{"id":66,"date":"2018-03-19T17:42:07","date_gmt":"2018-03-19T17:42:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/?post_type=chapter&#038;p=66"},"modified":"2024-04-26T22:00:34","modified_gmt":"2024-04-26T22:00:34","slug":"addition-subtraction-multiplication-and-division-with-whole-numbers","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/chapter\/addition-subtraction-multiplication-and-division-with-whole-numbers\/","title":{"raw":"Adding, Subtracting, Multiplying, and Dividing Whole Numbers","rendered":"Adding, Subtracting, Multiplying, and Dividing Whole Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use addition, subtraction, multiplication, and division when evaluating whole number expressions<\/li>\r\n<\/ul>\r\n<\/div>\r\nWorking with whole numbers and performing basic calculations is the backbone of all math. We're going to assume you remember how to do single digit addition, subtraction, multiplication, and division. You will often have a calculator on hand to do these calculations, but a quick refresher will help you better understand how to work with numbers so that complex equations are less daunting.\r\n<h2>Addition<\/h2>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nAdd: [latex]28+61[\/latex]\r\n\r\nSolution\r\nTo add numbers with more than one digit, it is often easier to write the numbers vertically in columns.\r\n<table id=\"eip-id1168288293873\" class=\"unnumbered unstyled\" style=\"width: 70%;\" summary=\"a\">\r\n<tbody>\r\n<tr>\r\n<td>Write the numbers so the ones and tens digits line up vertically.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 28\\\\ \\\\ \\hfill \\underset{\\text{____}}{+61}\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Then add the digits in each place value.\r\n\r\nAdd the ones: [latex]8+1=9[\/latex]\r\n\r\nAdd the tens: [latex]2+6=8[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 28\\\\ \\\\ \\hfill \\underset{\\text{____}}{+61}\\\\ \\hfill 89\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<\/div>\r\nIn the previous example, the sum of the ones and the sum of the tens were both less than [latex]10[\/latex]. But what happens if the sum is [latex]10[\/latex] or more? Let\u2019s use our base-[latex]10[\/latex] model to find out.\r\n\r\nThe graphic below\u00a0shows the addition of [latex]17[\/latex] and [latex]26[\/latex] again.\r\n\r\n&nbsp;\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215337\/CNX_BMath_Figure_01_02_001.png\" alt=\"17 plus 26. 17 is represented with one rod (one rod equals 10 ones) and seven one. 27 is represented with two rods and six ones. When you add 17 and 26, ten of the ones combine into a rod and the sum is represented with four rods and three ones, or 43\" width=\"619\" height=\"146\" data-media-type=\"image\/png\" \/>\r\n\r\n&nbsp;\r\n\r\nWhen we add the ones, [latex]7+6[\/latex], we get [latex]13[\/latex] ones. Because we have more than [latex]10[\/latex] ones, we can exchange [latex]10[\/latex] of the ones for [latex]1[\/latex] ten. Now we have [latex]4[\/latex] tens and [latex]3[\/latex] ones. Without using the model, we show this as a small red [latex]1[\/latex] above the digits in the tens place.\r\n\r\nWhen the sum in a place value column is greater than [latex]9[\/latex], we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, [latex]10[\/latex] ones for [latex]1[\/latex] ten or [latex]10[\/latex] tens for [latex]1[\/latex] hundred.\r\n<div class=\"textbox shaded\">\r\n<h3>Add whole numbers<\/h3>\r\n<ol id=\"eip-id1168288474750\" class=\"stepwise\" data-number-style=\"arabic\">\r\n \t<li>Write the numbers so each place value lines up vertically.<\/li>\r\n \t<li>Add the digits in each place value. Work from right to left starting with the ones place. If a sum in a place value is more than [latex]9[\/latex], carry to the next place value.<\/li>\r\n \t<li>Continue adding each place value from right to left, adding each place value and carrying if needed.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nAdd: [latex]43+69[\/latex]\r\n[reveal-answer q=\"61333\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"61333\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168287505854\" class=\"unnumbered unstyled\" style=\"width: 70%;\" summary=\"a\">\r\n<tbody>\r\n<tr>\r\n<td>Write the numbers so the digits line up vertically.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 43\\\\ \\\\ \\hfill \\underset{\\text{____}}{+69}\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the digits in each place.\r\n\r\nAdd the ones: [latex]3+9=12[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the [latex]2[\/latex] in the ones place in the sum.\r\n\r\nAdd the [latex]1[\/latex] ten to the tens place.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{1}{4}3\\\\ \\hfill \\underset{\\text{____}}{+69}\\\\ \\hfill 2\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Now add the tens: [latex]1+4+6=11[\/latex]\r\n\r\nWrite the 11 in the sum.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{1}{4}3\\\\ \\hfill \\underset{\\text{____}}{+69}\\\\ \\hfill 112\\end{array}[\/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]147154[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]147156[\/ohm_question]\r\n\r\n<\/div>\r\nWhen the addends have different numbers of digits, be careful to line up the corresponding place values starting with the ones and moving toward the left.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nAdd: [latex]1,683+479[\/latex].\r\n[reveal-answer q=\"603010\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"603010\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168288687380\" class=\"unnumbered unstyled\" style=\"width: 80%;\" summary=\"a\">\r\n<tbody>\r\n<tr>\r\n<td>Write the numbers so the digits line up vertically.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 1,683\\\\ \\\\ \\hfill \\underset{\\text{______}}{+479}\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the digits in each place value.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the ones: [latex]3+9=12[\/latex].\r\n\r\nWrite the [latex]2[\/latex] in the ones place of the sum and carry the [latex]1[\/latex] ten to the tens place.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 1,6\\stackrel{1}{8}3\\\\ \\\\ \\hfill \\underset{\\text{______}}{+479}\\\\ \\hfill 2\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the tens: [latex]1+7+8=16[\/latex]\r\n\r\nWrite the [latex]6[\/latex] in the tens place and carry the [latex]1[\/latex] hundred to the hundreds place.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 1,\\stackrel{1}{6}\\stackrel{1}{8}3\\\\ \\\\ \\hfill \\underset{\\text{______}}{+479}\\\\ \\hfill 62\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the hundreds: [latex]1+6+4=11[\/latex]\r\n\r\nWrite the [latex]1[\/latex] in the hundreds place and carry the [latex]1[\/latex] thousand to the thousands place.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{1}{1},\\stackrel{1}{6}\\stackrel{1}{8}3\\\\ \\\\ \\hfill \\underset{\\text{______}}{+479}\\\\ \\hfill 162\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the thousands [latex]1+1=2[\/latex] .\r\n\r\nWrite the [latex]2[\/latex] in the thousands place of the sum.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{1}{1},\\stackrel{1}{6}\\stackrel{1}{8}3\\\\ \\\\ \\hfill \\underset{\\text{______}}{+479}\\\\ \\hfill 2,162\\end{array}[\/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]156996[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the video below for another example of how to add three whole numbers by lining up place values.\r\nhttps:\/\/youtu.be\/N3I6OiO5mKI\r\n<h2>Subtraction<\/h2>\r\nAddition and subtraction are inverse operations. Addition undoes subtraction, and subtraction undoes addition.\r\nWe know [latex]7 - 3=4[\/latex] because [latex]4+3=7[\/latex]. Knowing all the addition number facts will help with subtraction. Then we can check subtraction by adding. In the examples above, our subtractions can be checked by addition.\r\n<table class=\"unnumbered unstyled\" summary=\"This image consists of there columns\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]7-3=4[\/latex]<\/td>\r\n<td>because<\/td>\r\n<td>[latex]4+3=7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]13-8=5[\/latex]<\/td>\r\n<td>because<\/td>\r\n<td>[latex]5+8=13[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]43-26=17[\/latex]<\/td>\r\n<td>because<\/td>\r\n<td>[latex]17+26=43[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo subtract numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition. Align the digits by place value, and then subtract each column starting with the ones and then working to the left.\r\n<div class=\"textbox exercises\">\r\n<h3>Exercise<\/h3>\r\nSubtract and then check by adding: [latex]89 - 61[\/latex].\r\n[reveal-answer q=\"656368\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"656368\"]\r\n\r\nSolution\r\n<table id=\"eip-458\" class=\"unnumbered unstyled\" summary=\"a\">\r\n<tbody>\r\n<tr>\r\n<td>Write the numbers so the ones and tens digits line up vertically.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 89\\\\ \\hfill \\underset{\\text{____}}{-61}\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Subtract the digits in each place value.\r\n\r\nSubtract the ones: [latex]9 - 1=8[\/latex]\r\n\r\nSubtract the tens: [latex]8 - 6=2[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 89\\\\ \\hfill \\underset{\\text{____}}{-61}\\\\ \\hfill 28\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check using addition.\r\n\r\n[latex]\\begin{array}{c}\\hfill 28\\\\ \\hfill \\underset{\\text{____}}{+61}\\\\ \\hfill 89\\end{array}\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOur answer is correct.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=143322&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Subtract whole numbers<\/h3>\r\n<ol id=\"eip-id1168289617573\" class=\"stepwise\" data-number-style=\"arabic\">\r\n \t<li>Write the numbers so each place value lines up vertically.<\/li>\r\n \t<li>Subtract the digits in each place value. Work from right to left starting with the ones place. If the digit on top is less than the digit below, borrow as needed.<\/li>\r\n \t<li>Continue subtracting each place value from right to left, borrowing if needed.<\/li>\r\n \t<li>Check by adding.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>excercise<\/h3>\r\nSubtract: [latex]43 - 26[\/latex].\r\n[reveal-answer q=\"491725\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"491725\"]\r\n\r\nSolution\r\n<table id=\"eip-640\" class=\"unnumbered unstyled\" style=\"width: 915px;\" summary=\".\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 587.944px;\">Write the numbers so each place value lines up vertically.<\/td>\r\n<td style=\"width: 306.056px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215443\/CNX_BMath_Figure_01_03_029_img-01.png\" alt=\"Vertical subtraction for 43 minus 26\" width=\"94\" height=\"56\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 587.944px;\">Subtract the ones. We cannot subtract [latex]6[\/latex] from [latex]3[\/latex], so we borrow [latex]1[\/latex] ten. This makes [latex]3[\/latex] tens and [latex]13[\/latex] ones. We write these numbers above each place and cross out the original digits.<\/td>\r\n<td style=\"width: 306.056px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215444\/CNX_BMath_Figure_01_03_029_img-02.png\" alt=\"The 3 borrows from the 4, so the 4 becomes 3 and the 3 becomes 13.\" width=\"92\" height=\"73\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 587.944px;\">Now we can subtract the ones. [latex]13 - 6=7[\/latex]. We write the [latex]7[\/latex] in the ones place in the difference.<\/td>\r\n<td style=\"width: 306.056px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215445\/CNX_BMath_Figure_01_03_029_img-03.png\" alt=\"13 minus 6 equals 7.\" width=\"90\" height=\"102\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 587.944px;\">Now we subtract the tens. [latex]3 - 2=1[\/latex]. We write the[latex]1[\/latex] in the tens place in the difference.<\/td>\r\n<td style=\"width: 306.056px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215445\/CNX_BMath_Figure_01_03_029_img-04.png\" alt=\"3 minus 2 equals 1. In total, we have 43 minus 26 equals 17.\" width=\"89\" height=\"101\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 587.944px;\" data-align=\"left\">Check by adding.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215446\/CNX_BMath_Figure_01_03_029_img-05.png\" alt=\"17 plus 26 equals 43, therefore our check is succcessful and our answer correct.\" width=\"94\" height=\"88\" data-media-type=\"image\/png\" \/>\r\n\r\nOur answer is correct.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the example above, if we model subtracting [latex]26[\/latex] from [latex]43[\/latex], we would exchange [latex]1[\/latex] ten for [latex]10[\/latex] ones. When we do this without models, we say we borrow [latex]1[\/latex] from the tens place and add [latex]10[\/latex] to the ones place.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=143327&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercise<\/h3>\r\nSubtract and then check by adding: [latex]207 - 64[\/latex].\r\n[reveal-answer q=\"125839\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"125839\"]\r\n\r\nSolution\r\n<table id=\"eip-274\" class=\"unnumbered unstyled\" style=\"width: 913px;\" summary=\"a\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 593.059px;\">Write the numbers so each place value lines up vertically.<\/td>\r\n<td style=\"width: 299.941px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215447\/CNX_BMath_Figure_01_03_030-01.png\" alt=\"Vertical subtraction for 207 minus 64\" width=\"108\" height=\"59\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 593.059px;\">Subtract the ones. [latex]7 - 4=3[\/latex].\r\n\r\nWrite the [latex]3[\/latex] in the ones place in the difference. Write the [latex]3[\/latex] in the ones place in the difference.<\/td>\r\n<td style=\"width: 299.941px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215448\/CNX_BMath_Figure_01_03_030-02.png\" alt=\"7 minus 4 equals 3.\" width=\"109\" height=\"99\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 593.059px;\">Subtract the tens. We cannot subtract [latex]6[\/latex] from [latex]0[\/latex] so we borrow [latex]1[\/latex] hundred and add [latex]10[\/latex] tens to the [latex]0[\/latex] tens we had. This makes a total of [latex]10[\/latex] tens. We write [latex]10[\/latex] above the tens place and cross out the [latex]0[\/latex]. Then we cross out the [latex]2[\/latex] in the hundreds place and write [latex]1[\/latex] above it.<\/td>\r\n<td style=\"width: 299.941px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215449\/CNX_BMath_Figure_01_03_030-03.png\" alt=\"The 0 borrows from the 2, so the 0 becomes 10 and the 2 becomes 1.\" width=\"108\" height=\"113\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 593.059px;\">Now we subtract the tens. [latex]10 - 6=4[\/latex]. We write the 4 in the tens place in the difference.<\/td>\r\n<td style=\"width: 299.941px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215450\/CNX_BMath_Figure_01_03_030-04.png\" alt=\"10 minus 6 equals 4.\" width=\"109\" height=\"114\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 593.059px;\">Finally, subtract the hundreds. There is no digit in the hundreds place in the bottom number so we can imagine a [latex]0[\/latex] in that place. Since [latex]1 - 0=1[\/latex], we write [latex]1[\/latex] in the hundreds place in the difference.<\/td>\r\n<td style=\"width: 299.941px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215451\/CNX_BMath_Figure_01_03_030-05.png\" alt=\"1 carries down into the difference. In total, we have 207 minus 64 equals 143.\" width=\"111\" height=\"116\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 593.059px;\" data-align=\"left\">Check by adding.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215451\/CNX_BMath_Figure_01_03_030-06.png\" alt=\"143 plus 64 equals 207, therefore our check is succcessful and our answer correct.\" width=\"106\" height=\"111\" data-media-type=\"image\/png\" \/>\r\n\r\nOur answer is correct.<\/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<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=143341&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Exercise<\/h3>\r\nSubtract and then check by adding: [latex]2,162 - 479[\/latex].\r\n[reveal-answer q=\"61197\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"61197\"]\r\n\r\nSolution\r\n<table id=\"eip-7544534\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image is two columns. The left column includes annotations and the right column includes math expressions. The first line reads \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 697.014px;\">Write the numbers so each place values line up vertically.<\/td>\r\n<td style=\"width: 196.986px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215458\/CNX_BMath_Figure_01_03_028_img-02.png\" alt=\"Vertical subtraction for 2,162 minus 479.\" width=\"103\" height=\"66\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 697.014px;\">Subtract the ones. Since we cannot subtract [latex]9[\/latex] from [latex]2[\/latex], borrow [latex]1[\/latex] ten and add [latex]10[\/latex] ones to the [latex]2[\/latex] ones to make [latex]12[\/latex] ones. Write [latex]5[\/latex] above the tens place and cross out the [latex]6[\/latex]. Write [latex]12[\/latex] above the ones place and cross out the [latex]2[\/latex].<\/td>\r\n<td style=\"width: 196.986px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215459\/CNX_BMath_Figure_01_03_028_img-03.png\" alt=\"The 2 borrows from the 6, so the 2 becomes 12 and the 6 becomes 5.\" width=\"100\" height=\"83\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 697.014px;\">Now we can subtract the ones.<\/td>\r\n<td style=\"width: 196.986px;\">[latex]12 - 9=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 697.014px;\">Write [latex]3[\/latex] in the ones place in the difference.<\/td>\r\n<td style=\"width: 196.986px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215500\/CNX_BMath_Figure_01_03_028_img-04.png\" alt=\"12 minus 9 equals 3.\" width=\"102\" height=\"122\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 697.014px;\">Subtract the tens. Since we cannot subtract [latex]7[\/latex] from [latex]5[\/latex], borrow [latex]1[\/latex] hundred and add [latex]10[\/latex] tens to the [latex]5[\/latex] tens to make [latex]15[\/latex] tens. Write [latex]0[\/latex] above the hundreds place and cross out the [latex]1[\/latex]. Write [latex]15[\/latex] above the tens place.<\/td>\r\n<td style=\"width: 196.986px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215501\/CNX_BMath_Figure_01_03_028_img-05.png\" alt=\"The 5 borrows from the 1, so the 5 becomes 15 and the 1 becomes 0.