chơi xổ số keno trực tuyến

{"appState":{"pageLoadApiCallsStatus":true},"categoryState":{"relatedCategories":{"headers":{"timestamp":"2025-03-04T08:01:10+00:00"},"categoryId":33722,"data":{"title":"Basic Math","slug":"basic-math","image":{"src":null,"width":0,"height":0},"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Basic Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"},"slug":"basic-math","categoryId":33722}],"parentCategory":{"categoryId":33720,"title":"Math","slug":"math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"}},"childCategories":[],"description":"Add and subtract negative numbers, calculate how much to tip, make sense of exponents, and generally get mathy, with our easy-to-read articles.","relatedArticles":{"self":"//dummies-api.coursofppt.com/v2/articles?category=33722&offset=0&size=5"},"hasArticle":true,"hasBook":true,"articleCount":151,"bookCount":4},"_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"}},"relatedCategoriesLoadedStatus":"success"},"listState":{"list":{"count":10,"total":154,"items":[{"headers":{"creationTime":"2017-03-27T16:46:41+00:00","modifiedTime":"2024-09-14T15:21:16+00:00","timestamp":"2024-09-14T18:01:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Basic Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"},"slug":"basic-math","categoryId":33722}],"title":"AS and A-Level Maths For Dummies Cheat Sheet","strippedTitle":"as and a-level maths for dummies cheat sheet","slug":"as-a-level-maths-for-dummies-cheat-sheet","canonicalUrl":"","寻找网络归类网页引挚改进":{"metaDescription":"Use this Cheat Sheet as a reference for differentiating and integrating powers of x, trigonometric functions, exponentials, and more.","noIndex":0,"noFollow":0},"content":"Some of the most important things to remember in AS-level and A-level maths are the rules for differentiating and integrating expressions. This cheat sheet is a handy reference for what happens when you differentiate or integrate powers of <i>x</i>, trigonometric functions, exponentials or logarithms – as well as the rules you need for what to do when they’re combined!","description":"Some of the most important things to remember in AS-level and A-level maths are the rules for differentiating and integrating expressions. This cheat sheet is a handy reference for what happens when you differentiate or integrate powers of <i>x</i>, trigonometric functions, exponentials or logarithms – as well as the rules you need for what to do when they’re combined!","blurb":"","authors":[],"primaryCategoryTaxonomy":{"categoryId":33722,"title":"Basic Math","slug":"basic-math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":291491,"title":"Teaching Your Kids New Math (K-5) For Dummies Cheat Sheet","slug":"teaching-your-kids-new-math-k-5-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/291491"}},{"articleId":253710,"title":"Pre-Algebra: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249996,"title":"Solving Simple Equations in Pre-Algebra Problems","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Rearranging Algebraic Equations to Isolate X","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":0,"slug":null,"isbn":null,"categoryList":null,"amazon":null,"image":null,"title":null,"testBankPinActivationLink":null,"bookOutOfPrint":false,"authorsInfo":null,"authors":null,"_links":null},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-65034a5eca2d0\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-65034a5ecac92\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":138462,"title":"Differentiating Simple Terms and Functions for AS & A-Level Maths","slug":"differentiating-simple-terms-and-functions-for-as-a-level-maths","categoryList":[],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/138462"}},{"articleId":138464,"title":"Differentiating Combinations for AS & A-Level Maths","slug":"differentiating-combinations-for-as-a-level-maths","categoryList":[],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/138464"}},{"articleId":138461,"title":"Integrating Simple Terms and Functions for AS & A Level