{"id":230,"date":"2023-02-13T13:16:08","date_gmt":"2023-02-13T13:16:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=230"},"modified":"2024-10-18T20:52:25","modified_gmt":"2024-10-18T20:52:25","slug":"historical-counting-systems-background-youll-need-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/historical-counting-systems-background-youll-need-2\/","title":{"raw":"Historical Counting Systems: Background You'll Need 2","rendered":"Historical Counting Systems: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Simplify using math words and symbols<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Translate Word Phrases of Addition to Math Notation<\/h2>\r\n<p>Mathematics often presents itself as a unique language with its own vocabulary and symbols. To streamline problem-solving, it's helpful to translate common mathematical terms into their symbolic counterparts. The following table serves as a guide to convert verbal expressions of addition into their equivalent mathematical symbols.<\/p>\r\n<table>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Operation<\/th>\r\n<th>Words<\/th>\r\n<th>Example<\/th>\r\n<th>Expression<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td rowspan=\"7\">Addition<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>plus<\/td>\r\n<td>[latex]1[\/latex] plus [latex]2[\/latex]<\/td>\r\n<td>[latex]1+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>sum<\/td>\r\n<td>the sum of [latex]3[\/latex] and [latex]4[\/latex]<\/td>\r\n<td>[latex]3+4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>increased by<\/td>\r\n<td>[latex]5[\/latex] increased by [latex]6[\/latex]<\/td>\r\n<td>[latex]5+6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>more than<\/td>\r\n<td>[latex]8[\/latex] more than [latex]7[\/latex]<\/td>\r\n<td>[latex]7+8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>total of<\/td>\r\n<td>the total of [latex]9[\/latex] and [latex]5[\/latex]<\/td>\r\n<td>[latex]9+5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>added to<\/td>\r\n<td>[latex]6[\/latex] added to [latex]4[\/latex]<\/td>\r\n<td>[latex]4+6[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox example\">Translate and simplify the following:<center>[latex]28[\/latex] increased by [latex]31[\/latex]<\/center>[reveal-answer q=\"852875\"]Show Solution[\/reveal-answer] [hidden-answer a=\"852875\"] The words <em>increased by<\/em> tell us to add. The numbers given are the addends.\r\n\r\n\r\n<table id=\"eip-id1168288520954\" class=\"unnumbered unstyled\" summary=\"a\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>[latex]28[\/latex] increased by [latex]31[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate.<\/td>\r\n<td>[latex]28+31[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add.<\/td>\r\n<td>[latex]59[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>So [latex]28[\/latex] increased by [latex]31[\/latex] is [latex]59[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2925[\/ohm2_question]<\/section>\r\n<section><\/section>\r\n<section>\r\n<h2>Translate Word Phrases of Subtraction to Math Notation<\/h2>\r\n\r\n\r\nIn everyday language, we often talk about taking away or comparing amounts. In math, this is known as subtraction. The words we use in conversation can be directly translated to subtraction symbols, which you'll see in math problems. The table below shows how common phrases that describe taking away or comparing quantities are expressed in the universal language of math.\r\n\r\n\r\n<table>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Operation<\/th>\r\n<th>Word Phrase<\/th>\r\n<th>Example<\/th>\r\n<th>Expression<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td rowspan=\"6\">Subtraction<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>minus<\/td>\r\n<td>[latex]5[\/latex] minus [latex]1[\/latex]<\/td>\r\n<td>[latex]5 - 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>difference<\/td>\r\n<td>the difference of [latex]9[\/latex] and [latex]4[\/latex]<\/td>\r\n<td>[latex]9 - 4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>decreased by<\/td>\r\n<td>[latex]7[\/latex] decreased by [latex]3[\/latex]<\/td>\r\n<td>[latex]7 - 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>less than<\/td>\r\n<td>[latex]5[\/latex] less than [latex]8[\/latex]<\/td>\r\n<td>[latex]8 - 5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>subtracted