\" width=\"98\" height=\"140\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 697.014px;\">Now we can subtract the tens.<\/td>\r\n<td style=\"width: 196.986px;\">[latex]15 - 7=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 697.014px;\">Write [latex]8[\/latex] in the tens place in the difference.<\/td>\r\n<td style=\"width: 196.986px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215502\/CNX_BMath_Figure_01_03_028_img-06.png\" alt=\"15 minus 7 equals 8.\" width=\"97\" height=\"116\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 697.014px;\">Now we can subtract the hundreds.<\/td>\r\n<td style=\"width: 196.986px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215503\/CNX_BMath_Figure_01_03_028_img-07.png\" alt=\"The 0 borrows from the 2, so the 0 becomes 10 and the 2 becomes 1.\" width=\"101\" height=\"144\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 697.014px;\">Write [latex]6[\/latex] in the hundreds place in the difference.<\/td>\r\n<td style=\"width: 196.986px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215504\/CNX_BMath_Figure_01_03_028_img-08.png\" alt=\"10 minus 4 equals 6.\" width=\"102\" height=\"119\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 697.014px;\">Subtract the thousands. There is no digit in the thousands place of the bottom number, so we imagine a [latex]0[\/latex]. [latex]1 - 0=1[\/latex]. Write [latex]1[\/latex] in the thousands place of the difference.<\/td>\r\n<td style=\"width: 196.986px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215505\/CNX_BMath_Figure_01_03_028_img-09.png\" alt=\"1 carries down into the difference. In total, we have 2,162 minus 479 equals 1,683.\" width=\"102\" height=\"122\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 697.014px;\">Check by adding.\r\n\r\n[latex]\\begin{array}{}\\\\ \\stackrel{1}{1},\\stackrel{1}{6}\\stackrel{1}{8}3\\hfill \\\\ \\underset{\\text{______}}{+479}\\hfill \\\\ 2,162\\quad\\checkmark \\hfill \\end{array}\\quad\\checkmark[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOur answer is correct.\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<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=143343&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"360\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nWatch the video below to see another example of subtracting whole numbers by lining up place values.\r\n\r\nhttps:\/\/youtu.be\/hneqy1EGACs\r\n<h2 data-type=\"title\">Multiplication<\/h2>\r\nIn order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section. The table below shows the multiplication facts.\r\n\r\nEach box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers.\r\n<table id=\"fs-id1563789\" class=\"column-header\" style=\"width: 398px;\" summary=\"This is a multiplication table with 11 columns and 11 rows. The first column has the values \">\r\n<thead>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<th style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/th>\r\n<th style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]1[\/latex]<\/th>\r\n<th style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]2[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]3[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]4[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]5[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]6[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]7[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]8[\/latex]<\/th>\r\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]9[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]7[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]10[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]12[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]14[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]16[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]18[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]9[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]12[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]15[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]18[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]21[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]24[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]27[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]4[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]12[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]16[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]20[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]24[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]28[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]32[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]36[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]10[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]15[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]20[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]25[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]30[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]35[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]40[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]45[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]6[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]12[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]18[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]24[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]30[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]36[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]42[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]48[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]54[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15.7812px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15.7812px;\" data-align=\"center\">[latex]7[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15.7812px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15.7812px;\" data-align=\"center\">[latex]7[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15.7812px;\" data-align=\"center\">[latex]14[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]21[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]28[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]35[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]42[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]49[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]56[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]63[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]16[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]24[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]32[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]40[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]48[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]56[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]64[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]72[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\" valign=\"middle\">\r\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]9[\/latex]<\/td>\r\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]9[\/latex]<\/td>\r\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]18[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]27[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]36[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]45[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]54[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]63[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]72[\/latex]<\/td>\r\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]81[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe know that changing the order of addition does not change the sum. We saw that [latex]8+9=17[\/latex] is the same as [latex]9+8=17[\/latex].\r\n\r\nIs this also true for multiplication? Let\u2019s look at a few pairs of factors.\r\n<p style=\"text-align: center;\">[latex]4\\cdot 7=28\\quad 7\\cdot 4=28[\/latex]\r\n[latex]9\\cdot 7=63\\quad 7\\cdot 9=63[\/latex]\r\n[latex]8\\cdot 9=72\\quad 9\\cdot 8=72[\/latex]<\/p>\r\nWhen the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.\r\n<div class=\"textbox shaded\">\r\n<h3>Commutative Property of Multiplication<\/h3>\r\nChanging the order of the factors does not change their product.\r\n<p style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply:\r\n\r\n[latex]8\\cdot 7[\/latex]\r\n[latex]7\\cdot 8[\/latex]\r\n\r\n[reveal-answer q=\"468629\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"468629\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168287497846\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td>[latex]8\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]56[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.<\/td>\r\n<td>[latex]7\\cdot 8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]56[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nChanging the order of the factors does not change the product.\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<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144424&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"340\"><\/iframe>\r\n\r\n<\/div>\r\nTo multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\hfill 27\\\\ \\hfill \\underset{\\text{___}}{\\times 3}\\end{array}[\/latex]<\/p>\r\nWe start by multiplying [latex]3[\/latex] by [latex]7[\/latex].\r\n<p style=\"text-align: center;\">[latex]3\\times 7=21[\/latex]<\/p>\r\nWe write the [latex]1[\/latex] in the ones place of the product. We carry the [latex]2[\/latex] tens by writing [latex]2[\/latex] above the tens 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\/24215526\/CNX_BMath_Figure_01_04_008_img.png\" alt=\"Vertical multiplication for 27 times 3. In the product, that is below the line, we have a 1, to which an arrow is pointing labeled &quot;Here is the 1 in 21&quot;. Above the 2 in 27, we have a smaller two, to which an arrow points labeled &quot;Here are the 2 tens in 21&quot;. \" width=\"165\" height=\"153\" data-media-type=\"image\/png\" \/>\r\nThen we multiply the [latex]3[\/latex] by the [latex]2[\/latex], and add the [latex]2[\/latex] above the tens place to the product. So [latex]3\\times 2=6[\/latex], and [latex]6+2=8[\/latex]. Write the [latex]8[\/latex] in the tens place of the product.\r\n<p style=\"text-align: center;\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215527\/CNX_BMath_Figure_01_04_010_img.png\" alt=\"We now have an 8 in the ten's place of our product, to which an arrow points labeled &quot;This comes from 3 times 2 plus the 2 we carried.&quot;\" width=\"221\" height=\"124\" data-media-type=\"image\/png\" \/>\r\nThe product is [latex]81[\/latex].<\/p>\r\n&nbsp;\r\n\r\nWhen we multiply two numbers with a different number of digits, it\u2019s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]15\\cdot 4[\/latex]\r\n\r\n[reveal-answer q=\"942773\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"942773\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168287565780\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Write the numbers so the digits [latex]5[\/latex] and [latex]4[\/latex] line up vertically.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill 15\\\\ \\hfill \\underset{\\text{_____}}{\\times 4}\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]4[\/latex] by the digit in the ones place of [latex]15[\/latex]. [latex]4\\cdot 5=20[\/latex].<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write [latex]0[\/latex] in the ones place of the product and carry the [latex]2[\/latex] tens.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{2}{1}5\\\\ \\hfill \\underset{\\text{_____}}{\\times 4}\\\\ \\hfill 0\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]4[\/latex] by the digit in the tens place of [latex]15[\/latex]. [latex]4\\cdot 1=4[\/latex] .\r\n\r\nAdd the [latex]2[\/latex] tens we carried. [latex]4+2=6[\/latex] .<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the [latex]6[\/latex] in the tens place of the product.<\/td>\r\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{2}{1}5\\\\ \\hfill \\underset{\\text{_____}}{\\times 4}\\\\ \\hfill 60\\end{array}[\/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<iframe id=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144427&amp;theme=oea&amp;iframe_resize_id=mom\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]286\\cdot 5[\/latex]\r\n\r\n[reveal-answer q=\"849168\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"849168\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168288632671\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 109.441px;\">\r\n<td style=\"height: 109.441px;\">Write the numbers so the digits [latex]5[\/latex] and [latex]6[\/latex] line up vertically.<\/td>\r\n<td style=\"height: 109.441px;\">[latex]\\begin{array}{c}\\hfill 286\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 88px;\">\r\n<td style=\"height: 88px;\">Multiply [latex]5[\/latex] by the digit in the ones place of [latex]286[\/latex].\r\n\r\n[latex]5\\cdot 6=30[\/latex]<\/td>\r\n<td style=\"height: 88px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 145px;\">\r\n<td style=\"height: 145px;\">Write the [latex]0[\/latex] in the ones place of the product and carry the [latex]3[\/latex] to the tens place.Multiply [latex]5[\/latex] by the digit in the tens place of [latex]286[\/latex].\r\n\r\n[latex]5\\cdot 8=40[\/latex]<\/td>\r\n<td style=\"height: 145px;\">[latex]\\begin{array}{}\\\\ \\hfill 2\\stackrel{3}{8}6\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\\\ \\hfill 0\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 161px;\">\r\n<td style=\"height: 161px;\">Add the [latex]3[\/latex] tens we carried to get [latex]40+3=43[\/latex] .\r\n\r\nWrite the [latex]3[\/latex] in the tens place of the product and carry the [latex]4[\/latex] to the hundreds place.<\/td>\r\n<td style=\"height: 161px;\">[latex]\\begin{array}{c}\\hfill \\stackrel{4}{2}\\stackrel{3}{8}6\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\\\ \\hfill 30\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 251px;\">\r\n<td style=\"height: 251px;\">Multiply [latex]5[\/latex] by the digit in the hundreds place of [latex]286[\/latex]. [latex]5\\cdot 2=10[\/latex].\r\n\r\nAdd the [latex]4[\/latex] hundreds we carried to get [latex]10+4=14[\/latex].\r\n\r\nWrite the [latex]4[\/latex] in the hundreds place of the product and the [latex]1[\/latex] to the thousands place.<\/td>\r\n<td style=\"height: 251px;\">[latex]\\begin{array}{c}\\hfill \\stackrel{4}{2}\\stackrel{3}{8}6\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\\\ \\hfill 1,430\\end{array}[\/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<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144429&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<\/div>\r\nWhen we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.\r\n<div class=\"textbox shaded\">\r\n<h3>MultiplIcation of whole numbers<\/h3>\r\n<ol id=\"eip-id1168286064302\" class=\"stepwise\" data-number-style=\"arabic\">\r\n \t<li>Write the numbers so each place value lines up vertically.<\/li>\r\n \t<li>Multiply the digits in each place value.\r\n<ul id=\"eip-id1168288663980\">\r\n \t<li>Work from right to left, starting with the ones place in the bottom number.\r\n<ul id=\"eip-id1168288663983\" data-bullet-style=\"bullet\">\r\n \t<li>Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.<\/li>\r\n \t<li>If a product in a place value is more than [latex]9[\/latex], carry to the next place value.<\/li>\r\n \t<li>Write the partial products, lining up the digits in the place values with the numbers above.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Repeat for the tens place in the bottom number, the hundreds place, and so on.<\/li>\r\n \t<li>Insert a zero as a placeholder with each additional partial product.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Add the partial products.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]62\\left(87\\right)[\/latex]\r\n\r\n[reveal-answer q=\"698602\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"698602\"]\r\n\r\nSolution\r\n<table id=\"eip-279\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"No alt text\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 585.903px;\">Write the numbers so each place lines up vertically.<\/td>\r\n<td style=\"width: 308.097px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215528\/CNX_BMath_Figure_01_04_020_img-02.png\" alt=\"Vertical multiplication for 62 times 87.\" width=\"42\" height=\"46\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 585.903px;\">Start by multiplying 7 by 62. Multiply 7 by the digit in the ones place of 62.\r\n\r\n[latex]7\\cdot 2=14[\/latex].\r\n\r\nWrite the 4 in the ones place of the product and carry the 1 to the tens place.<\/td>\r\n<td style=\"width: 308.097px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215529\/CNX_BMath_Figure_01_04_020_img-03.png\" alt=\"2 times 7 is 14, so we carry over our 1 to the ten's place and the 4 is the one's digit of the first product.\" width=\"42\" height=\"92\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 585.903px;\">Multiply 7 by the digit in the tens place of 62. [latex]7\\cdot 6=42[\/latex]. Add the 1 ten we carried.\r\n\r\n[latex]42+1=43[\/latex]latex].\r\n\r\nWrite the 3 in the tens place of the product and the 4 in the hundreds place.<\/td>\r\n<td style=\"width: 308.097px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215530\/CNX_BMath_Figure_01_04_020_img-04.png\" alt=\"6 times 7 is 42 plus 1 equals 43, which becomes are hundred's and ten's place of our first product.\" width=\"42\" height=\"92\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 585.903px;\">The first partial product is [latex]434[\/latex].<\/td>\r\n<td style=\"width: 308.097px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 585.903px;\">Now, write a [latex]0[\/latex] under the [latex]4[\/latex] in the ones place of the next partial product as a placeholder since we now multiply the digit in the tens place of [latex]87[\/latex] by [latex]62[\/latex].\r\n\r\nMultiply [latex]8[\/latex] by the digit in the ones place of [latex]62[\/latex]\r\n\r\n[latex]8\\cdot 2=16[\/latex]. Write the [latex]6[\/latex] in the next place of the product, which is the tens place. Carry the [latex]1[\/latex] to the tens place.<\/td>\r\n<td style=\"width: 308.097px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215530\/CNX_BMath_Figure_01_04_020_img-05.png\" alt=\"Begin with a 0 in the one's place of our second product. 2 times 8 equals 16, so we carry over the 1 to the ten's place and the 6 is the ten's digit of the second product.\" width=\"42\" height=\"127\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 585.903px;\">Multiply [latex]8[\/latex] by [latex]6[\/latex], the digit in the tens place of [latex]62[\/latex], then add the [latex]1[\/latex] ten we carried to get [latex]49[\/latex].