Maths","slug":"integrating-simple-terms-and-functions-for-as-a-level-maths","categoryList":[],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/138461"}},{"articleId":138463,"title":"Integrating Combinations for AS & A-Level Maths","slug":"integrating-combinations-for-as-a-level-maths","categoryList":[],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/138463"}}],"content":[{"title":"Differentiating simple terms and functions","thumb":null,"image":null,"content":"<p>This is a handy guide to the building blocks of differentiation for A-level maths: it gives you all of the things you’re expected to know to differentiate (powers of <i>x</i>, exponentials, logarithms and trigonometric functions) and what you get when you do the calculus!</p>\n<p>In the following table, <i>a</i> represents any constant.</p>\n<p><img loading=\"lazy\" src=\"//coursofppt.com/wp-content/uploads/503353.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"168\" /></p>\n<p class=\"Tip\">Note that ln(<i>ax</i>) differentiates to <i>a</i> / <i>ax</i>, and after you cancel the <i>a</i>s you’re left with 1/<i>x</i>.</p>\n"},{"title":"Differentiating combinations","thumb":null,"image":null,"content":"<p>Once you’ve mastered the building blocks of differentiation for A-level maths, the next step is to get comfortable with the various rules – what happens when you have a function with something complicated inside the brackets? How about if you have expressions involving addition, subtraction, multiplication or division? This table reminds you what to do in these situations, including the dreaded chain rule, quotient rule and product rule from Core 3.</p>\n<p><img loading=\"lazy\" src=\"//coursofppt.com/wp-content/uploads/503355.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"168\" /></p>\n<p class=\"Remember\">If you’re doing <i>p</i><i>arametric differentiation</i><i> </i>in Core 4, then <i>d</i><i>y</i>/<i>d</i><i>x</i> = (<i>d</i><i>y</i>/<i>d</i><i>t</i>)/(<i>d</i><i>x</i>/<i>d</i><i>t</i>).</p>\n"},{"title":"Integrating simple terms and functions","thumb":null,"image":null,"content":"<p>If you want to get a good grade in A-level maths, one of the skills you need is the integration of simple terms – powers of <i>x</i>, exponentials and trigonometric functions.</p>\n<p class=\"Remember\">Any time you integrate, you have to remember to add a constant! If you are working with limits, this cancels out, and if you have additional information, you may be able to work the constant out.</p>\n<p><img loading=\"lazy\" src=\"//coursofppt.com/wp-content/uploads/503357.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"144\" /></p>\n"},{"title":"Integrating combinations","thumb":null,"image":null,"content":"<p>When you have to integrate something more complicated than a simple expression for A-level maths, the chances are you’ll need to use one of the rules in this table, which shows you how to deal with a function when it has a linear argument, two terms added to or subtracted from each other, and the formula for integration by parts. In this table, a capital letter means “the integral of the lower-case letter” – so <i>F</i> is the integral of <i>f</i>, <i>U</i> is the integral of <i>u</i>, and <i>V</i> is the integral of <i>v</i>.</p>\n<p><img loading=\"lazy\" src=\"//coursofppt.com/wp-content/uploads/503359.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"120\" /></p>\n<p>You also need to know about calculating a volume of revolution: Rotating the curve <i>y</i>=<i>f</i>(<i>x</i>) around the <i>x</i>-axis between limits of <i>a</i> and <i>b</i> gives you a volume of</p>\n<p><img loading=\"lazy\" src=\"//coursofppt.com/wp-content/uploads/503360.image1.png\" alt=\"image1.png\" width=\"69\" height=\"33\" /></p>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2024-02-16T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207393},{"headers":{"creationTime":"2019-02-13T02:02:54+00:00","modifiedTime":"2024-08-07T14:56:15+00:00","timestamp":"2024-08-07T15:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Basic Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"},"slug":"basic-math","categoryId":33722}],"title":"Pre-Algebra