from<\/td>\r\n<td>[latex]1[\/latex] subtracted from [latex]6[\/latex]<\/td>\r\n<td>[latex]6 - 1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox example\">Translate the following into numerical form and then simplify:\r\n\r\n\r\n<ol id=\"eip-id1168287211491\" class=\"circled\">\r\n\t<li>The difference of [latex]13[\/latex] and [latex]8[\/latex]<\/li>\r\n\t<li>Subtract [latex]24[\/latex] from [latex]43[\/latex]<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"41059\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"41059\"]<\/p>\r\n<ol>\r\n\t<li>The word <em>difference<\/em> tells us to subtract the two numbers. The numbers stay in the same order as in the phrase.<br \/>\r\n<table id=\"eip-id1168289460345\" class=\"unnumbered unstyled\" style=\"width: 90%;\" summary=\"a\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>The difference of [latex]13[\/latex] and [latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate.<\/td>\r\n<td>[latex]13 - 8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<br \/>\r\n<\/li>\r\n\t<li>The words <em>subtract<\/em> <em>from<\/em> tells us to take the second number away from the first. We must be careful to get the order correct.<br \/>\r\n<table id=\"eip-id1168287585454\" class=\"unnumbered unstyled\" style=\"width: 90%;\" summary=\"a\">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>Subtract [latex]24[\/latex] from [latex]43[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate.<\/td>\r\n<td>[latex]43 - 24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]19[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2926[\/ohm2_question]<\/section>\r\n<section>\r\n<h2>Translate Word Phrases of Multiplication to Math Notation<\/h2>\r\n\r\n\r\nTurning words into math symbols is like finding shortcuts to solve problems faster. For multiplication, we use different phrases to mean 'put together in groups.' This table helps you to see how phrases like 'times' and 'twice' can be written using multiplication signs or dots, making them ready for quick calculations.\r\n\r\n\r\n<table>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Operation<\/th>\r\n<th>Word Phrase<\/th>\r\n<th>Example<\/th>\r\n<th>Expression<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"width: 25%;\" rowspan=\"4\">Multiplication<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>times<\/td>\r\n<td>[latex]3[\/latex] times [latex]8[\/latex]<\/td>\r\n<td>[latex]3\\times 8, 3\\cdot 8, (3)(8)[\/latex], [latex](3)8, \\text{ or } 3(8)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>product<\/td>\r\n<td>the product of [latex]3[\/latex] and [latex]8[\/latex]<\/td>\r\n<td>[latex]3\\times 8, 3\\cdot 8, (3)(8)[\/latex], [latex](3)8, \\text{ or } 3(8)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>twice<\/td>\r\n<td>twice [latex]4[\/latex]<\/td>\r\n<td>[latex]2\\cdot 4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox example\">\r\n<p>Translate the following to math notation and simplify:<\/p>\r\n<center>the product of [latex]12[\/latex] and [latex]27[\/latex].<\/center>[reveal-answer q=\"674781\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"674781\"]The word <em>product<\/em> tells us to multiply. The words <em>of<\/em> [latex]12[\/latex] <em>and<\/em> [latex]27[\/latex] tell us the two factors.\r\n\r\n\r\n<table id=\"eip-id1168287170388\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>the product of [latex]12[\/latex] and [latex]27[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate.<\/td>\r\n<td>[latex]12\\cdot 27[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]324[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section><\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2930[\/ohm2_question]<\/section>\r\n<section>\r\n<h2>Translate Word Phrases of Division to Math Notation<\/h2>\r\n\r\n\r\nWhen we share things equally or find out how many times one number fits into another, we're doing division. The words 'divided by' and 'quotient of' are keys that tell us to divide. This table shows how these words change into division symbols.