\r\n\r\nWrite the [latex]9[\/latex] in the hundreds place of the product and the [latex]4[\/latex] in the thousands place.<\/td>\r\n<td style=\"width: 308.097px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215531\/CNX_BMath_Figure_01_04_020_img-06.png\" alt=\"6 times 8 is 48 plus 1 equals 49, which becomes the hundred's and thousand's place of the second product.\" width=\"42\" height=\"127\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 585.903px;\">The second partial product is [latex]4960[\/latex]. Add the partial products.<\/td>\r\n<td style=\"width: 308.097px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215532\/CNX_BMath_Figure_01_04_020_img-07.png\" alt=\"In total, 62 times 87 equals 434 plus 4960 which equals 5394.\" width=\"42\" height=\"155\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe product is [latex]5,394[\/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<iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144432&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"280\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWhen there are three or more factors, we multiply the first two and then multiply their product by the next factor. For example:\r\n<table id=\"eip-0\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>to multiply<\/td>\r\n<td>[latex]8\\cdot 3\\cdot 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>first multiply [latex]8\\cdot 3[\/latex]<\/td>\r\n<td>[latex]24\\cdot 2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>then multiply [latex]24\\cdot 2[\/latex]<\/td>\r\n<td>[latex]48[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the video below, we summarize the concepts presented on this page including the multiplication property of zero, the identity property of multiplication, and the commutative property of multiplication.m\r\n\r\nhttps:\/\/youtu.be\/kW7JBfplJGE\r\n<h2 data-type=\"title\">Division<\/h2>\r\nWe said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know [latex]12\\div 4=3[\/latex] because [latex]3\\cdot 4=12[\/latex]. Knowing all the multiplication number facts is very important when doing division.\r\n\r\nWe check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. We know [latex]24\\div 8=3[\/latex] is correct because [latex]3\\cdot 8=24[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide. Then check by multiplying.\r\n<ol>\r\n \t<li>[latex]42\\div 6[\/latex]<\/li>\r\n \t<li>[latex]\\frac{72}{9}[\/latex]<\/li>\r\n \t<li>[latex]7\\overline{)63}[\/latex]<\/li>\r\n<\/ol>\r\nSolution:\r\n<table id=\"eip-id1168287031935\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]42\\div 6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide [latex]42[\/latex] by [latex]6[\/latex].<\/td>\r\n<td>[latex]7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]7\\cdot 6[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]42\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id11682870335\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\frac{72}{9}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide [latex]72[\/latex] by [latex]9[\/latex].<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]8\\cdot 9[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]72\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id11670335\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>3.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]7\\overline{)63}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide [latex]63[\/latex] by [latex]7[\/latex].<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]9\\cdot 7[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]63\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/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]144463[\/ohm_question]\r\n\r\n<\/div>\r\nWhat is the quotient when you divide a number by itself?\r\n<p style=\"text-align: center;\">[latex]\\frac{15}{15}=1\\text{ because }1\\cdot 15=15[\/latex]<\/p>\r\nDividing any number [latex]\\text{(except 0)}[\/latex] by itself produces a quotient of [latex]1[\/latex]. Also, any number divided by [latex]1[\/latex] produces a quotient of the number. These two ideas are stated in the Division Properties of One.\r\n<div class=\"textbox shaded\">\r\n<h3>Division Properties of One<\/h3>\r\n<table id=\"eip-735\" summary=\"a\" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"height: 14px;\">Any number (except 0) divided by itself is one.<\/td>\r\n<td style=\"height: 14px;\">[latex]a\\div a=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14.4585px;\">\r\n<td style=\"height: 14.4585px;\">Any number divided by one is the same number.<\/td>\r\n<td style=\"height: 14.4585px;\">[latex]a\\div 1=a[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide. Then check by multiplying:\r\n<ol id=\"eip-id1168288568257\" class=\"circled\" data-number-style=\"arabic\">\r\n \t<li>[latex]11\\div 11[\/latex]<\/li>\r\n \t<li>[latex]\\frac{19}{1}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"519474\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"519474\"]\r\n\r\nSolution:\r\n<table id=\"eip-id1168288480300\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]11\\div 11[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A number divided by itself is [latex]1[\/latex].<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]1\\cdot 11[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]11\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<table id=\"eip-id1168288536381\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\frac{19}{1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>A number divided by [latex]1[\/latex] equals itself.<\/td>\r\n<td>[latex]19[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]19\\cdot 1[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]19\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\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]144635[\/ohm_question]\r\n\r\n<\/div>\r\nSuppose we have [latex]\\text{\\$0}[\/latex], and want to divide it among [latex]3[\/latex] people. How much would each person get? Each person would get [latex]\\text{\\$0}[\/latex]. Zero divided by any number is [latex]0[\/latex].\r\n\r\nNow suppose that we want to divide [latex]\\text{\\$10}[\/latex] by [latex]0[\/latex]. That means we would want to find a number that we multiply by [latex]0[\/latex] to get [latex]10[\/latex]. This cannot happen because [latex]0[\/latex] times any number is [latex]0[\/latex]. Division by zero is said to be <em data-effect=\"italics\">undefined<\/em>.\r\n\r\nThese two ideas make up the Division Properties of Zero.\r\n<div class=\"textbox shaded\">\r\n<h3>Division Properties of Zero<\/h3>\r\n<table id=\"eip-158\" summary=\"a\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>Zero divided by any number is [latex]0[\/latex].<\/td>\r\n<td>[latex]0\\div a=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Dividing a number by zero is undefined.<\/td>\r\n<td>[latex]a\\div 0[\/latex] undefined<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nAnother way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away [latex]0[\/latex] from [latex]10?[\/latex] Because subtracting [latex]0[\/latex] will never change the total, we will never get an answer. So we cannot divide a number by [latex]0[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide. Check by multiplying:\r\n<ol>\r\n \t<li>[latex]0\\div 3[\/latex]<\/li>\r\n \t<li>[latex]\\frac{10}{0}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"208505\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"208505\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168288542869\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]0\\div 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Zero divided by any number is zero.<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Check by multiplying.\r\n\r\n[latex]0\\cdot 3[\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<table id=\"eip-id1168288419363\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]10\/0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Division by zero is undefined.<\/td>\r\n<td>undefined<\/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]144478[\/ohm_question]\r\n\r\n<\/div>\r\nWhen the divisor or the dividend has more than one digit, it is usually easier to use the [latex]4\\overline{)12}[\/latex] notation. This process is called long division. Let\u2019s work through the process by dividing [latex]78[\/latex] by [latex]3[\/latex].\r\n<table id=\"eip-244\" class=\"unnumbered unstyled\" style=\"width: 970.438px;\" summary=\"This image has 2 columns. the left column contains instructions and the right column contains expressions. The exercises being worked out is 78 divided by 3. The first line reads: Divide the first digit of dividend 7, by the divisor, 3. The next line reads: the divisor 3 can go into 7 two times since 2 times 3 equals 6. Write the 2 above the 7 in the quotient. Next to this shows the expression 3 divided by 78, with the two above the seven in the quotient. The next line reads \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 599px;\">Divide the first digit of the dividend, [latex]7[\/latex], by the divisor, [latex]3[\/latex].<\/td>\r\n<td style=\"width: 337px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">The divisor [latex]3[\/latex] can go into [latex]7[\/latex] two times since [latex]2\\times 3=6[\/latex] . Write the [latex]2[\/latex] above the [latex]7[\/latex] in the quotient.<\/td>\r\n<td style=\"width: 337px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215615\/CNX_BMath_Figure_01_05_043_img-02.png\" alt=\"Long division for 78 divided by 3. 3 goes into 7 2 times, so 2 becomes the ten's digit of our quotient.\" width=\"38\" height=\"46\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">Multiply the [latex]2[\/latex] in the quotient by [latex]2[\/latex] and write the product, [latex]6[\/latex], under the[latex]7[\/latex].<\/td>\r\n<td style=\"width: 337px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215616\/CNX_BMath_Figure_01_05_043_img-03.png\" alt=\"Thus, we subtract 6 from 7.\" width=\"35\" height=\"64\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">Subtract that product from the first digit in the dividend. Subtract [latex]7 - 6[\/latex] . Write the difference, 1, under the first digit in the dividend.<\/td>\r\n<td style=\"width: 337px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215616\/CNX_BMath_Figure_01_05_043_img-04.png\" alt=\"7 minus 6 equals 1.\" width=\"35\" height=\"78\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">Bring down the next digit of the dividend. Bring down the [latex]8[\/latex].<\/td>\r\n<td style=\"width: 337px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215617\/CNX_BMath_Figure_01_05_043_img-05.png\" alt=\"We carry the one's digit, 8, down to the 1, resulting in 18.\" width=\"35\" height=\"85\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">Divide [latex]18[\/latex] by the divisor, [latex]3[\/latex]. The divisor [latex]3[\/latex] goes into [latex]18[\/latex] six times.<\/td>\r\n<td style=\"width: 337px;\" rowspan=\"2\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215617\/CNX_BMath_Figure_01_05_043_img-06.png\" alt=\"3 goes into 18 6 times, so 6 becomes the one's digit in our quotient.\" width=\"35\" height=\"80\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\" data-valign=\"bottom\">Write [latex]6[\/latex] in the quotient above the [latex]8[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 599px;\">Multiply the [latex]6[\/latex] in the quotient by the divisor and write the product, [latex]18[\/latex], under the dividend. Subtract [latex]18[\/latex] from [latex]18[\/latex].<\/td>\r\n<td style=\"width: 337px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215618\/CNX_BMath_Figure_01_05_043_img-07.png\" alt=\"18 minus 18 equals 0. We are left with a remainder of 0. In total 78 divided by 3 equals 26.\" width=\"35\" height=\"117\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished.\r\n<p style=\"text-align: center;\">[latex]\\text{So }78\\div 3=26[\/latex].<\/p>\r\nCheck by multiplying the quotient times the divisor to get the dividend. Multiply [latex]26\\times 3[\/latex] to make sure that product equals the dividend, [latex]78[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\hfill \\stackrel{1}{2}6\\\\ \\hfill \\underset{\\text{___}}{\\times 3}\\\\ \\hfill 78 \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">It does, so our answer is correct.\u00a0[latex]\\checkmark[\/latex]<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Division of whole numbers<\/h3>\r\n<ol id=\"eip-id1168288534169\" class=\"stepwise\" data-number-style=\"arabic\">\r\n \t<li>Divide the first digit of the dividend by the divisor.If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on.<\/li>\r\n \t<li>Write the quotient above the dividend.<\/li>\r\n \t<li>Multiply the quotient by the divisor and write the product under the dividend.<\/li>\r\n \t<li>Subtract that product from the dividend.<\/li>\r\n \t<li>Bring down the next digit of the dividend.<\/li>\r\n \t<li>Repeat from Step 1 until there are no more digits in the dividend to bring down.<\/li>\r\n \t<li>Check by multiplying the quotient times the divisor.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the video below we show another example of using long division.\r\n\r\nhttps:\/\/youtu.be\/KvVhaB5mqr8\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide [latex]2,596\\div 4[\/latex]. Check by multiplying:\r\n[reveal-answer q=\"252445\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"252445\"]\r\n\r\nSolution\r\n<table id=\"eip-287\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows long division for solving the expression 4 divided by 2,596. The first line says to divide the first digit of the dividend, 2, by the divisor, 4. The expression shows the 2 in red. The next line says that since 4 does not go into 2, we use the first two digits of the dividend and divide 25 by 4. the divisor 4 goes into 25 six times. We write the six in the quotient above the 5. The expression now shows the 6 above the 5 in the quotient. Next, multiply the 6 in the quotient by the divisor 4 and write the product, 24, under the first two digits in the dividend. The expression shows the 24 under the 25. Subtract 25 minus 4. Write the difference, 1, under the second digit in the dividend. The expression shows in long division, 25 minus 24 equals 1. Now bring the 9 down and repeat these steps. There are 4 fours in 19. Write the 4 over the 9. Multiply the 4 by 4 and subtract this product from 19. The expression now shows in long division, the 9 in the dividend brought down next to the 1, to make 19, and the 4 in the quotient above the 9. Under the 19 is 16 with a difference of 3. The next line says to bring down the 6 and repeat these steps. there are 9 fours in 36. Write 9 over the 6. Multiply the 9 by 4 and subtract this product from 36. The expression shows in long division the 6 brought down next to the 3 to make 36. The 9 in the quotient above the 6 for an answer of 649.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 676.753px;\">Let's rewrite the problem to set it up for long division.<\/td>\r\n<td style=\"width: 218.247px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215619\/CNX_BMath_Figure_01_05_044_img-01.png\" alt=\"Long division for 2,596 divided by 4.\" width=\"84\" height=\"27\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 676.753px;\">Divide the first digit of the dividend, [latex]2[\/latex], by the divisor, [latex]4[\/latex].<\/td>\r\n<td style=\"width: 218.247px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215620\/CNX_BMath_Figure_01_05_044_img-02.png\" alt=\"First, we must consider how many times 4 goes into 2.\" width=\"84\" height=\"27\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 676.753px;\" data-valign=\"bottom\">Since [latex]4[\/latex] does not go into [latex]2[\/latex], we use the first two digits of the dividend and divide [latex]25[\/latex] by [latex]4[\/latex]. The divisor [latex]4[\/latex] goes into [latex]25[\/latex] six times.<\/td>\r\n<td style=\"width: 218.247px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 676.753px;\">We write the [latex]6[\/latex] in the quotient above the [latex]5[\/latex].<\/td>\r\n<td style=\"width: 218.247px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215620\/CNX_BMath_Figure_01_05_044_img-03.png\" alt=\"4 does not go into 2, so we consider how many times 4 goes into 25. 4 goes into 25 6 times, so 6 becomes the hundred's place in our quotient.\" width=\"84\" height=\"46\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 676.753px;\">Multiply the [latex]6[\/latex]in the quotient by the divisor [latex]4[\/latex] and write the product, [latex]24[\/latex], under the first two digits in the dividend.<\/td>\r\n<td style=\"width: 218.247px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215621\/CNX_BMath_Figure_01_05_044_img-04.png\" alt=\"We subtract 24 from 25.\" width=\"84\" height=\"64\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 676.753px;\">Subtract that product from the first two digits in the dividend. Subtract [latex]25 - 24[\/latex] . Write the difference, [latex]1[\/latex], under the second digit in the dividend.<\/td>\r\n<td style=\"width: 218.247px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215622\/CNX_BMath_Figure_01_05_044_img-05.png\" alt=\"25 minus 24 equals 1\" width=\"84\" height=\"85\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 676.753px;\">Now bring down the [latex]9[\/latex] and repeat these steps. There are [latex]4[\/latex] fours in [latex]19[\/latex]. Write the [latex]4[\/latex] over the [latex]9[\/latex]. Multiply the [latex]4[\/latex] by [latex]4[\/latex] and subtract this product from [latex]19[\/latex].<\/td>\r\n<td style=\"width: 218.247px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215624\/CNX_BMath_Figure_01_05_044_img-06.png\" alt=\"We carry the ten's digit, 9 down to the 1, resulting in 19. Consider how many times 4 goes into 19. 4 goes into 19 4 times, so 4 becomes our ten's digit in the quotient. We subtract 16 from 19. 16 minus 19 equals 3.\" width=\"84\" height=\"125\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 676.753px;\">Bring down the [latex]6[\/latex] and repeat these steps. There are [latex]9[\/latex] fours in [latex]36[\/latex]. Write the [latex]9[\/latex] over the [latex]6[\/latex]. Multiply the [latex]9[\/latex] by [latex]4[\/latex] and subtract this product from [latex]36[\/latex].<\/td>\r\n<td style=\"width: 218.247px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215625\/CNX_BMath_Figure_01_05_044_img-07.png\" alt=\"We carry the one's digit, 6 down to the 3, resulting in 36. Consider how many times 4 goes into 36. 4 goes into 36 9 times, so 9 becomes the one's digit in our quotient. We subtract 36 from 36. 