has been added after the 3. The last line shows 563.\" data-label=\"\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<\/tr>\n<tr>\n<td>[latex]56.3\\left(10\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>There is [latex]1[\/latex] zero in [latex]10[\/latex], so move the decimal point [latex]1[\/latex] place to the right.<\/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\/24221607\/CNX_BMath_Figure_05_02_023a_img-01.png\" alt=\"5.63. Move the decimal point 1 place to the right.\" width=\"77\" height=\"56\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>[latex]56.3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466121817\" class=\"unnumbered unstyled\" summary=\"The top line says 5.63 times 100. The next line says,\" data-label=\"\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<\/tr>\n<tr>\n<td>[latex]5.63\\left(100\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>There are [latex]2[\/latex] zeros in [latex]100[\/latex], so move the decimal point [latex]2[\/latex] places to the right.<\/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\/24221608\/CNX_BMath_Figure_05_02_023b_img-01.png\" alt=\"5.63. Move the decimal point 2 places to the right.\" width=\"88\" height=\"60\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>[latex]563[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469496249\" class=\"unnumbered unstyled\" summary=\"The top line says 5.63 times 1000. The next line says,\" data-label=\"\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<\/tr>\n<tr>\n<td>[latex]5.63\\left(1000\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>There are [latex]3[\/latex] zeros in [latex]1000[\/latex], so move the decimal point [latex]3[\/latex] places to the right.<\/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\/24221609\/CNX_BMath_Figure_05_02_023c_img-01.png\" alt=\"5.63. Move the decimal point 3 places to the right.\" width=\"91\" height=\"56\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>A zero must be added at the end.<\/td>\n<td>[latex]5,630[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146599\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146599&theme=oea&iframe_resize_id=ohm146599&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2 data-type=\"title\">Multiplying Decimals<\/h2>\n<p data-type=\"title\">Multiplying decimals is very much like multiplying whole numbers\u2014we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions.<\/p>\n<p data-type=\"title\">Do you remember how to multiply fractions? To multiply fractions, you multiply the numerators and then multiply the denominators.<\/p>\n<p data-type=\"title\">So let\u2019s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side below. Look for a pattern.<\/p>\n<table id=\"fs-id1620547\" summary=\"A table is shown with 3 columns and 5 rows. The first row is a header row. The first column is not labeled. The other columns are labeled\">\n<thead>\n<tr valign=\"top\">\n<th><\/th>\n<th data-align=\"center\">A<\/th>\n<th data-align=\"center\">B<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td><\/td>\n<td data-align=\"center\">[latex]\\left(0.3\\right)\\left(0.7\\right)[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\left(0.2\\right)\\left(0.46\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">Convert to fractions.<\/td>\n<td data-align=\"center\">[latex]\\left({\\Large\\frac{3}{10}}\\right)\\left({\\Large\\frac{7}{10}}\\right)[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\left({\\Large\\frac{2}{10}}\\right)\\left({\\Large\\frac{46}{100}}\\right)[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">Multiply.<\/td>\n<td data-align=\"center\">[latex]{\\Large\\frac{21}{100}}[\/latex]<\/td>\n<td data-align=\"center\">[latex]{\\Large\\frac{92}{1000}}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">Convert back to decimals.<\/td>\n<td data-align=\"center\">[latex]0.21[\/latex]<\/td>\n<td data-align=\"center\">[latex]0.092[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>There is a pattern that we can use. In A, we multiplied two numbers that each had one decimal place, and the product had two decimal places. In B, we multiplied a number with one decimal place by a number with two decimal places, and the product had three decimal places.<\/p>\n<p>How many decimal places would you expect for the product of [latex]\\left(0.01\\right)\\left(0.004\\right)?[\/latex] If you said &#8220;five&#8221;, you recognized the pattern. When we multiply two numbers with decimals, we count all the decimal places in the factors\u2014in this case two plus three\u2014to get the number of decimal places in the product\u2014in this case five.<\/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\/24221548\/CNX_BMath_Figure_05_02_018_img.png\" alt=\"The top line says 0.01 times 0.004 equals 0.00004. Below the 0.01, it says 2 places. Below the 0.004, it says 3 places. Below the 0.00004, it says 5 places. The bottom line says 1 over 100 times 4 over 1000 equals 4 over 100,000.\" data-media-type=\"image\/png\" \/><br \/>\nOnce we know how to determine the number of digits after the decimal point, we can multiply decimal numbers without converting them to fractions first. The number of decimal places in the product is the sum of the number of decimal places in the factors. The rules for multiplying positive and negative numbers apply to decimals too.<\/p>\n<div class=\"textbox shaded\">\n<h3>Multiply decimal numbers<\/h3>\n<ol id=\"eip-id1168469803517\" class=\"stepwise\" data-number-style=\"arabic\">\n<li>Determine the sign of the product.<\/li>\n<li>Write the numbers in vertical format, lining up the numbers on the right.<\/li>\n<li>Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/li>\n<li>Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.<\/li>\n<li>Write the product with the appropriate sign.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(3.9\\right)\\left(4.075\\right)[\/latex]<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168466119810\" class=\"unnumbered unstyled\" summary=\"The top line says 3.9 times 4.075. The next line says,\" data-label=\"\">\n<tbody>\n<tr>\n<td>[latex]\\left(3.9\\right)\\left(4.075\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Determine the sign of the product. The signs are the same.<\/td>\n<td>The product will be positive.<\/td>\n<\/tr>\n<tr>\n<td>Write the numbers in vertical format, lining up the numbers on the right.<\/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\/24221549\/CNX_BMath_Figure_05_02_019_img-01.png\" alt=\"Vertical multiplication for 4.075 times 3.9.\" width=\"149\" height=\"55\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.<\/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\/24221550\/CNX_BMath_Figure_05_02_019_img-02.png\" alt=\"4.075 times 3.9 equals 36675 plus 122250 which equals 158925.\" width=\"149\" height=\"109\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>Place the decimal point. Add the number of decimal places in the factors [latex]\\left(1+3\\right)[\/latex]. Place the decimal point 4 places from the right.