\r\n\r\n\r\n<table>\r\n<thead>\r\n<tr valign=\"top\">\r\n<th>Operation<\/th>\r\n<th>Word Phrase<\/th>\r\n<th>Example<\/th>\r\n<th>Expression<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td rowspan=\"4\">Division<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>divided by<\/td>\r\n<td>[latex]12[\/latex] divided by [latex]4[\/latex]<\/td>\r\n<td>[latex]12\\div 4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>quotient of<\/td>\r\n<td>the quotient of [latex]12[\/latex] and [latex]4[\/latex]<\/td>\r\n<td>[latex]\\frac{12}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>divided into<\/td>\r\n<td>[latex]4[\/latex] divided into [latex]12[\/latex]<\/td>\r\n<td>[latex]4\\overline{)12}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox example\">Translate the following into numerical notation and simplify:<center>the quotient of [latex]51[\/latex] and [latex]17[\/latex].<\/center>[reveal-answer q=\"754097\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"754097\"]The word <em>quotient<\/em> tells us to divide. The words <em>of<\/em> [latex]51[\/latex] <em>and<\/em> [latex]17[\/latex] tell us the two factors.\r\n\r\n\r\n<table id=\"eip-id1168287170388\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\" \">\r\n<tbody>\r\n<tr>\r\n<td>&nbsp;<\/td>\r\n<td>the quotient of [latex]51[\/latex] and [latex]17[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Translate.<\/td>\r\n<td>[latex]51\\div 17[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Divide.<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p><strong>Note:<\/strong> We could just as correctly have translated <em>the quotient of<\/em> [latex]51[\/latex] <em>and<\/em>\u00a0 [latex]17[\/latex] using the notation:<\/p>\r\n<center>[latex]17\\overline{)51}[\/latex] \u00a0 or \u00a0[latex]\\frac{51}{17}[\/latex].<center><\/center><\/center>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section><\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]2931[\/ohm2_question]<\/section>\r\n<\/section>\r\n<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Simplify using math words and symbols<\/li>\n<\/ul>\n<\/section>\n<h2>Translate Word Phrases of Addition to Math Notation<\/h2>\n<p>Mathematics often presents itself as a unique language with its own vocabulary and symbols. To streamline problem-solving, it&#8217;s helpful to translate common mathematical terms into their symbolic counterparts. The following table serves as a guide to convert verbal expressions of addition into their equivalent mathematical symbols.<\/p>\n<table>\n<thead>\n<tr valign=\"top\">\n<th>Operation<\/th>\n<th>Words<\/th>\n<th>Example<\/th>\n<th>Expression<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td rowspan=\"7\">Addition<\/td>\n<\/tr>\n<tr>\n<td>plus<\/td>\n<td>[latex]1[\/latex] plus [latex]2[\/latex]<\/td>\n<td>[latex]1+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>sum<\/td>\n<td>the sum of [latex]3[\/latex] and [latex]4[\/latex]<\/td>\n<td>[latex]3+4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>increased by<\/td>\n<td>[latex]5[\/latex] increased by [latex]6[\/latex]<\/td>\n<td>[latex]5+6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>more than<\/td>\n<td>[latex]8[\/latex] more than [latex]7[\/latex]<\/td>\n<td>[latex]7+8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>total of<\/td>\n<td>the total of [latex]9[\/latex] and [latex]5[\/latex]<\/td>\n<td>[latex]9+5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>added to<\/td>\n<td>[latex]6[\/latex] added to [latex]4[\/latex]<\/td>\n<td>[latex]4+6[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox example\">Translate and simplify the following:<\/p>\n<div style=\"text-align: center;\">[latex]28[\/latex] increased by [latex]31[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q852875\">Show Solution<\/button> <\/p>\n<div id=\"q852875\" class=\"hidden-answer\" style=\"display: none\"> The words <em>increased by<\/em> tell us to add. The numbers given are the addends.<\/p>\n<table id=\"eip-id1168288520954\" class=\"unnumbered unstyled\" summary=\"a\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>[latex]28[\/latex] increased by [latex]31[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>Translate.<\/td>\n<td>[latex]28+31[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add.