36 minus 36 equals 0. We are left with a remainder of 0. In total, we have 2596 divided by 4 equals 649.\" width=\"84\" height=\"161\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 676.753px;\">So [latex]2,596\\div 4=649[\/latex] .<\/td>\r\n<td style=\"width: 218.247px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 676.753px;\">Check by multiplying.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215626\/CNX_BMath_Figure_01_05_044_img-08.png\" alt=\"649 times 4 equals 2596, therefore our check is succcessful and our answer correct.\" width=\"84\" height=\"84\" data-media-type=\"image\/png\" \/><\/td>\r\n<td style=\"width: 218.247px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt equals the dividend, so our answer is correct.\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]144636[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide [latex]4,506\\div 6[\/latex]. Check by multiplying:\r\n[reveal-answer q=\"474096\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"474096\"]\r\n\r\nSolution\r\n<table id=\"eip-483\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows how 4,506 divided by six is worked out in long division. The first line reads first we try to divide 6 into 4. Since that won't work, we try 6 into 45. There are 7 sixes in 45. We write the 7 over the 5. The expression shows the 7 above the 5 in the quotient. Multiply the 7 by the 6 and subtract the product from 45. The expression shows in long division, 45 minus 42, equals 3. The next line reads \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 672.847px;\">Let's rewrite the problem to set it up for long division.<\/td>\r\n<td style=\"width: 222.153px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215627\/CNX_BMath_Figure_01_05_045_img-01.png\" alt=\"Long division for 4506 divided by 6.\" width=\"84\" height=\"32\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">First we try to divide [latex]6[\/latex] into [latex]4[\/latex].<\/td>\r\n<td style=\"width: 222.153px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215628\/CNX_BMath_Figure_01_05_045_img-02.png\" alt=\"Consider how many times 6 goes into 4.\" width=\"84\" height=\"27\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">Since that won't work, we try [latex]6[\/latex] into [latex]45[\/latex].\r\n\r\nThere are [latex]7[\/latex] sixes in [latex]45[\/latex]. We write the [latex]7[\/latex] over the [latex]5[\/latex].<\/td>\r\n<td style=\"width: 222.153px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215628\/CNX_BMath_Figure_01_05_045_img-03.png\" alt=\"6 does not go into 4, so we consider how many times 6 goes into 45. 6 goes into 45 7 times, so 7 becomes the hundred's place in the quotient. \" width=\"84\" height=\"46\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">Multiply the [latex]7[\/latex] by [latex]6[\/latex] and subtract this product from [latex]45[\/latex].<\/td>\r\n<td style=\"width: 222.153px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215629\/CNX_BMath_Figure_01_05_045_img-04.png\" alt=\"We subtract 42 from 45. 45 minus 42 equals 3.\" width=\"84\" height=\"85\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">Now bring down the [latex]0[\/latex] and repeat these steps. There are [latex]5[\/latex] sixes in [latex]30[\/latex]. Write the [latex]5[\/latex] over the [latex]0[\/latex]. Multiply the [latex]5[\/latex] by [latex]6[\/latex] and subtract this product from [latex]30[\/latex].<\/td>\r\n<td style=\"width: 222.153px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215631\/CNX_BMath_Figure_01_05_045_img-05.png\" alt=\"We carry the ten's digit, 0, down to 3, resulting in 30. Consider how many times 6 goes into 30. 6 goes into 30 5 times, so 5 becomes the ten's digit in our quotient. We subtract 30 from 30, which equals 0.\" width=\"84\" height=\"125\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">Now bring down the [latex]6[\/latex] and repeat these steps. There is [latex]1[\/latex] six in [latex]6[\/latex]. Write the [latex]1[\/latex] over the [latex]6[\/latex]. Multiply [latex]1[\/latex] by [latex]6[\/latex] and subtract this product from [latex]6[\/latex]<\/td>\r\n<td style=\"width: 222.153px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215632\/CNX_BMath_Figure_01_05_045_img-06.png\" alt=\"We carry down the one's digit, 6, to the 0, resulting in 6. Consider how many times 6 goes into 6. 6 goes into 6 1 time, so 1 becomes the one's digit in our quotient. Subtract 6 from 6 which is 0. We have a remainder of 0. In total 4506 divided by 6 equals 751\" width=\"84\" height=\"164\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 672.847px;\">Check by multiplying.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215633\/CNX_BMath_Figure_01_05_045_img-07.png\" alt=\"751 times 6 equals 4506, therefore our check is succcessful and our answer correct.\" width=\"84\" height=\"78\" data-media-type=\"image\/png\" \/><\/td>\r\n<td style=\"width: 222.153px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt equals the dividend, so our answer is correct.\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]144640[\/ohm_question]\r\n\r\n<\/div>\r\nWatch this video for another example of how to use long division to divide a four digit whole number by a two digit whole number.\r\n\r\nhttps:\/\/youtu.be\/V7Korf09iWI\r\n\r\nSo far all the division problems have worked out evenly. For example, if we had [latex]24[\/latex] cookies and wanted to make bags of [latex]8[\/latex] cookies, we would have [latex]3[\/latex] bags. But what if there were [latex]28[\/latex] cookies and we wanted to make bags of [latex]8?[\/latex] Start with the [latex]28[\/latex] cookies.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215640\/CNX_BMath_Figure_01_05_027.png\" alt=\"An image of 28 cookies placed at random.\" data-media-type=\"image\/png\" \/>\r\nTry to put the cookies in groups of eight.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215642\/CNX_BMath_Figure_01_05_028.png\" alt=\"An image of 28 cookies. There are 3 circles, each containing 8 cookies, leaving 3 cookies outside the circles.\" data-media-type=\"image\/png\" \/>\r\nThere are [latex]3[\/latex] groups of eight cookies, and [latex]4[\/latex] cookies left over. We call the [latex]4[\/latex] cookies that are left over the remainder and show it by writing R4 next to the [latex]3[\/latex]. (The R stands for remainder.)\r\n\r\nTo check this division we multiply [latex]3[\/latex] times [latex]8[\/latex] to get [latex]24[\/latex], and then add the remainder of [latex]4[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\hfill 3\\\\ \\hfill \\underset{\\text{___}}{\\times 8}\\\\ \\hfill 24\\\\ \\hfill \\underset{\\text{___}}{+4}\\\\ \\hfill 28\\end{array}[\/latex]<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide [latex]1,439\\div 4[\/latex]. Check by multiplying.\r\n[reveal-answer q=\"498101\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"498101\"]\r\n\r\nSolution\r\n<table id=\"eip-879\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows 1,439 divided by 4. The first line says to try to divide 4 into 1. Since this won't work, try 4 into 14. There are 3 fours in 14. Write the 3 over the 4. the expression shows the quotient 3 above the 4. Next, multiply the 3 by the 4 and subtract this product from 14. The expressions shows this in long division with the quotient 5 above the 3. The next line says to bring down the 3 and repeat these steps. There are 5 fours in 23. Write the 5 over the 3. Multiply the 5 by 4 and subtract the product from 23. The expression shows this in long division. The next line says to bring down the 9 and repeat these steps. There are 9 fours in 39. Write the 9 over the 9. Multiply the 9 by 4 and subtract the product from 39. There are no more numbers to bring down. There is a remainder of 3. The expression shows the answer of 359 remainder 3.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 695.851px;\">Let's rewrite the problem to set it up for long division.<\/td>\r\n<td style=\"width: 199.149px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215643\/CNX_BMath_Figure_01_05_047_img-01.png\" alt=\"Long division for 1439 divided by 4.\" width=\"171\" height=\"28\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 695.851px;\">First we try to divide [latex]4[\/latex] into [latex]1[\/latex]. Since that won't work, we try [latex]4[\/latex] into [latex]14[\/latex]. There are [latex]3[\/latex] fours in [latex]14[\/latex]. We write the [latex]3[\/latex] over the [latex]4[\/latex].<\/td>\r\n<td style=\"width: 199.149px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215644\/CNX_BMath_Figure_01_05_047_img-02.png\" alt=\"4 goes into 14 3 times, so 3 becomes the hundred's place of the quotient.\" width=\"171\" height=\"45\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 695.851px;\">Multiply the [latex]3[\/latex] by [latex]4[\/latex] and subtract this product from [latex]14[\/latex].<\/td>\r\n<td style=\"width: 199.149px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215645\/CNX_BMath_Figure_01_05_047_img-03.png\" alt=\"14 minus 12 equals 2.\" width=\"171\" height=\"85\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 695.851px;\">Now bring down the [latex]3[\/latex] and repeat these steps. There are [latex]5[\/latex] fours in [latex]23[\/latex]. Write the [latex]5[\/latex] over the [latex]3[\/latex]. Multiply the [latex]5[\/latex] by [latex]4[\/latex] and subtract this product from [latex]23[\/latex].<\/td>\r\n<td style=\"width: 199.149px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215646\/CNX_BMath_Figure_01_05_047_img-04.png\" alt=\"We carry down the ten's digit, 3, to the 2, resulting in 23. Consider how many times 4 goes into 23. 4 goes into 23 5 times, so 5 becomes the ten's place in our quotient. 23 minus 20 equals 3.\" width=\"171\" height=\"125\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 695.851px;\">Now bring down the [latex]9[\/latex] and repeat these steps. There are [latex]9[\/latex] fours in [latex]39[\/latex]. Write the [latex]9[\/latex] over the [latex]9[\/latex]. Multiply the [latex]9[\/latex] by [latex]4[\/latex] and subtract this product from [latex]39[\/latex]. There are no more numbers to bring down, so we are done. The remainder is [latex]3[\/latex].<\/td>\r\n<td style=\"width: 199.149px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215647\/CNX_BMath_Figure_01_05_047_img-05.png\" alt=\"We carry down the one's digit, 9, to the 3, resulting in 39. Consider how many times 4 goes into 39. 4 goes into 39 9 times, so 9 becomes the one's digit in our quotient. 39 minus 36 equals 3. We have a remainder of 3, so we write R 3 in the quotient. In total, 1439 divided by 4 equals 359 with a remainder of 3.\" width=\"171\" height=\"164\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 695.851px;\">Check by multiplying.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215648\/CNX_BMath_Figure_01_05_047_img-06.png\" alt=\"To check, we multiple our quotient, 359, by our divisor, 4, which equals 1436. Then, we add our remainder, 3, which equals 1439, the original dividend.\" width=\"171\" height=\"125\" data-media-type=\"image\/png\" \/><\/td>\r\n<td style=\"width: 199.149px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo [latex]1,439\\div 4[\/latex] is [latex]359[\/latex] with a remainder of [latex]3[\/latex]. Our answer is correct.\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]144643[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDivide and then check by multiplying: [latex]1,461\\div 13[\/latex].\r\n[reveal-answer q=\"174689\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"174689\"]\r\n\r\nSolution\r\n<table id=\"eip-708\" class=\"unnumbered unstyled\" style=\"width: 915px;\" summary=\"This image shows 1,461 divided by 13. The first line says First we try to divide 13 into 1. Since that won\u2019t work, we try 13 into 14. There is 1 thirteen in 14. We write the 1 over the 4. Multiply the 1 by 13 and subtract this product from 14. The expression shows the quotient 1 above the 4, and 13 under 14 with the difference as 1. The next line says to bring down the 6 and repeat these steps. There is 1 thirteen in 16. Write the 1 over the 6. Multiply the 1 by 13 and subtract this product from 16. The expression shows the 6 in the dividend with a blue arrow pointing downward to show the 6 is brought down next to the 1. In long division, 16 minus 3 shows a difference of 3. Next it says, now bring down the 1 and repeat these steps. There are 2 thirteens in 31. Write the 2 over the 1. Multiply the 2 by 13 and subtract this product from 31. The expression shows in long division the 1 in the dividend with a blue arrow pointing downward to show the one brought down to make 31. A 2 is placed above the 1 in the dividend. Thirty-one minus 26 is 5, but there are no more numbers to bring down, so we are done. The remainder is 5. The answer is 112 remainder 5. The next line says to check by multiplying in which the expression shows 112 with the word \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 701.698px;\">Let's rewrite the problem to set it up for long division.<\/td>\r\n<td style=\"width: 192.302px;\">[latex]13\\overline{)1,461}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 701.698px;\">First we try to divide [latex]13[\/latex] into [latex]1[\/latex]. Since that won't work, we try [latex]13[\/latex] into [latex]14[\/latex]. There is [latex]1[\/latex] thirteen in [latex]14[\/latex]. We write the [latex]1[\/latex] over the [latex]4[\/latex].<\/td>\r\n<td style=\"width: 192.302px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215649\/CNX_BMath_Figure_01_05_048_img-02.png\" alt=\"Long division for 1461 divided by 13. 13 goes into 14 1 time, so 1 becomes the hundred's digit in the quotient.\" width=\"173\" height=\"43\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 701.698px;\">Multiply the [latex]1[\/latex] by [latex]13[\/latex] and subtract this product from [latex]14[\/latex].<\/td>\r\n<td style=\"width: 192.302px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215650\/CNX_BMath_Figure_01_05_048_img-03.png\" alt=\"14 minus 13 equals 1.\" width=\"173\" height=\"85\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 701.698px;\">Now bring down the [latex]6[\/latex] and repeat these steps. There is [latex]1[\/latex] thirteen in [latex]16[\/latex]. Write the [latex]1[\/latex] over the [latex]6[\/latex]. Multiply the [latex]1[\/latex] by [latex]13[\/latex] and subtract this product from [latex]16[\/latex].<\/td>\r\n<td style=\"width: 192.302px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215651\/CNX_BMath_Figure_01_05_048_img-04.png\" alt=\"We carry down the ten's digit, 6, to the 1, resulting in 16. 13 goes into 16 1 time, so 1 becomes the ten's digit in our quotient. 16 minus 13 equals 3.\" width=\"173\" height=\"125\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 701.698px;\">Now bring down the [latex]1[\/latex] and repeat these steps. There are [latex]2[\/latex] thirteens in [latex]31[\/latex]. Write the [latex]2[\/latex] over the [latex]1[\/latex]. Multiply the [latex]2[\/latex] by [latex]13[\/latex] and subtract this product from [latex]31[\/latex]. There are no more numbers to bring down, so we are done. The remainder is [latex]5[\/latex]. [latex]1,462\\div 13[\/latex] is [latex]112[\/latex] with a remainder of [latex]5[\/latex].<\/td>\r\n<td style=\"width: 192.302px;\"><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215652\/CNX_BMath_Figure_01_05_048_img-05.png\" alt=\"We carry down the one's digit, 1, to the 3, resulting in 31. 13 goes into 31 twice, so 2 becomes the one's digit in our quotient. 31 minus 26 equals 5. We have a remainder of 5 so we write R 5 in the quotient. In total, 1461 divided by 13 equals 112 with a remainder of 5. \" width=\"173\" height=\"164\" data-media-type=\"image\/png\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 701.698px;\">Check by multiplying.\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215653\/CNX_BMath_Figure_01_05_048_img-06.png\" alt=\"To check, we multiple our quotient, 112, by our divisor, 13, which equals 336 plus 1120, and we add our remainder, 5. All 3 equal 1461, our original dividend.\" width=\"173\" height=\"136\" data-media-type=\"image\/png\" \/><\/td>\r\n<td style=\"width: 192.302px;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOur answer is correct.\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]144644[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the video below for another example of how to use long division to divide whole numbers when there is a remainder.\r\n\r\nhttps:\/\/youtu.be\/UPUcShGCBOs","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use addition, subtraction, multiplication, and division when evaluating whole number expressions<\/li>\n<\/ul>\n<\/div>\n<p>Working with whole numbers and performing basic calculations is the backbone of all math. We&#8217;re going to assume you remember how to do single digit addition, subtraction, multiplication, and division. You will often have a calculator on hand to do these calculations, but a quick refresher will help you better understand how to work with numbers so that complex equations are less daunting.<\/p>\n<h2>Addition<\/h2>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Add: [latex]28+61[\/latex]<\/p>\n<p>Solution<br \/>\nTo add numbers with more than one digit, it is often easier to write the numbers vertically in columns.<\/p>\n<table id=\"eip-id1168288293873\" class=\"unnumbered unstyled\" style=\"width: 70%;\" summary=\"a\">\n<tbody>\n<tr>\n<td>Write the numbers so the ones and tens digits line up vertically.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 28\\\\ \\\\ \\hfill \\underset{\\text{____}}{+61}\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Then add the digits in each place value.<\/p>\n<p>Add the ones: [latex]8+1=9[\/latex]<\/p>\n<p>Add the tens: [latex]2+6=8[\/latex]<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 28\\\\ \\\\ \\hfill \\underset{\\text{____}}{+61}\\\\ \\hfill 89\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<p>In the previous example, the sum of the ones and the sum of the tens were both less than [latex]10[\/latex]. But what happens if the sum is [latex]10[\/latex] or more? Let\u2019s use our base-[latex]10[\/latex] model to find out.<\/p>\n<p>The graphic below\u00a0shows the addition of [latex]17[\/latex] and [latex]26[\/latex] again.<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215337\/CNX_BMath_Figure_01_02_001.png\" alt=\"17 plus 26. 17 is represented with one rod (one rod equals 10 ones) and seven one. 27 is represented with two rods and six ones. When you add 17 and 26, ten of the ones combine into a rod and the sum is represented with four rods and three ones, or 43\" width=\"619\" height=\"146\" data-media-type=\"image\/png\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>When we add the ones, [latex]7+6[\/latex], we get [latex]13[\/latex] ones. Because we have more than [latex]10[\/latex] ones, we can exchange [latex]10[\/latex] of the ones for [latex]1[\/latex] ten. Now we have [latex]4[\/latex] tens and [latex]3[\/latex] ones. Without using the model, we show this as a small red [latex]1[\/latex] above the digits in the tens place.