<\/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\/24221550\/CNX_BMath_Figure_05_02_019_img-03.png\" alt=\"In our calculation, 4.075 has 3 decimal places, and 3.9 has 1 decimal place. Considering 3 plus 1 equals 4, our solution has 4 decimal places, such that the decimal comes after the first 5 in 158925.\" width=\"149\" height=\"120\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td>The product is positive.<\/td>\n<td>[latex]\\left(3.9\\right)\\left(4.075\\right)=15.8925[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146596\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146596&theme=oea&iframe_resize_id=ohm146596&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show another example of how to multiply two decimals.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Examples 2: Multiplication of Decimals\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/55OtS_Dil1Y?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Dividing Decimals<\/h2>\n<p>Just as with multiplication, division of decimals is very much like dividing whole numbers \u2014 we 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.\" data-media-type=\"image\/png\" \/><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.\" data-media-type=\"image\/png\" \/><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\" data-number-style=\"arabic\">\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,\" data-label=\"\">\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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Long division for 0.12 divided by 3. Our decimal point is located directly above the decimal point in 0.12\" width=\"123\" height=\"40\" data-media-type=\"image\/png\" \/><\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Consider how many time 3 goes into 12. 3 goes into 12 4 times, so 4 becomes our hundredths digit in the quotient. 12 minus 12 is 0, so we have a remainder of 0. In total 0.12 divided by 3 equals 0.04.\" width=\"123\" height=\"85\" data-media-type=\"image\/png\" \/><\/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<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-3\" 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<p>Most commonly, this calculation will be done with money while shopping. Prices of products tend to be presented as a combination of dollars and cents (i.e. [latex]\\$5.75[\/latex], [latex]2.99[\/latex], or [latex]\\$25.60[\/latex]). You can then divide by the number of items or the number of units to determine the cost per item or unit.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>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 Answer<\/span><\/p>\n<div id=\"q929706\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469496456\" class=\"unnumbered unstyled\" style=\"height: 255px;\" summary=\"The top line says &gt;.99 divided by 24. The first step says,\" data-label=\"\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 676.265625px;\"><\/td>\n<td style=\"height: 12px; width: 145.03125px;\">[latex]$3.99\\div 24[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 38px;\">\n<td style=\"height: 38px; width: 676.265625px;\">Place the decimal point in the quotient above the decimal point in the dividend.<\/td>\n<td style=\"height: 38px; width: 145.03125px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Long division for 3.99 divided by 24. Our decimal point is located directly above the decimal point in 3.99.\" width=\"153\" height=\"38\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr style=\"height: 157px;\">\n<td style=\"height: 157px; width: 676.265625px;\">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 style=\"height: 157px; width: 145.03125px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"24 goes into 39 1 time, so 1 becomes our tenths digit in the quotient. 39 minus 24 equals 15. We carry down the 9, so 15 becomes 159. 24 goes into 159 6 times, so 6 becomes the hundredths digit in the quotient. 159 minus 144 equals 15. We carry down the trailing 0 so 15 becomes 150. 24 goes into 150 6 times, so 6 becomes the thousandths digit in the quotient. 150 minus 144 equals 6.\" width=\"153\" height=\"164\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px; width: 676.265625px;\">Round to the nearest cent.<\/td>\n<td style=\"height: 24px; width: 145.03125px;\" data-align=\"right\">[latex]$0.166\\approx $0.17[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px; width: 676.265625px;\"><\/td>\n<td style=\"height: 24px; width: 145.03125px;\" data-align=\"right\">[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<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<p><iframe loading=\"lazy\" id=\"ohm156951\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=156951&theme=oea&iframe_resize_id=ohm156951&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Next, we will divide a whole number by a decimal.<\/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 Answer<\/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,\" data-label=\"\">\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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Long division for 4.00 divided by 0.05. Accounting for the 2 decimal places in 0.05, we shift both 0.05 and 4.00 2 decimal places to the right.\" width=\"197\" height=\"36\" data-media-type=\"image\/png\" \/><\/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 loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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=\"Our equation becomes 400 divided by 5. 5 goes into 40 8 times, so eight becomes the tens place in the quotient. 40 minus 40 equals 0, and we carry down the 0, so 0 stays as 0. 5 does not go into 0, so 0 becomes the ones place in the quotient and we have a remainder of 0. In total, 4.00 divided by 0.05 equals 80.\" width=\"197\" height=\"104\" data-media-type=\"image\/png\" \/><\/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<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-4\" 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-325\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/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":17533,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"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\"}]","CANDELA_OUTCOMES_GUID":"01e6499a-37ac-4c1c-b810-3012ec1008d9, 505481be-85c2-4109-a284-b1d34efd16c5","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-325","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/325","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/325\/revisions"}],"predecessor-version":[{"id":3986,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/325\/revisions\/3986"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/325\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/media?parent=325"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapter-type?post=325"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/contributor?post=325"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/license?post=325"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}