<\/td>\n<td>[latex]59[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>&nbsp;<\/td>\n<td>So [latex]28[\/latex] increased by [latex]31[\/latex] is [latex]59[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2925\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2925&theme=lumen&iframe_resize_id=ohm2925&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section><\/section>\n<section>\n<h2>Translate Word Phrases of Subtraction to Math Notation<\/h2>\n<p>In everyday language, we often talk about taking away or comparing amounts. In math, this is known as subtraction. The words we use in conversation can be directly translated to subtraction symbols, which you&#8217;ll see in math problems. The table below shows how common phrases that describe taking away or comparing quantities are expressed in the universal language of math.<\/p>\n<table>\n<thead>\n<tr valign=\"top\">\n<th>Operation<\/th>\n<th>Word Phrase<\/th>\n<th>Example<\/th>\n<th>Expression<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td rowspan=\"6\">Subtraction<\/td>\n<\/tr>\n<tr>\n<td>minus<\/td>\n<td>[latex]5[\/latex] minus [latex]1[\/latex]<\/td>\n<td>[latex]5 - 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>difference<\/td>\n<td>the difference of [latex]9[\/latex] and [latex]4[\/latex]<\/td>\n<td>[latex]9 - 4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>decreased by<\/td>\n<td>[latex]7[\/latex] decreased by [latex]3[\/latex]<\/td>\n<td>[latex]7 - 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>less than<\/td>\n<td>[latex]5[\/latex] less than [latex]8[\/latex]<\/td>\n<td>[latex]8 - 5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>subtracted from<\/td>\n<td>[latex]1[\/latex] subtracted from [latex]6[\/latex]<\/td>\n<td>[latex]6 - 1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox example\">Translate the following into numerical form and then simplify:<\/p>\n<ol id=\"eip-id1168287211491\" class=\"circled\">\n<li>The difference of [latex]13[\/latex] and [latex]8[\/latex]<\/li>\n<li>Subtract [latex]24[\/latex] from [latex]43[\/latex]<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q41059\">Show Answer<\/button><\/p>\n<div id=\"q41059\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The word <em>difference<\/em> tells us to subtract the two numbers. The numbers stay in the same order as in the phrase.<br \/>\n<table id=\"eip-id1168289460345\" class=\"unnumbered unstyled\" style=\"width: 90%;\" summary=\"a\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>The difference of [latex]13[\/latex] and [latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Translate.<\/td>\n<td>[latex]13 - 8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n<\/li>\n<li>The words <em>subtract<\/em> <em>from<\/em> tells us to take the second number away from the first. We must be careful to get the order correct.<br \/>\n<table id=\"eip-id1168287585454\" class=\"unnumbered unstyled\" style=\"width: 90%;\" summary=\"a\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>Subtract [latex]24[\/latex] from [latex]43[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Translate.<\/td>\n<td>[latex]43 - 24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]19[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2926\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2926&theme=lumen&iframe_resize_id=ohm2926&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<h2>Translate Word Phrases of Multiplication to Math Notation<\/h2>\n<p>Turning words into math symbols is like finding shortcuts to solve problems faster. For multiplication, we use different phrases to mean &#8216;put together in groups.&#8217; This table helps you to see how phrases like &#8216;times&#8217; and &#8216;twice&#8217; can be written using multiplication signs or dots, making them ready for quick calculations.<\/p>\n<table>\n<thead>\n<tr valign=\"top\">\n<th>Operation<\/th>\n<th>Word Phrase<\/th>\n<th>Example<\/th>\n<th>Expression<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 25%;\" rowspan=\"4\">Multiplication<\/td>\n<\/tr>\n<tr>\n<td>times<\/td>\n<td>[latex]3[\/latex] times [latex]8[\/latex]<\/td>\n<td>[latex]3\\times 8, 3\\cdot 8, (3)(8)[\/latex], [latex](3)8, \\text{ or } 3(8)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>product<\/td>\n<td>the product of [latex]3[\/latex] and [latex]8[\/latex]<\/td>\n<td>[latex]3\\times 8, 3\\cdot 8, (3)(8)[\/latex], [latex](3)8, \\text{ or } 3(8)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>twice<\/td>\n<td>twice [latex]4[\/latex]<\/td>\n<td>[latex]2\\cdot 4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox example\">\n<p>Translate the following to math notation and simplify:<\/p>\n<div style=\"text-align: center;\">the product of [latex]12[\/latex] and [latex]27[\/latex].