<\/p>\n<p>When the sum in a place value column is greater than [latex]9[\/latex], we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, [latex]10[\/latex] ones for [latex]1[\/latex] ten or [latex]10[\/latex] tens for [latex]1[\/latex] hundred.<\/p>\n<div class=\"textbox shaded\">\n<h3>Add whole numbers<\/h3>\n<ol id=\"eip-id1168288474750\" class=\"stepwise\" data-number-style=\"arabic\">\n<li>Write the numbers so each place value lines up vertically.<\/li>\n<li>Add the digits in each place value. Work from right to left starting with the ones place. If a sum in a place value is more than [latex]9[\/latex], carry to the next place value.<\/li>\n<li>Continue adding each place value from right to left, adding each place value and carrying if needed.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Add: [latex]43+69[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q61333\">Show Answer<\/span><\/p>\n<div id=\"q61333\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168287505854\" class=\"unnumbered unstyled\" style=\"width: 70%;\" summary=\"a\">\n<tbody>\n<tr>\n<td>Write the numbers so the digits line up vertically.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 43\\\\ \\\\ \\hfill \\underset{\\text{____}}{+69}\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the digits in each place.<\/p>\n<p>Add the ones: [latex]3+9=12[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write the [latex]2[\/latex] in the ones place in the sum.<\/p>\n<p>Add the [latex]1[\/latex] ten to the tens place.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{1}{4}3\\\\ \\hfill \\underset{\\text{____}}{+69}\\\\ \\hfill 2\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Now add the tens: [latex]1+4+6=11[\/latex]<\/p>\n<p>Write the 11 in the sum.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{1}{4}3\\\\ \\hfill \\underset{\\text{____}}{+69}\\\\ \\hfill 112\\end{array}[\/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=\"ohm147154\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147154&theme=oea&iframe_resize_id=ohm147154&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147156\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147156&theme=oea&iframe_resize_id=ohm147156&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>When the addends have different numbers of digits, be careful to line up the corresponding place values starting with the ones and moving toward the left.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Add: [latex]1,683+479[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q603010\">Show Answer<\/span><\/p>\n<div id=\"q603010\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168288687380\" class=\"unnumbered unstyled\" style=\"width: 80%;\" summary=\"a\">\n<tbody>\n<tr>\n<td>Write the numbers so the digits line up vertically.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 1,683\\\\ \\\\ \\hfill \\underset{\\text{______}}{+479}\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the digits in each place value.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Add the ones: [latex]3+9=12[\/latex].<\/p>\n<p>Write the [latex]2[\/latex] in the ones place of the sum and carry the [latex]1[\/latex] ten to the tens place.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 1,6\\stackrel{1}{8}3\\\\ \\\\ \\hfill \\underset{\\text{______}}{+479}\\\\ \\hfill 2\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the tens: [latex]1+7+8=16[\/latex]<\/p>\n<p>Write the [latex]6[\/latex] in the tens place and carry the [latex]1[\/latex] hundred to the hundreds place.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 1,\\stackrel{1}{6}\\stackrel{1}{8}3\\\\ \\\\ \\hfill \\underset{\\text{______}}{+479}\\\\ \\hfill 62\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the hundreds: [latex]1+6+4=11[\/latex]<\/p>\n<p>Write the [latex]1[\/latex] in the hundreds place and carry the [latex]1[\/latex] thousand to the thousands place.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{1}{1},\\stackrel{1}{6}\\stackrel{1}{8}3\\\\ \\\\ \\hfill \\underset{\\text{______}}{+479}\\\\ \\hfill 162\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the thousands [latex]1+1=2[\/latex] .<\/p>\n<p>Write the [latex]2[\/latex] in the thousands place of the sum.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{1}{1},\\stackrel{1}{6}\\stackrel{1}{8}3\\\\ \\\\ \\hfill \\underset{\\text{______}}{+479}\\\\ \\hfill 2,162\\end{array}[\/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=\"ohm156996\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=156996&theme=oea&iframe_resize_id=ohm156996&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the video below for another example of how to add three whole numbers by lining up place values.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-1\" title=\"Example:  Adding Whole Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/N3I6OiO5mKI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Subtraction<\/h2>\n<p>Addition and subtraction are inverse operations. Addition undoes subtraction, and subtraction undoes addition.<br \/>\nWe know [latex]7 - 3=4[\/latex] because [latex]4+3=7[\/latex]. Knowing all the addition number facts will help with subtraction. Then we can check subtraction by adding. In the examples above, our subtractions can be checked by addition.<\/p>\n<table class=\"unnumbered unstyled\" summary=\"This image consists of there columns\">\n<tbody>\n<tr>\n<td>[latex]7-3=4[\/latex]<\/td>\n<td>because<\/td>\n<td>[latex]4+3=7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]13-8=5[\/latex]<\/td>\n<td>because<\/td>\n<td>[latex]5+8=13[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]43-26=17[\/latex]<\/td>\n<td>because<\/td>\n<td>[latex]17+26=43[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To subtract numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition. Align the digits by place value, and then subtract each column starting with the ones and then working to the left.<\/p>\n<div class=\"textbox exercises\">\n<h3>Exercise<\/h3>\n<p>Subtract and then check by adding: [latex]89 - 61[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q656368\">Show Answer<\/span><\/p>\n<div id=\"q656368\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-458\" class=\"unnumbered unstyled\" summary=\"a\">\n<tbody>\n<tr>\n<td>Write the numbers so the ones and tens digits line up vertically.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 89\\\\ \\hfill \\underset{\\text{____}}{-61}\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Subtract the digits in each place value.<\/p>\n<p>Subtract the ones: [latex]9 - 1=8[\/latex]<\/p>\n<p>Subtract the tens: [latex]8 - 6=2[\/latex]<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 89\\\\ \\hfill \\underset{\\text{____}}{-61}\\\\ \\hfill 28\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check using addition.<\/p>\n<p>[latex]\\begin{array}{c}\\hfill 28\\\\ \\hfill \\underset{\\text{____}}{+61}\\\\ \\hfill 89\\end{array}\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Our answer is correct.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=143322&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Subtract whole numbers<\/h3>\n<ol id=\"eip-id1168289617573\" class=\"stepwise\" data-number-style=\"arabic\">\n<li>Write the numbers so each place value lines up vertically.<\/li>\n<li>Subtract the digits in each place value. Work from right to left starting with the ones place. If the digit on top is less than the digit below, borrow as needed.<\/li>\n<li>Continue subtracting each place value from right to left, borrowing if needed.<\/li>\n<li>Check by adding.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>excercise<\/h3>\n<p>Subtract: [latex]43 - 26[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q491725\">Show Answer<\/span><\/p>\n<div id=\"q491725\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-640\" class=\"unnumbered unstyled\" style=\"width: 915px;\" summary=\".\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 587.944px;\">Write the numbers so each place value lines up vertically.<\/td>\n<td style=\"width: 306.056px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215443\/CNX_BMath_Figure_01_03_029_img-01.png\" alt=\"Vertical subtraction for 43 minus 26\" width=\"94\" height=\"56\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.944px;\">Subtract the ones. We cannot subtract [latex]6[\/latex] from [latex]3[\/latex], so we borrow [latex]1[\/latex] ten. This makes [latex]3[\/latex] tens and [latex]13[\/latex] ones. We write these numbers above each place and cross out the original digits.<\/td>\n<td style=\"width: 306.056px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215444\/CNX_BMath_Figure_01_03_029_img-02.png\" alt=\"The 3 borrows from the 4, so the 4 becomes 3 and the 3 becomes 13.\" width=\"92\" height=\"73\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.944px;\">Now we can subtract the ones. [latex]13 - 6=7[\/latex]. We write the [latex]7[\/latex] in the ones place in the difference.<\/td>\n<td style=\"width: 306.056px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215445\/CNX_BMath_Figure_01_03_029_img-03.png\" alt=\"13 minus 6 equals 7.\" width=\"90\" height=\"102\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.944px;\">Now we subtract the tens. [latex]3 - 2=1[\/latex]. We write the[latex]1[\/latex] in the tens place in the difference.<\/td>\n<td style=\"width: 306.056px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215445\/CNX_BMath_Figure_01_03_029_img-04.png\" alt=\"3 minus 2 equals 1. In total, we have 43 minus 26 equals 17.\" width=\"89\" height=\"101\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 587.944px;\" data-align=\"left\">Check by adding.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215446\/CNX_BMath_Figure_01_03_029_img-05.png\" alt=\"17 plus 26 equals 43, therefore our check is succcessful and our answer correct.\" width=\"94\" height=\"88\" data-media-type=\"image\/png\" \/><\/p>\n<p>Our answer is correct.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>In the example above, if we model subtracting [latex]26[\/latex] from [latex]43[\/latex], we would exchange [latex]1[\/latex] ten for [latex]10[\/latex] ones. When we do this without models, we say we borrow [latex]1[\/latex] from the tens place and add [latex]10[\/latex] to the ones place.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=143327&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercise<\/h3>\n<p>Subtract and then check by adding: [latex]207 - 64[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q125839\">Show Answer<\/span><\/p>\n<div id=\"q125839\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-274\" class=\"unnumbered unstyled\" style=\"width: 913px;\" summary=\"a\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 593.059px;\">Write the numbers so each place value lines up vertically.<\/td>\n<td style=\"width: 299.941px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215447\/CNX_BMath_Figure_01_03_030-01.png\" alt=\"Vertical subtraction for 207 minus 64\" width=\"108\" height=\"59\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 593.059px;\">Subtract the ones. [latex]7 - 4=3[\/latex].<\/p>\n<p>Write the [latex]3[\/latex] in the ones place in the difference. Write the [latex]3[\/latex] in the ones place in the difference.<\/td>\n<td style=\"width: 299.941px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215448\/CNX_BMath_Figure_01_03_030-02.png\" alt=\"7 minus 4 equals 3.\" width=\"109\" height=\"99\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 593.059px;\">Subtract the tens. We cannot subtract [latex]6[\/latex] from [latex]0[\/latex] so we borrow [latex]1[\/latex] hundred and add [latex]10[\/latex] tens to the [latex]0[\/latex] tens we had. This makes a total of [latex]10[\/latex] tens. We write [latex]10[\/latex] above the tens place and cross out the [latex]0[\/latex]. Then we cross out the [latex]2[\/latex] in the hundreds place and write [latex]1[\/latex] above it.<\/td>\n<td style=\"width: 299.941px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215449\/CNX_BMath_Figure_01_03_030-03.png\" alt=\"The 0 borrows from the 2, so the 0 becomes 10 and the 2 becomes 1.\" width=\"108\" height=\"113\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 593.059px;\">Now we subtract the tens. [latex]10 - 6=4[\/latex]. We write the 4 in the tens place in the difference.<\/td>\n<td style=\"width: 299.941px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215450\/CNX_BMath_Figure_01_03_030-04.png\" alt=\"10 minus 6 equals 4.\" width=\"109\" height=\"114\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 593.059px;\">Finally, subtract the hundreds. There is no digit in the hundreds place in the bottom number so we can imagine a [latex]0[\/latex] in that place. Since [latex]1 - 0=1[\/latex], we write [latex]1[\/latex] in the hundreds place in the difference.<\/td>\n<td style=\"width: 299.941px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215451\/CNX_BMath_Figure_01_03_030-05.png\" alt=\"1 carries down into the difference. In total, we have 207 minus 64 equals 143.\" width=\"111\" height=\"116\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 593.059px;\" data-align=\"left\">Check by adding.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215451\/CNX_BMath_Figure_01_03_030-06.png\" alt=\"143 plus 64 equals 207, therefore our check is succcessful and our answer correct.\" width=\"106\" height=\"111\" data-media-type=\"image\/png\" \/><\/p>\n<p>Our answer is correct.<\/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=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=143341&amp;theme=oea&amp;iframe_resize_id=mom4\" width=\"100%\" height=\"280\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Exercise<\/h3>\n<p>Subtract and then check by adding: [latex]2,162 - 479[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q61197\">Show Answer<\/span><\/p>\n<div id=\"q61197\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-7544534\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image is two columns. The left column includes annotations and the right column includes math expressions. The first line reads\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 697.014px;\">Write the numbers so each place values line up vertically.<\/td>\n<td style=\"width: 196.986px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215458\/CNX_BMath_Figure_01_03_028_img-02.png\" alt=\"Vertical subtraction for 2,162 minus 479.\" width=\"103\" height=\"66\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 697.014px;\">Subtract the ones. Since we cannot subtract [latex]9[\/latex] from [latex]2[\/latex], borrow [latex]1[\/latex] ten and add [latex]10[\/latex] ones to the [latex]2[\/latex] ones to make [latex]12[\/latex] ones. Write [latex]5[\/latex] above the tens place and cross out the [latex]6[\/latex]. Write [latex]12[\/latex] above the ones place and cross out the [latex]2[\/latex].<\/td>\n<td style=\"width: 196.986px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215459\/CNX_BMath_Figure_01_03_028_img-03.png\" alt=\"The 2 borrows from the 6, so the 2 becomes 12 and the 6 becomes 5.\" width=\"100\" height=\"83\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 697.014px;\">Now we can subtract the ones.<\/td>\n<td style=\"width: 196.986px;\">[latex]12 - 9=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 697.014px;\">Write [latex]3[\/latex] in the ones place in the difference.<\/td>\n<td style=\"width: 196.986px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215500\/CNX_BMath_Figure_01_03_028_img-04.png\" alt=\"12 minus 9 equals 3.\" width=\"102\" height=\"122\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 697.014px;\">Subtract the tens. Since we cannot subtract [latex]7[\/latex] from [latex]5[\/latex], borrow [latex]1[\/latex] hundred and add [latex]10[\/latex] tens to the [latex]5[\/latex] tens to make [latex]15[\/latex] tens. Write [latex]0[\/latex] above the hundreds place and cross out the [latex]1[\/latex]. Write [latex]15[\/latex] above the tens place.<\/td>\n<td style=\"width: 196.986px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215501\/CNX_BMath_Figure_01_03_028_img-05.png\" alt=\"The 5 borrows from the 1, so the 5 becomes 15 and the 1 becomes 0.\" width=\"98\" height=\"140\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 697.014px;\">Now we can subtract the tens.<\/td>\n<td style=\"width: 196.986px;\">[latex]15 - 7=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 697.014px;\">Write [latex]8[\/latex] in the tens place in the difference.<\/td>\n<td style=\"width: 196.986px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215502\/CNX_BMath_Figure_01_03_028_img-06.png\" alt=\"15 minus 7 equals 8.\" width=\"97\" height=\"116\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 697.014px;\">Now we can subtract the hundreds.<\/td>\n<td style=\"width: 196.986px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215503\/CNX_BMath_Figure_01_03_028_img-07.png\" alt=\"The 0 borrows from the 2, so the 0 becomes 10 and the 2 becomes 1.\" width=\"101\" height=\"144\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 697.014px;\">Write [latex]6[\/latex] in the hundreds place in the difference.<\/td>\n<td style=\"width: 196.986px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215504\/CNX_BMath_Figure_01_03_028_img-08.png\" alt=\"10 minus 4 equals 6.\" width=\"102\" height=\"119\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 697.014px;\">Subtract the thousands. There is no digit in the thousands place of the bottom number, so we imagine a [latex]0[\/latex]. [latex]1 - 0=1[\/latex]. Write [latex]1[\/latex] in the thousands place of the difference.<\/td>\n<td style=\"width: 196.986px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215505\/CNX_BMath_Figure_01_03_028_img-09.png\" alt=\"1 carries down into the difference. In total, we have 2,162 minus 479 equals 1,683.\" width=\"102\" height=\"122\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 697.014px;\">Check by adding.<\/p>\n<p>[latex]\\begin{array}{}\\\\ \\stackrel{1}{1},\\stackrel{1}{6}\\stackrel{1}{8}3\\hfill \\\\ \\underset{\\text{______}}{+479}\\hfill \\\\ 2,162\\quad\\checkmark \\hfill \\end{array}\\quad\\checkmark[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Our answer is correct.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=143343&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"360\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Watch the video below to see another example of subtracting whole numbers by lining up place values.