Practice Questions: Finding the Volume of Prisms and Cylinders","strippedTitle":"pre-algebra practice questions: finding the volume of prisms and cylinders","slug":"pre-algebra-practice-questions-finding-volume-prisms-cylinders","canonicalUrl":"","寻找网络归类网页引挚改进":{"metaDescription":"To find the volume of a prism or cylinder, you can use the following formula, where A b is the area of the base and h is the height: V = A b x h --> Practice qu","noIndex":0,"noFollow":0},"content":"To find the volume of a prism or cylinder, you can use the following formula, where <em>A<sub>b</sub></em> is the area of the base and <em>h</em> is the height:\r\n<pre><em>V</em> = <em>A<sub>b</sub></em> x <em>h</em></pre>\r\n<!--<img class=\"alignnone size-full wp-image-249965\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3401.gif\" alt=\"PREALGEBRA_3401\" width=\"75\" height=\"27\" />-->\r\n<h2 id=\"tab1\" >Practice questions</h2>\r\n<ol>\r\n \t<li>Find the volume of a prism with a base that has an area of 6 square centimeters and a height of 3 centimeters.</li>\r\n \t<li>Figure out the approximate volume of a cylinder whose base has a radius of 7 millimeters and whose height is 16 millimeters.</li>\r\n</ol>\r\n<h2 id=\"tab2\" >Answers and explanations</h2>\r\n<ol>\r\n \t<li>18 cubic centimeters\r\n \t<pre><em>V</em> = <em>A<sub>b</sub></em> x <em>h</em> = 6cm<sup>2</sup> x 3cm = 18cm<sup>3</sup></pre>\r\n<!--<img class=\"alignnone size-full wp-image-249966\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3402.gif\" alt=\"PREALGEBRA_3402\" width=\"244\" height=\"28\" />--></li>\r\n \t<li>Approximately 2,461.76 cubic millimeters\r\n<p>First, use the area formula for a circle to find the area of the base:</p>\r\n \t <pre><em>A<sub>b</sub></em> = <em>π</em> x <em>r<sup>2</sup>\r\n ≅ 3.14 x (7mm)<sup>2</sup>\r\n = 3.14 x 49mm<sup>2</sup>\r\n = 153.86mm<sup>2</sup></pre>\r\n<!--<img class=\"alignnone size-full wp-image-249967\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3403.gif\" alt=\"PREALGEBRA_3403\" width=\"144\" height=\"120\" />-->\r\n\r\nPlug this result into the formula for the volume of a cylinder:\r\n \t<pre><em>V</em> = <em>A<sub>b</sub></em> x <em>h\r\n = 153.86mm<sup>2</sup> x 16mm\r\n = 2,461.76mm<sup>3</sup></pre>\r\n<!--<img class=\"alignnone size-full wp-image-249968\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3404.gif\" alt=\"PREALGEBRA_3404\" width=\"185\" height=\"84\" />--></li>\r\n</ol>","description":"To find the volume of a prism or cylinder, you can use the following formula, where <em>A<sub>b</sub></em> is the area of the base and <em>h</em> is the height:\r\n<pre><em>V</em> = <em>A<sub>b</sub></em> x <em>h</em></pre>\r\n<!--<img class=\"alignnone size-full wp-image-249965\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3401.gif\" alt=\"PREALGEBRA_3401\" width=\"75\" height=\"27\" />-->\r\n<h2 id=\"tab1\" >Practice questions</h2>\r\n<ol>\r\n \t<li>Find the volume of a prism with a base that has an area of 6 square centimeters and a height of 3 centimeters.</li>\r\n \t<li>Figure out the approximate volume of a cylinder whose base has a radius of 7 millimeters and whose height is 16 millimeters.</li>\r\n</ol>\r\n<h2 id=\"tab2\" >Answers and explanations</h2>\r\n<ol>\r\n \t<li>18 cubic centimeters\r\n \t<pre><em>V</em> = <em>A<sub>b</sub></em> x <em>h</em> = 6cm<sup>2</sup> x 3cm = 18cm<sup>3</sup></pre>\r\n<!--<img class=\"alignnone size-full wp-image-249966\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3402.gif\" alt=\"PREALGEBRA_3402\" width=\"244\" height=\"28\" />--></li>\r\n \t<li>Approximately 2,461.76 cubic millimeters\r\n<p>First, use the area formula for a circle to find the area of the base:</p>\r\n \t <pre><em>A<sub>b</sub></em> = <em>π</em> x <em>r<sup>2</sup>\r\n ≅ 3.14 x (7mm)<sup>2</sup>\r\n = 3.14 x 49mm<sup>2</sup>\r\n = 153.86mm<sup>2</sup></pre>\r\n<!--<img class=\"alignnone size-full wp-image-249967\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3403.gif\" alt=\"PREALGEBRA_3403\" width=\"144\" height=\"120\" />-->\r\n\r\nPlug this result into the formula for the volume of a cylinder:\r\n \t<pre><em>V</em> = <em>A<sub>b</sub></em> x <em>h\r\n = 153.86mm<sup>2</sup> x 16mm\r\n = 2,461.76mm<sup>3</sup></pre>\r\n<!