<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q674781\">Show Answer<\/button><\/p>\n<div id=\"q674781\" class=\"hidden-answer\" style=\"display: none\">The word <em>product<\/em> tells us to multiply. The words <em>of<\/em> [latex]12[\/latex] <em>and<\/em> [latex]27[\/latex] tell us the two factors.<\/p>\n<table id=\"eip-id1168287170388\" class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>the product of [latex]12[\/latex] and [latex]27[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Translate.<\/td>\n<td>[latex]12\\cdot 27[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]324[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2930\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2930&theme=lumen&iframe_resize_id=ohm2930&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<h2>Translate Word Phrases of Division to Math Notation<\/h2>\n<p>When we share things equally or find out how many times one number fits into another, we&#8217;re doing division. The words &#8216;divided by&#8217; and &#8216;quotient of&#8217; are keys that tell us to divide. This table shows how these words change into division symbols.<\/p>\n<table>\n<thead>\n<tr valign=\"top\">\n<th>Operation<\/th>\n<th>Word Phrase<\/th>\n<th>Example<\/th>\n<th>Expression<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td rowspan=\"4\">Division<\/td>\n<\/tr>\n<tr>\n<td>divided by<\/td>\n<td>[latex]12[\/latex] divided by [latex]4[\/latex]<\/td>\n<td>[latex]12\\div 4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>quotient of<\/td>\n<td>the quotient of [latex]12[\/latex] and [latex]4[\/latex]<\/td>\n<td>[latex]\\frac{12}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>divided into<\/td>\n<td>[latex]4[\/latex] divided into [latex]12[\/latex]<\/td>\n<td>[latex]4\\overline{)12}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox example\">Translate the following into numerical notation and simplify:<\/p>\n<div style=\"text-align: center;\">the quotient of [latex]51[\/latex] and [latex]17[\/latex].<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q754097\">Show Answer<\/button><\/p>\n<div id=\"q754097\" class=\"hidden-answer\" style=\"display: none\">The word <em>quotient<\/em> tells us to divide. The words <em>of<\/em> [latex]51[\/latex] <em>and<\/em> [latex]17[\/latex] tell us the two factors.<\/p>\n<table class=\"unnumbered unstyled\" style=\"width: 75%;\" summary=\"\">\n<tbody>\n<tr>\n<td>&nbsp;<\/td>\n<td>the quotient of [latex]51[\/latex] and [latex]17[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Translate.<\/td>\n<td>[latex]51\\div 17[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Divide.<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Note:<\/strong> We could just as correctly have translated <em>the quotient of<\/em> [latex]51[\/latex] <em>and<\/em>\u00a0 [latex]17[\/latex] using the notation:<\/p>\n<div style=\"text-align: center;\">[latex]17\\overline{)51}[\/latex] \u00a0 or \u00a0[latex]\\frac{51}{17}[\/latex].<\/p>\n<div style=\"text-align: center;\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm2931\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=2931&theme=lumen&iframe_resize_id=ohm2931&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/section>\n<\/section>\n<\/section>\n","protected":false},"author":16,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":40,"module-header":"background_you_need","content_attributions":[{"type":"cc-attribution","description":"Prealgebra","author":"","organization":"OpenStax","url":"","project":"","license":"cc-by","license_terms":"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/230"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/16"}],"version-history":[{"count":48,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/230\/revisions"}],"predecessor-version":[{"id":15087,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/230\/revisions\/15087"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/40"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/230\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=230"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=230"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=230"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=230"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}