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Example:  Subtracting Whole Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/hneqy1EGACs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 data-type=\"title\">Multiplication<\/h2>\n<p>In order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section. The table below shows the multiplication facts.<\/p>\n<p>Each box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers.<\/p>\n<table id=\"fs-id1563789\" class=\"column-header\" style=\"width: 398px;\" summary=\"This is a multiplication table with 11 columns and 11 rows. The first column has the values\">\n<thead>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<th style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]x[\/latex]<\/th>\n<th style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/th>\n<th style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]1[\/latex]<\/th>\n<th style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]2[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]3[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]4[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]5[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]6[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]7[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]8[\/latex]<\/th>\n<th style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]9[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]1[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]1[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]2[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]3[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]4[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]5[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]6[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]7[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]8[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]9[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]2[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]2[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]4[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]6[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]8[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]10[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]12[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]14[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]16[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]18[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]3[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]3[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]6[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]9[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]12[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]15[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]18[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]21[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]24[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]27[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]4[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]4[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]8[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]12[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]16[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]20[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]24[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]28[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]32[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]36[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]5[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]5[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]10[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]15[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]20[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]25[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]30[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]35[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]40[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]45[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]6[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]6[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]12[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]18[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]24[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]30[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]36[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]42[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]48[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]54[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15.7812px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15.7812px;\" data-align=\"center\">[latex]7[\/latex]<\/td>\n<td style=\"width: 122px; height: 15.7812px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15.7812px;\" data-align=\"center\">[latex]7[\/latex]<\/td>\n<td style=\"width: 16px; height: 15.7812px;\" data-align=\"center\">[latex]14[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]21[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]28[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]35[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]42[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]49[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]56[\/latex]<\/td>\n<td style=\"width: 17px; height: 15.7812px;\" data-align=\"center\">[latex]63[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]8[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]8[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]16[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]24[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]32[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]40[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]48[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]56[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]64[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]72[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\" valign=\"middle\">\n<td style=\"width: 11px; height: 15px;\" data-align=\"center\">[latex]9[\/latex]<\/td>\n<td style=\"width: 122px; height: 15px;\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"width: 9px; height: 15px;\" data-align=\"center\">[latex]9[\/latex]<\/td>\n<td style=\"width: 16px; height: 15px;\" data-align=\"center\">[latex]18[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]27[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]36[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]45[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]54[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]63[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]72[\/latex]<\/td>\n<td style=\"width: 17px; height: 15px;\" data-align=\"center\">[latex]81[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We know that changing the order of addition does not change the sum. We saw that [latex]8+9=17[\/latex] is the same as [latex]9+8=17[\/latex].<\/p>\n<p>Is this also true for multiplication? Let\u2019s look at a few pairs of factors.<\/p>\n<p style=\"text-align: center;\">[latex]4\\cdot 7=28\\quad 7\\cdot 4=28[\/latex]<br \/>\n[latex]9\\cdot 7=63\\quad 7\\cdot 9=63[\/latex]<br \/>\n[latex]8\\cdot 9=72\\quad 9\\cdot 8=72[\/latex]<\/p>\n<p>When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.<\/p>\n<div class=\"textbox shaded\">\n<h3>Commutative Property of Multiplication<\/h3>\n<p>Changing the order of the factors does not change their product.<\/p>\n<p style=\"text-align: center;\">[latex]a\\cdot b=b\\cdot a[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply:<\/p>\n<p>[latex]8\\cdot 7[\/latex]<br \/>\n[latex]7\\cdot 8[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q468629\">Show Answer<\/span><\/p>\n<div id=\"q468629\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168287497846\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\" data-label=\"\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td>[latex]8\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]56[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2.<\/td>\n<td>[latex]7\\cdot 8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]56[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Changing the order of the factors does not change the product.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144424&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"340\"><\/iframe><\/p>\n<\/div>\n<p>To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\hfill 27\\\\ \\hfill \\underset{\\text{___}}{\\times 3}\\end{array}[\/latex]<\/p>\n<p>We start by multiplying [latex]3[\/latex] by [latex]7[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3\\times 7=21[\/latex]<\/p>\n<p>We write the [latex]1[\/latex] in the ones place of the product. We carry the [latex]2[\/latex] tens by writing [latex]2[\/latex] above the tens place.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215526\/CNX_BMath_Figure_01_04_008_img.png\" alt=\"Vertical multiplication for 27 times 3. In the product, that is below the line, we have a 1, to which an arrow is pointing labeled &quot;Here is the 1 in 21&quot;. Above the 2 in 27, we have a smaller two, to which an arrow points labeled &quot;Here are the 2 tens in 21&quot;.\" width=\"165\" height=\"153\" data-media-type=\"image\/png\" \/><br \/>\nThen we multiply the [latex]3[\/latex] by the [latex]2[\/latex], and add the [latex]2[\/latex] above the tens place to the product. So [latex]3\\times 2=6[\/latex], and [latex]6+2=8[\/latex]. Write the [latex]8[\/latex] in the tens place of the product.<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215527\/CNX_BMath_Figure_01_04_010_img.png\" alt=\"We now have an 8 in the ten's place of our product, to which an arrow points labeled &quot;This comes from 3 times 2 plus the 2 we carried.&quot;\" width=\"221\" height=\"124\" data-media-type=\"image\/png\" \/><br \/>\nThe product is [latex]81[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>When we multiply two numbers with a different number of digits, it\u2019s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]15\\cdot 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q942773\">Show Answer<\/span><\/p>\n<div id=\"q942773\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168287565780\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\" data-label=\"\">\n<tbody>\n<tr>\n<td>Write the numbers so the digits [latex]5[\/latex] and [latex]4[\/latex] line up vertically.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill 15\\\\ \\hfill \\underset{\\text{_____}}{\\times 4}\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]4[\/latex] by the digit in the ones place of [latex]15[\/latex]. [latex]4\\cdot 5=20[\/latex].<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write [latex]0[\/latex] in the ones place of the product and carry the [latex]2[\/latex] tens.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{2}{1}5\\\\ \\hfill \\underset{\\text{_____}}{\\times 4}\\\\ \\hfill 0\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]4[\/latex] by the digit in the tens place of [latex]15[\/latex]. [latex]4\\cdot 1=4[\/latex] .<\/p>\n<p>Add the [latex]2[\/latex] tens we carried. [latex]4+2=6[\/latex] .<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>Write the [latex]6[\/latex] in the tens place of the product.<\/td>\n<td>[latex]\\begin{array}{c}\\hfill \\stackrel{2}{1}5\\\\ \\hfill \\underset{\\text{_____}}{\\times 4}\\\\ \\hfill 60\\end{array}[\/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=\"mom4\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144427&amp;theme=oea&amp;iframe_resize_id=mom\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]286\\cdot 5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q849168\">Show Answer<\/span><\/p>\n<div id=\"q849168\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168288632671\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\" data-label=\"\">\n<tbody>\n<tr style=\"height: 109.441px;\">\n<td style=\"height: 109.441px;\">Write the numbers so the digits [latex]5[\/latex] and [latex]6[\/latex] line up vertically.<\/td>\n<td style=\"height: 109.441px;\">[latex]\\begin{array}{c}\\hfill 286\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 88px;\">\n<td style=\"height: 88px;\">Multiply [latex]5[\/latex] by the digit in the ones place of [latex]286[\/latex].<\/p>\n<p>[latex]5\\cdot 6=30[\/latex]<\/td>\n<td style=\"height: 88px;\"><\/td>\n<\/tr>\n<tr style=\"height: 145px;\">\n<td style=\"height: 145px;\">Write the [latex]0[\/latex] in the ones place of the product and carry the [latex]3[\/latex] to the tens place.Multiply [latex]5[\/latex] by the digit in the tens place of [latex]286[\/latex].<\/p>\n<p>[latex]5\\cdot 8=40[\/latex]<\/td>\n<td style=\"height: 145px;\">[latex]\\begin{array}{}\\\\ \\hfill 2\\stackrel{3}{8}6\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\\\ \\hfill 0\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 161px;\">\n<td style=\"height: 161px;\">Add the [latex]3[\/latex] tens we carried to get [latex]40+3=43[\/latex] .<\/p>\n<p>Write the [latex]3[\/latex] in the tens place of the product and carry the [latex]4[\/latex] to the hundreds place.<\/td>\n<td style=\"height: 161px;\">[latex]\\begin{array}{c}\\hfill \\stackrel{4}{2}\\stackrel{3}{8}6\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\\\ \\hfill 30\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 251px;\">\n<td style=\"height: 251px;\">Multiply [latex]5[\/latex] by the digit in the hundreds place of [latex]286[\/latex]. [latex]5\\cdot 2=10[\/latex].<\/p>\n<p>Add the [latex]4[\/latex] hundreds we carried to get [latex]10+4=14[\/latex].<\/p>\n<p>Write the [latex]4[\/latex] in the hundreds place of the product and the [latex]1[\/latex] to the thousands place.<\/td>\n<td style=\"height: 251px;\">[latex]\\begin{array}{c}\\hfill \\stackrel{4}{2}\\stackrel{3}{8}6\\\\ \\hfill \\underset{\\text{_____}}{\\times 5}\\\\ \\hfill 1,430\\end{array}[\/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=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144429&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<\/div>\n<p>When we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.<\/p>\n<div class=\"textbox shaded\">\n<h3>MultiplIcation of whole numbers<\/h3>\n<ol id=\"eip-id1168286064302\" class=\"stepwise\" data-number-style=\"arabic\">\n<li>Write the numbers so each place value lines up vertically.<\/li>\n<li>Multiply the digits in each place value.\n<ul id=\"eip-id1168288663980\">\n<li>Work from right to left, starting with the ones place in the bottom number.\n<ul id=\"eip-id1168288663983\" data-bullet-style=\"bullet\">\n<li>Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.<\/li>\n<li>If a product in a place value is more than [latex]9[\/latex], carry to the next place value.<\/li>\n<li>Write the partial products, lining up the digits in the place values with the numbers above.<\/li>\n<\/ul>\n<\/li>\n<li>Repeat for the tens place in the bottom number, the hundreds place, and so on.<\/li>\n<li>Insert a zero as a placeholder with each additional partial product.<\/li>\n<\/ul>\n<\/li>\n<li>Add the partial products.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]62\\left(87\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q698602\">Show Answer<\/span><\/p>\n<div id=\"q698602\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-279\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"No alt text\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 585.903px;\">Write the numbers so each place lines up vertically.<\/td>\n<td style=\"width: 308.097px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215528\/CNX_BMath_Figure_01_04_020_img-02.png\" alt=\"Vertical multiplication for 62 times 87.\" width=\"42\" height=\"46\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 585.903px;\">Start by multiplying 7 by 62. Multiply 7 by the digit in the ones place of 62.<\/p>\n<p>[latex]7\\cdot 2=14[\/latex].<\/p>\n<p>Write the 4 in the ones place of the product and carry the 1 to the tens place.<\/td>\n<td style=\"width: 308.097px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215529\/CNX_BMath_Figure_01_04_020_img-03.png\" alt=\"2 times 7 is 14, so we carry over our 1 to the ten's place and the 4 is the one's digit of the first product.\" width=\"42\" height=\"92\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 585.903px;\">Multiply 7 by the digit in the tens place of 62. [latex]7\\cdot 6=42[\/latex]. Add the 1 ten we carried.<\/p>\n<p>[latex]42+1=43[\/latex]latex].<\/p>\n<p>Write the 3 in the tens place of the product and the 4 in the hundreds place.<\/td>\n<td style=\"width: 308.097px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215530\/CNX_BMath_Figure_01_04_020_img-04.png\" alt=\"6 times 7 is 42 plus 1 equals 43, which becomes are hundred's and ten's place of our first product.\" width=\"42\" height=\"92\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 585.903px;\">The first partial product is [latex]434[\/latex].<\/td>\n<td style=\"width: 308.097px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 585.903px;\">Now, write a [latex]0[\/latex] under the [latex]4[\/latex] in the ones place of the next partial product as a placeholder since we now multiply the digit in the tens place of [latex]87[\/latex] by [latex]62[\/latex].<\/p>\n<p>Multiply [latex]8[\/latex] by the digit in the ones place of [latex]62[\/latex]<\/p>\n<p>[latex]8\\cdot 2=16[\/latex]. Write the [latex]6[\/latex] in the next place of the product, which is the tens place. Carry the [latex]1[\/latex] to the tens place.<\/td>\n<td style=\"width: 308.097px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215530\/CNX_BMath_Figure_01_04_020_img-05.png\" alt=\"Begin with a 0 in the one's place of our second product. 2 times 8 equals 16, so we carry over the 1 to the ten's place and the 6 is the ten's digit of the second product.\" width=\"42\" height=\"127\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 585.903px;\">Multiply [latex]8[\/latex] by [latex]6[\/latex], the digit in the tens place of [latex]62[\/latex], then add the [latex]1[\/latex] ten we carried to get [latex]49[\/latex].<\/p>\n<p>Write the [latex]9[\/latex] in the hundreds place of the product and the [latex]4[\/latex] in the thousands place.<\/td>\n<td style=\"width: 308.097px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215531\/CNX_BMath_Figure_01_04_020_img-06.png\" alt=\"6 times 8 is 48 plus 1 equals 49, which becomes the hundred's and thousand's place of the second product.\" width=\"42\" height=\"127\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 585.903px;\">The second partial product is [latex]4960[\/latex]. Add the partial products.<\/td>\n<td style=\"width: 308.097px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215532\/CNX_BMath_Figure_01_04_020_img-07.png\" alt=\"In total, 62 times 87 equals 434 plus 4960 which equals 5394.\" width=\"42\" height=\"155\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The product is [latex]5,394[\/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=\"mom6\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144432&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"280\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>When there are three or more factors, we multiply the first two and then multiply their product by the next factor. For example:<\/p>\n<table id=\"eip-0\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\" data-label=\"\">\n<tbody>\n<tr>\n<td>to multiply<\/td>\n<td>[latex]8\\cdot 3\\cdot 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>first multiply [latex]8\\cdot 3[\/latex]<\/td>\n<td>[latex]24\\cdot 2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>then multiply [latex]24\\cdot 2[\/latex]<\/td>\n<td>[latex]48[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the video below, we summarize the concepts presented on this page including the multiplication property of zero, the identity property of multiplication, and the commutative property of multiplication.m<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Multiplying Whole Numbers\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/kW7JBfplJGE?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 data-type=\"title\">Division<\/h2>\n<p>We said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know [latex]12\\div 4=3[\/latex] because [latex]3\\cdot 4=12[\/latex]. Knowing all the multiplication number facts is very important when doing division.<\/p>\n<p>We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. We know [latex]24\\div 8=3[\/latex] is correct because [latex]3\\cdot 8=24[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide. Then check by multiplying.<\/p>\n<ol>\n<li>[latex]42\\div 6[\/latex]<\/li>\n<li>[latex]\\frac{72}{9}[\/latex]<\/li>\n<li>[latex]7\\overline{)63}[\/latex]<\/li>\n<\/ol>\n<p>Solution:<\/p>\n<table id=\"eip-id1168287031935\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\" data-label=\"\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]42\\div 6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide [latex]42[\/latex] by [latex]6[\/latex].<\/td>\n<td>[latex]7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]7\\cdot 6[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]42\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id11682870335\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\" data-label=\"\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{72}{9}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide [latex]72[\/latex] by [latex]9[\/latex].<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]8\\cdot 9[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]72\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id11670335\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"a\" data-label=\"\">\n<tbody>\n<tr>\n<td>3.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]7\\overline{)63}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide [latex]63[\/latex] by [latex]7[\/latex].<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]9\\cdot 7[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]63\\quad\\checkmark[\/latex]<\/td>\n<td><\/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=\"ohm144463\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144463&theme=oea&iframe_resize_id=ohm144463&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>What is the quotient when you divide a number by itself?<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{15}{15}=1\\text{ because }1\\cdot 15=15[\/latex]<\/p>\n<p>Dividing any number [latex]\\text{(except 0)}[\/latex] by itself produces a quotient of [latex]1[\/latex]. Also, any number divided by [latex]1[\/latex] produces a quotient of the number. These two ideas are stated in the Division Properties of One.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division Properties of One<\/h3>\n<table id=\"eip-735\" summary=\"a\" data-label=\"\">\n<tbody>\n<tr style=\"height: 14px;\">\n<td style=\"height: 14px;\">Any number (except 0) divided by itself is one.<\/td>\n<td style=\"height: 14px;\">[latex]a\\div a=1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 14.4585px;\">\n<td style=\"height: 14.4585px;\">Any number divided by one is the same number.<\/td>\n<td style=\"height: 14.4585px;\">[latex]a\\div 1=a[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide. Then check by multiplying:<\/p>\n<ol id=\"eip-id1168288568257\" class=\"circled\" data-number-style=\"arabic\">\n<li>[latex]11\\div 11[\/latex]<\/li>\n<li>[latex]\\frac{19}{1}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q519474\">Show Answer<\/span><\/p>\n<div id=\"q519474\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution:<\/p>\n<table id=\"eip-id1168288480300\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\" data-label=\"\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]11\\div 11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>A number divided by itself is [latex]1[\/latex].<\/td>\n<td>[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]1\\cdot 11[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]11\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table id=\"eip-id1168288536381\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"t\" data-label=\"\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]\\frac{19}{1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>A number divided by [latex]1[\/latex] equals itself.<\/td>\n<td>[latex]19[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]19\\cdot 1[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]19\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144635\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144635&theme=oea&iframe_resize_id=ohm144635&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Suppose we have [latex]\\text{\\$0}[\/latex], and want to divide it among [latex]3[\/latex] people. How much would each person get? Each person would get [latex]\\text{\\$0}[\/latex]. Zero divided by any number is [latex]0[\/latex].<\/p>\n<p>Now suppose that we want to divide [latex]\\text{\\$10}[\/latex] by [latex]0[\/latex]. That means we would want to find a number that we multiply by [latex]0[\/latex] to get [latex]10[\/latex]. This cannot happen because [latex]0[\/latex] times any number is [latex]0[\/latex]. Division by zero is said to be <em data-effect=\"italics\">undefined<\/em>.<\/p>\n<p>These two ideas make up the Division Properties of Zero.<\/p>\n<div class=\"textbox shaded\">\n<h3>Division Properties of Zero<\/h3>\n<table id=\"eip-158\" summary=\"a\" data-label=\"\">\n<tbody>\n<tr>\n<td>Zero divided by any number is [latex]0[\/latex].<\/td>\n<td>[latex]0\\div a=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Dividing a number by zero is undefined.<\/td>\n<td>[latex]a\\div 0[\/latex] undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Another way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away [latex]0[\/latex] from [latex]10?[\/latex] Because subtracting [latex]0[\/latex] will never change the total, we will never get an answer. So we cannot divide a number by [latex]0[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide. Check by multiplying:<\/p>\n<ol>\n<li>[latex]0\\div 3[\/latex]<\/li>\n<li>[latex]\\frac{10}{0}[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q208505\">Show Answer<\/span><\/p>\n<div id=\"q208505\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168288542869\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\" data-label=\"\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]0\\div 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Zero divided by any number is zero.<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Check by multiplying.<\/p>\n<p>[latex]0\\cdot 3[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]0\\quad\\checkmark[\/latex]<\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<table id=\"eip-id1168288419363\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\" data-label=\"\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]10\/0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Division by zero is undefined.<\/td>\n<td>undefined<\/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=\"ohm144478\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144478&theme=oea&iframe_resize_id=ohm144478&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>When the divisor or the dividend has more than one digit, it is usually easier to use the [latex]4\\overline{)12}[\/latex] notation. This process is called long division. Let\u2019s work through the process by dividing [latex]78[\/latex] by [latex]3[\/latex].<\/p>\n<table id=\"eip-244\" class=\"unnumbered unstyled\" style=\"width: 970.438px;\" summary=\"This image has 2 columns. the left column contains instructions and the right column contains expressions. The exercises being worked out is 78 divided by 3. The first line reads: Divide the first digit of dividend 7, by the divisor, 3. The next line reads: the divisor 3 can go into 7 two times since 2 times 3 equals 6. Write the 2 above the 7 in the quotient. Next to this shows the expression 3 divided by 78, with the two above the seven in the quotient. The next line reads\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 599px;\">Divide the first digit of the dividend, [latex]7[\/latex], by the divisor, [latex]3[\/latex].<\/td>\n<td style=\"width: 337px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">The divisor [latex]3[\/latex] can go into [latex]7[\/latex] two times since [latex]2\\times 3=6[\/latex] . Write the [latex]2[\/latex] above the [latex]7[\/latex] in the quotient.<\/td>\n<td style=\"width: 337px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215615\/CNX_BMath_Figure_01_05_043_img-02.png\" alt=\"Long division for 78 divided by 3. 3 goes into 7 2 times, so 2 becomes the ten's digit of our quotient.\" width=\"38\" height=\"46\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">Multiply the [latex]2[\/latex] in the quotient by [latex]2[\/latex] and write the product, [latex]6[\/latex], under the[latex]7[\/latex].<\/td>\n<td style=\"width: 337px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215616\/CNX_BMath_Figure_01_05_043_img-03.png\" alt=\"Thus, we subtract 6 from 7.\" width=\"35\" height=\"64\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">Subtract that product from the first digit in the dividend. Subtract [latex]7 - 6[\/latex] . Write the difference, 1, under the first digit in the dividend.<\/td>\n<td style=\"width: 337px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215616\/CNX_BMath_Figure_01_05_043_img-04.png\" alt=\"7 minus 6 equals 1.\" width=\"35\" height=\"78\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">Bring down the next digit of the dividend. Bring down the [latex]8[\/latex].<\/td>\n<td style=\"width: 337px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215617\/CNX_BMath_Figure_01_05_043_img-05.png\" alt=\"We carry the one's digit, 8, down to the 1, resulting in 18.\" width=\"35\" height=\"85\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">Divide [latex]18[\/latex] by the divisor, [latex]3[\/latex]. The divisor [latex]3[\/latex] goes into [latex]18[\/latex] six times.<\/td>\n<td style=\"width: 337px;\" rowspan=\"2\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215617\/CNX_BMath_Figure_01_05_043_img-06.png\" alt=\"3 goes into 18 6 times, so 6 becomes the one's digit in our quotient.\" width=\"35\" height=\"80\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\" data-valign=\"bottom\">Write [latex]6[\/latex] in the quotient above the [latex]8[\/latex].<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 599px;\">Multiply the [latex]6[\/latex] in the quotient by the divisor and write the product, [latex]18[\/latex], under the dividend. Subtract [latex]18[\/latex] from [latex]18[\/latex].<\/td>\n<td style=\"width: 337px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215618\/CNX_BMath_Figure_01_05_043_img-07.png\" alt=\"18 minus 18 equals 0. We are left with a remainder of 0. In total 78 divided by 3 equals 26.\" width=\"35\" height=\"117\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished.<\/p>\n<p style=\"text-align: center;\">[latex]\\text{So }78\\div 3=26[\/latex].<\/p>\n<p>Check by multiplying the quotient times the divisor to get the dividend. Multiply [latex]26\\times 3[\/latex] to make sure that product equals the dividend, [latex]78[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\hfill \\stackrel{1}{2}6\\\\ \\hfill \\underset{\\text{___}}{\\times 3}\\\\ \\hfill 78 \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">It does, so our answer is correct.\u00a0[latex]\\checkmark[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<h3>Division of whole numbers<\/h3>\n<ol id=\"eip-id1168288534169\" class=\"stepwise\" data-number-style=\"arabic\">\n<li>Divide the first digit of the dividend by the divisor.If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on.<\/li>\n<li>Write the quotient above the dividend.<\/li>\n<li>Multiply the quotient by the divisor and write the product under the dividend.<\/li>\n<li>Subtract that product from the dividend.<\/li>\n<li>Bring down the next digit of the dividend.<\/li>\n<li>Repeat from Step 1 until there are no more digits in the dividend to bring down.<\/li>\n<li>Check by multiplying the quotient times the divisor.<\/li>\n<\/ol>\n<\/div>\n<p>In the video below we show another example of using long division.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex: Long Division - Two Digit Divided by One Digit (No Remainder)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KvVhaB5mqr8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide [latex]2,596\\div 4[\/latex]. Check by multiplying:<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q252445\">Show Answer<\/span><\/p>\n<div id=\"q252445\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-287\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows long division for solving the expression 4 divided by 2,596. The first line says to divide the first digit of the dividend, 2, by the divisor, 4. The expression shows the 2 in red. The next line says that since 4 does not go into 2, we use the first two digits of the dividend and divide 25 by 4. the divisor 4 goes into 25 six times. We write the six in the quotient above the 5. The expression now shows the 6 above the 5 in the quotient. Next, multiply the 6 in the quotient by the divisor 4 and write the product, 24, under the first two digits in the dividend. The expression shows the 24 under the 25. Subtract 25 minus 4. Write the difference, 1, under the second digit in the dividend. The expression shows in long division, 25 minus 24 equals 1. Now bring the 9 down and repeat these steps. There are 4 fours in 19. Write the 4 over the 9. Multiply the 4 by 4 and subtract this product from 19. The expression now shows in long division, the 9 in the dividend brought down next to the 1, to make 19, and the 4 in the quotient above the 9. Under the 19 is 16 with a difference of 3. The next line says to bring down the 6 and repeat these steps. there are 9 fours in 36. Write 9 over the 6. Multiply the 9 by 4 and subtract this product from 36. The expression shows in long division the 6 brought down next to the 3 to make 36. The 9 in the quotient above the 6 for an answer of 649.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 676.753px;\">Let&#8217;s rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 218.247px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215619\/CNX_BMath_Figure_01_05_044_img-01.png\" alt=\"Long division for 2,596 divided by 4.\" width=\"84\" height=\"27\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">Divide the first digit of the dividend, [latex]2[\/latex], by the divisor, [latex]4[\/latex].<\/td>\n<td style=\"width: 218.247px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215620\/CNX_BMath_Figure_01_05_044_img-02.png\" alt=\"First, we must consider how many times 4 goes into 2.\" width=\"84\" height=\"27\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\" data-valign=\"bottom\">Since [latex]4[\/latex] does not go into [latex]2[\/latex], we use the first two digits of the dividend and divide [latex]25[\/latex] by [latex]4[\/latex]. The divisor [latex]4[\/latex] goes into [latex]25[\/latex] six times.<\/td>\n<td style=\"width: 218.247px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">We write the [latex]6[\/latex] in the quotient above the [latex]5[\/latex].<\/td>\n<td style=\"width: 218.247px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215620\/CNX_BMath_Figure_01_05_044_img-03.png\" alt=\"4 does not go into 2, so we consider how many times 4 goes into 25. 4 goes into 25 6 times, so 6 becomes the hundred's place in our quotient.\" width=\"84\" height=\"46\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">Multiply the [latex]6[\/latex]in the quotient by the divisor [latex]4[\/latex] and write the product, [latex]24[\/latex], under the first two digits in the dividend.<\/td>\n<td style=\"width: 218.247px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215621\/CNX_BMath_Figure_01_05_044_img-04.png\" alt=\"We subtract 24 from 25.\" width=\"84\" height=\"64\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">Subtract that product from the first two digits in the dividend. Subtract [latex]25 - 24[\/latex] . Write the difference, [latex]1[\/latex], under the second digit in the dividend.<\/td>\n<td style=\"width: 218.247px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215622\/CNX_BMath_Figure_01_05_044_img-05.png\" alt=\"25 minus 24 equals 1\" width=\"84\" height=\"85\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">Now bring down the [latex]9[\/latex] and repeat these steps. There are [latex]4[\/latex] fours in [latex]19[\/latex]. Write the [latex]4[\/latex] over the [latex]9[\/latex]. Multiply the [latex]4[\/latex] by [latex]4[\/latex] and subtract this product from [latex]19[\/latex].<\/td>\n<td style=\"width: 218.247px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215624\/CNX_BMath_Figure_01_05_044_img-06.png\" alt=\"We carry the ten's digit, 9 down to the 1, resulting in 19. Consider how many times 4 goes into 19. 4 goes into 19 4 times, so 4 becomes our ten's digit in the quotient. We subtract 16 from 19. 16 minus 19 equals 3.\" width=\"84\" height=\"125\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">Bring down the [latex]6[\/latex] and repeat these steps. There are [latex]9[\/latex] fours in [latex]36[\/latex]. Write the [latex]9[\/latex] over the [latex]6[\/latex]. Multiply the [latex]9[\/latex] by [latex]4[\/latex] and subtract this product from [latex]36[\/latex].<\/td>\n<td style=\"width: 218.247px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215625\/CNX_BMath_Figure_01_05_044_img-07.png\" alt=\"We carry the one's digit, 6 down to the 3, resulting in 36. Consider how many times 4 goes into 36. 4 goes into 36 9 times, so 9 becomes the one's digit in our quotient. We subtract 36 from 36. 36 minus 36 equals 0. We are left with a remainder of 0. In total, we have 2596 divided by 4 equals 649.\" width=\"84\" height=\"161\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">So [latex]2,596\\div 4=649[\/latex] .<\/td>\n<td style=\"width: 218.247px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 676.753px;\">Check by multiplying.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215626\/CNX_BMath_Figure_01_05_044_img-08.png\" alt=\"649 times 4 equals 2596, therefore our check is succcessful and our answer correct.\" width=\"84\" height=\"84\" data-media-type=\"image\/png\" \/><\/td>\n<td style=\"width: 218.247px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It equals the dividend, so our answer is correct.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144636\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144636&theme=oea&iframe_resize_id=ohm144636&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide [latex]4,506\\div 6[\/latex]. Check by multiplying:<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q474096\">Show Answer<\/span><\/p>\n<div id=\"q474096\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-483\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows how 4,506 divided by six is worked out in long division. The first line reads first we try to divide 6 into 4. Since that won't work, we try 6 into 45. There are 7 sixes in 45. We write the 7 over the 5. The expression shows the 7 above the 5 in the quotient. Multiply the 7 by the 6 and subtract the product from 45. The expression shows in long division, 45 minus 42, equals 3. The next line reads\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 672.847px;\">Let&#8217;s rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 222.153px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215627\/CNX_BMath_Figure_01_05_045_img-01.png\" alt=\"Long division for 4506 divided by 6.\" width=\"84\" height=\"32\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">First we try to divide [latex]6[\/latex] into [latex]4[\/latex].<\/td>\n<td style=\"width: 222.153px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215628\/CNX_BMath_Figure_01_05_045_img-02.png\" alt=\"Consider how many times 6 goes into 4.\" width=\"84\" height=\"27\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Since that won&#8217;t work, we try [latex]6[\/latex] into [latex]45[\/latex].<\/p>\n<p>There are [latex]7[\/latex] sixes in [latex]45[\/latex]. We write the [latex]7[\/latex] over the [latex]5[\/latex].<\/td>\n<td style=\"width: 222.153px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215628\/CNX_BMath_Figure_01_05_045_img-03.png\" alt=\"6 does not go into 4, so we consider how many times 6 goes into 45. 6 goes into 45 7 times, so 7 becomes the hundred's place in the quotient.\" width=\"84\" height=\"46\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Multiply the [latex]7[\/latex] by [latex]6[\/latex] and subtract this product from [latex]45[\/latex].<\/td>\n<td style=\"width: 222.153px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215629\/CNX_BMath_Figure_01_05_045_img-04.png\" alt=\"We subtract 42 from 45. 45 minus 42 equals 3.\" width=\"84\" height=\"85\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Now bring down the [latex]0[\/latex] and repeat these steps. There are [latex]5[\/latex] sixes in [latex]30[\/latex]. Write the [latex]5[\/latex] over the [latex]0[\/latex]. Multiply the [latex]5[\/latex] by [latex]6[\/latex] and subtract this product from [latex]30[\/latex].<\/td>\n<td style=\"width: 222.153px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215631\/CNX_BMath_Figure_01_05_045_img-05.png\" alt=\"We carry the ten's digit, 0, down to 3, resulting in 30. Consider how many times 6 goes into 30. 6 goes into 30 5 times, so 5 becomes the ten's digit in our quotient. We subtract 30 from 30, which equals 0.\" width=\"84\" height=\"125\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Now bring down the [latex]6[\/latex] and repeat these steps. There is [latex]1[\/latex] six in [latex]6[\/latex]. Write the [latex]1[\/latex] over the [latex]6[\/latex]. Multiply [latex]1[\/latex] by [latex]6[\/latex] and subtract this product from [latex]6[\/latex]<\/td>\n<td style=\"width: 222.153px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215632\/CNX_BMath_Figure_01_05_045_img-06.png\" alt=\"We carry down the one's digit, 6, to the 0, resulting in 6. Consider how many times 6 goes into 6. 6 goes into 6 1 time, so 1 becomes the one's digit in our quotient. Subtract 6 from 6 which is 0. We have a remainder of 0. In total 4506 divided by 6 equals 751\" width=\"84\" height=\"164\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 672.847px;\">Check by multiplying.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215633\/CNX_BMath_Figure_01_05_045_img-07.png\" alt=\"751 times 6 equals 4506, therefore our check is succcessful and our answer correct.\" width=\"84\" height=\"78\" data-media-type=\"image\/png\" \/><\/td>\n<td style=\"width: 222.153px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It equals the dividend, so our answer is correct.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144640\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144640&theme=oea&iframe_resize_id=ohm144640&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video for another example of how to use long division to divide a four digit whole number by a two digit whole number.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Example:  Dividing Whole Numbers without a Remainder\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/V7Korf09iWI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>So far all the division problems have worked out evenly. For example, if we had [latex]24[\/latex] cookies and wanted to make bags of [latex]8[\/latex] cookies, we would have [latex]3[\/latex] bags. But what if there were [latex]28[\/latex] cookies and we wanted to make bags of [latex]8?[\/latex] Start with the [latex]28[\/latex] cookies.<\/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\/24215640\/CNX_BMath_Figure_01_05_027.png\" alt=\"An image of 28 cookies placed at random.\" data-media-type=\"image\/png\" \/><br \/>\nTry to put the cookies in groups of eight.<\/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\/24215642\/CNX_BMath_Figure_01_05_028.png\" alt=\"An image of 28 cookies. There are 3 circles, each containing 8 cookies, leaving 3 cookies outside the circles.\" data-media-type=\"image\/png\" \/><br \/>\nThere are [latex]3[\/latex] groups of eight cookies, and [latex]4[\/latex] cookies left over. We call the [latex]4[\/latex] cookies that are left over the remainder and show it by writing R4 next to the [latex]3[\/latex]. (The R stands for remainder.)<\/p>\n<p>To check this division we multiply [latex]3[\/latex] times [latex]8[\/latex] to get [latex]24[\/latex], and then add the remainder of [latex]4[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\hfill 3\\\\ \\hfill \\underset{\\text{___}}{\\times 8}\\\\ \\hfill 24\\\\ \\hfill \\underset{\\text{___}}{+4}\\\\ \\hfill 28\\end{array}[\/latex]<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide [latex]1,439\\div 4[\/latex]. Check by multiplying.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q498101\">Show Answer<\/span><\/p>\n<div id=\"q498101\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-879\" class=\"unnumbered unstyled\" style=\"width: 917px;\" summary=\"This image shows 1,439 divided by 4. The first line says to try to divide 4 into 1. Since this won't work, try 4 into 14. There are 3 fours in 14. Write the 3 over the 4. the expression shows the quotient 3 above the 4. Next, multiply the 3 by the 4 and subtract this product from 14. The expressions shows this in long division with the quotient 5 above the 3. The next line says to bring down the 3 and repeat these steps. There are 5 fours in 23. Write the 5 over the 3. Multiply the 5 by 4 and subtract the product from 23. The expression shows this in long division. The next line says to bring down the 9 and repeat these steps. There are 9 fours in 39. Write the 9 over the 9. Multiply the 9 by 4 and subtract the product from 39. There are no more numbers to bring down. There is a remainder of 3. The expression shows the answer of 359 remainder 3.\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 695.851px;\">Let&#8217;s rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 199.149px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215643\/CNX_BMath_Figure_01_05_047_img-01.png\" alt=\"Long division for 1439 divided by 4.\" width=\"171\" height=\"28\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">First we try to divide [latex]4[\/latex] into [latex]1[\/latex]. Since that won&#8217;t work, we try [latex]4[\/latex] into [latex]14[\/latex]. There are [latex]3[\/latex] fours in [latex]14[\/latex]. We write the [latex]3[\/latex] over the [latex]4[\/latex].<\/td>\n<td style=\"width: 199.149px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215644\/CNX_BMath_Figure_01_05_047_img-02.png\" alt=\"4 goes into 14 3 times, so 3 becomes the hundred's place of the quotient.\" width=\"171\" height=\"45\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">Multiply the [latex]3[\/latex] by [latex]4[\/latex] and subtract this product from [latex]14[\/latex].<\/td>\n<td style=\"width: 199.149px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215645\/CNX_BMath_Figure_01_05_047_img-03.png\" alt=\"14 minus 12 equals 2.\" width=\"171\" height=\"85\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">Now bring down the [latex]3[\/latex] and repeat these steps. There are [latex]5[\/latex] fours in [latex]23[\/latex]. Write the [latex]5[\/latex] over the [latex]3[\/latex]. Multiply the [latex]5[\/latex] by [latex]4[\/latex] and subtract this product from [latex]23[\/latex].<\/td>\n<td style=\"width: 199.149px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215646\/CNX_BMath_Figure_01_05_047_img-04.png\" alt=\"We carry down the ten's digit, 3, to the 2, resulting in 23. Consider how many times 4 goes into 23. 4 goes into 23 5 times, so 5 becomes the ten's place in our quotient. 23 minus 20 equals 3.\" width=\"171\" height=\"125\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">Now bring down the [latex]9[\/latex] and repeat these steps. There are [latex]9[\/latex] fours in [latex]39[\/latex]. Write the [latex]9[\/latex] over the [latex]9[\/latex]. Multiply the [latex]9[\/latex] by [latex]4[\/latex] and subtract this product from [latex]39[\/latex]. There are no more numbers to bring down, so we are done. The remainder is [latex]3[\/latex].<\/td>\n<td style=\"width: 199.149px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215647\/CNX_BMath_Figure_01_05_047_img-05.png\" alt=\"We carry down the one's digit, 9, to the 3, resulting in 39. Consider how many times 4 goes into 39. 4 goes into 39 9 times, so 9 becomes the one's digit in our quotient. 39 minus 36 equals 3. We have a remainder of 3, so we write R 3 in the quotient. In total, 1439 divided by 4 equals 359 with a remainder of 3.\" width=\"171\" height=\"164\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 695.851px;\">Check by multiplying.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215648\/CNX_BMath_Figure_01_05_047_img-06.png\" alt=\"To check, we multiple our quotient, 359, by our divisor, 4, which equals 1436. Then, we add our remainder, 3, which equals 1439, the original dividend.\" width=\"171\" height=\"125\" data-media-type=\"image\/png\" \/><\/td>\n<td style=\"width: 199.149px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>So [latex]1,439\\div 4[\/latex] is [latex]359[\/latex] with a remainder of [latex]3[\/latex]. Our answer is correct.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144643\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144643&theme=oea&iframe_resize_id=ohm144643&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Divide and then check by multiplying: [latex]1,461\\div 13[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q174689\">Show Answer<\/span><\/p>\n<div id=\"q174689\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-708\" class=\"unnumbered unstyled\" style=\"width: 915px;\" summary=\"This image shows 1,461 divided by 13. The first line says First we try to divide 13 into 1. Since that won\u2019t work, we try 13 into 14. There is 1 thirteen in 14. We write the 1 over the 4. Multiply the 1 by 13 and subtract this product from 14. The expression shows the quotient 1 above the 4, and 13 under 14 with the difference as 1. The next line says to bring down the 6 and repeat these steps. There is 1 thirteen in 16. Write the 1 over the 6. Multiply the 1 by 13 and subtract this product from 16. The expression shows the 6 in the dividend with a blue arrow pointing downward to show the 6 is brought down next to the 1. In long division, 16 minus 3 shows a difference of 3. Next it says, now bring down the 1 and repeat these steps. There are 2 thirteens in 31. Write the 2 over the 1. Multiply the 2 by 13 and subtract this product from 31. The expression shows in long division the 1 in the dividend with a blue arrow pointing downward to show the one brought down to make 31. A 2 is placed above the 1 in the dividend. Thirty-one minus 26 is 5, but there are no more numbers to bring down, so we are done. The remainder is 5. The answer is 112 remainder 5. The next line says to check by multiplying in which the expression shows 112 with the word\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 701.698px;\">Let&#8217;s rewrite the problem to set it up for long division.<\/td>\n<td style=\"width: 192.302px;\">[latex]13\\overline{)1,461}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">First we try to divide [latex]13[\/latex] into [latex]1[\/latex]. Since that won&#8217;t work, we try [latex]13[\/latex] into [latex]14[\/latex]. There is [latex]1[\/latex] thirteen in [latex]14[\/latex]. We write the [latex]1[\/latex] over the [latex]4[\/latex].<\/td>\n<td style=\"width: 192.302px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215649\/CNX_BMath_Figure_01_05_048_img-02.png\" alt=\"Long division for 1461 divided by 13. 13 goes into 14 1 time, so 1 becomes the hundred's digit in the quotient.\" width=\"173\" height=\"43\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">Multiply the [latex]1[\/latex] by [latex]13[\/latex] and subtract this product from [latex]14[\/latex].<\/td>\n<td style=\"width: 192.302px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215650\/CNX_BMath_Figure_01_05_048_img-03.png\" alt=\"14 minus 13 equals 1.\" width=\"173\" height=\"85\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">Now bring down the [latex]6[\/latex] and repeat these steps. There is [latex]1[\/latex] thirteen in [latex]16[\/latex]. Write the [latex]1[\/latex] over the [latex]6[\/latex]. Multiply the [latex]1[\/latex] by [latex]13[\/latex] and subtract this product from [latex]16[\/latex].<\/td>\n<td style=\"width: 192.302px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215651\/CNX_BMath_Figure_01_05_048_img-04.png\" alt=\"We carry down the ten's digit, 6, to the 1, resulting in 16. 13 goes into 16 1 time, so 1 becomes the ten's digit in our quotient. 16 minus 13 equals 3.\" width=\"173\" height=\"125\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">Now bring down the [latex]1[\/latex] and repeat these steps. There are [latex]2[\/latex] thirteens in [latex]31[\/latex]. Write the [latex]2[\/latex] over the [latex]1[\/latex]. Multiply the [latex]2[\/latex] by [latex]13[\/latex] and subtract this product from [latex]31[\/latex]. There are no more numbers to bring down, so we are done. The remainder is [latex]5[\/latex]. [latex]1,462\\div 13[\/latex] is [latex]112[\/latex] with a remainder of [latex]5[\/latex].<\/td>\n<td style=\"width: 192.302px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215652\/CNX_BMath_Figure_01_05_048_img-05.png\" alt=\"We carry down the one's digit, 1, to the 3, resulting in 31. 13 goes into 31 twice, so 2 becomes the one's digit in our quotient. 31 minus 26 equals 5. We have a remainder of 5 so we write R 5 in the quotient. In total, 1461 divided by 13 equals 112 with a remainder of 5.\" width=\"173\" height=\"164\" data-media-type=\"image\/png\" \/><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 701.698px;\">Check by multiplying.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24215653\/CNX_BMath_Figure_01_05_048_img-06.png\" alt=\"To check, we multiple our quotient, 112, by our divisor, 13, which equals 336 plus 1120, and we add our remainder, 5. All 3 equal 1461, our original dividend.\" width=\"173\" height=\"136\" data-media-type=\"image\/png\" \/><\/td>\n<td style=\"width: 192.302px;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Our answer is correct.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm144644\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=144644&theme=oea&iframe_resize_id=ohm144644&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch the video below for another example of how to use long division to divide whole numbers when there is a remainder.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Example:  Dividing Whole Numbers with a Remainder\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/UPUcShGCBOs?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-66\">\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><\/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":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"423440b4-346f-4436-865b-55b917e695f9, a5c2f0ab-43bf-4b41-9c61-56c0c2a495f1","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-66","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/66","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":19,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/66\/revisions"}],"predecessor-version":[{"id":3980,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapters\/66\/revisions\/3980"}],"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\/66\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/media?parent=66"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/pressbooks\/v2\/chapter-type?post=66"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/contributor?post=66"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-accountingformanagers\/wp-json\/wp\/v2\/license?post=66"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}