--<img class=\"alignnone size-full wp-image-249968\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3404.gif\" alt=\"PREALGEBRA_3404\" width=\"185\" height=\"84\" />--></li>\r\n</ol>","blurb":"","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> is a math teacher and tutor with degrees in math and English from Rutgers University. He is the author of a dozen books, including <i>Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, and Calculus II For Dummies</i>. Through online tutoring, he teaches multiplication and beyond to preschoolers in a way that sets them up for school success while keeping the natural magic of math alive. Contact Mark at markzegarelli.com. ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"primaryCategoryTaxonomy":{"categoryId":33722,"title":"Basic Math","slug":"basic-math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"}},"secondaryCategoryTaxonomy":{"categoryId":33726,"title":"Pre-Algebra","slug":"pre-algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33726"}},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Practice questions","target":"#tab1"},{"label":"Answers and explanations","target":"#tab2"}],"relatedArticles":{"fromBook":[{"articleId":253710,"title":"Pre-Algebra: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249996,"title":"Solving Simple Equations in Pre-Algebra Problems","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Rearranging Algebraic Equations to Isolate X","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}},{"articleId":249971,"title":"Pre-Algebra: Cross-Multiply to Solve Equations","slug":"pre-algebra-practice-questions-cross-multiply-solve-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249971"}}],"fromCategory":[{"articleId":291491,"title":"Teaching Your Kids New Math (K-5) For Dummies Cheat Sheet","slug":"teaching-your-kids-new-math-k-5-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/291491"}},{"articleId":253710,"title":"Pre-Algebra: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249996,"title":"Solving Simple Equations in Pre-Algebra Problems","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Rearranging Algebraic Equations to Isolate X","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281980,"slug":"basic-math-and-pre-algebra-workbook-for-dummies-with-online-practice-3rd-edition","isbn":"9781119357513","categoryList":["academics-the-arts","math","basic-math"],"amazon":{"default":"//www.amazon.com/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119357519-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/basic-math-and-pre-algebra-workbook-for-dummies-3rd-edition-cover-9781119357513-204x255.jpg","width":204,"height":255},"title":"Basic Math & Pre-Algebra Workbook For Dummies with Online Practice","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"9399\">Mark Zegarelli</b> is a math and test prep teacher who has written a wide variety of basic math and pre-algebra books in the <i>For Dummies</i> series. </p>","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> is a math teacher and tutor with degrees in math and English from Rutgers University. He is the author of a dozen books, including <i>Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, and Calculus II For Dummies</i>. Through online tutoring, he teaches multiplication and beyond to preschoolers in a way that sets them up for school success while keeping the natural magic of math alive. Contact Mark at markzegarelli.com. ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119357513&quot;]}]\" id=\"du-slot-64d1072f6f6ab\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119357513&quot;]}]\" id=\"du-slot-64d1072f6fb69\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2024-08-07T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":249964},{"headers":{"creationTime":"2017-03-26T11:06:08+00:00","modifiedTime":"2024-07-10T19:00:07+00:00","timestamp":"2024-07-10T21:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Basic Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"},"slug":"basic-math","categoryId":33722}],"title":"Evaluating an Expression with Only Multiplication & Division","strippedTitle":"evaluating an expression with only multiplication & division","slug":"applying-order-of-operations-to-expressions-with-only-multiplication-and-division","canonicalUrl":"","寻找网络归类网页引挚改进":{"metaDescription":"Some expressions contain only multiplication and division. When this is the case, the rule for evaluating the expression is pretty straightforward. When an expr","noIndex":0,"noFollow":0},"content":"Some expressions contain only multiplication and division. When this is the case, the rule for evaluating the expression is pretty straightforward. When an expression contains only multiplication and division, evaluate it step by step from left to right.\r\n<h2 id=\"tab1\" >The Three Types of Big Four Expressions</h2>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Expression</th>\r\n<th>Example</th>\r\n<th>Rule</th>\r\n</tr>\r\n<tr>\r\n<td>Contains only addition and subtraction</td>\r\n<td>12 + 7 – 6 – 3 + 8</td>\r\n<td>Evaluate left to right.</td>\r\n</tr>\r\n<tr>\r\n<td>Contains only multiplication and division</td>\r\n<td>18 ÷ 3 x 7 ÷ 14</td>\r\n<td>Evaluate left to right.</td>\r\n</tr>\r\n<tr>\r\n<td>Mixed-operator expression: contains a combination of\r\naddition/subtraction and multiplication/division</td>\r\n<td>9 + 6 ÷ 3</td>\r\n<td>1. Evaluate multiplication and division left to right.\r\n2. Evaluate addition and subtraction left to right.</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<p class=\"Remember\">Suppose you want to evaluate this expression:</p>\r\n\r\n<blockquote>9 × 2 ÷ 6 ÷ 3 × 2</blockquote>\r\nAgain, the expression contains only multiplication and division, so you can move from left to right, starting with 9 x 2:\r\n<blockquote>= 18 ÷ 6 ÷ 3 × 2\r\n= 3 ÷ 3 × 2\r\n= 1 × 2\r\n= 2</blockquote>\r\nNotice that the expression shrinks one number at a time until all that’s left is 2. So\r\n<blockquote>9 × 2 ÷ 6 ÷ 3 × 2 = 2</blockquote>\r\nHere’s another quick example:\r\n<blockquote>−2 × 6 ÷ −4</blockquote>\r\nEven though this expression has some negative numbers, the only operations it contains are multiplication and division. So you can evaluate it in two steps from left to right (remembering the rules for multiplying and dividing with negative numbers):\r\n<blockquote>= −2 × 6 ÷ −4\r\n= −12 ÷ −4\r\n= 3</blockquote>\r\nThus,\r\n<blockquote>−2 × 6 ÷ −4 = 3</blockquote>","description":"Some expressions contain only multiplication and division. When this is the case, the rule for evaluating the expression is pretty straightforward. When an expression contains only multiplication and division, evaluate it step by step from left to right.\r\n<h2 id=\"tab1\" >The Three Types of Big Four Expressions</h2>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>Expression</th>\r\n<th>Example</th>\r\n<th>Rule</th>\r\n</tr>\r\n<tr>\r\n<td>Contains only addition and subtraction</td>\r\n<td>12 + 7 – 6 – 3 + 8</td>\r\n<td>Evaluate left to right.</td>\r\n</tr>\r\n<tr>\r\n<td>Contains only multiplication and division</td>\r\n<td>18 ÷ 3 x 7 ÷ 14</td>\r\n<td>Evaluate left to right.</td>\r\n</tr>\r\n<tr>\r\n<td>Mixed-operator expression: contains a combination of\r\naddition/subtraction and multiplication/division</td>\r\n<td>9 + 6 ÷ 3</td>\r\n<td>1. Evaluate multiplication and division left to right.\r\n2. Evaluate addition and subtraction left to right.</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\n<p class=\"Remember\">Suppose you want to evaluate this expression:</p>\r\n\r\n<blockquote>9 × 2 ÷ 6 ÷ 3 × 2</blockquote>\r\nAgain, the expression contains only multiplication and division, so you can move from left to right, starting with 9 x 2:\r\n<blockquote>= 18 ÷ 6 ÷ 3 × 2\r\n= 3 ÷ 3 × 2\r\n= 1 × 2\r\n= 2</blockquote>\r\nNotice that the expression shrinks one number at a time until all that’s left is 2. So\r\n<blockquote>9 × 2 ÷ 6 ÷ 3 × 2 = 2</blockquote>\r\nHere’s another quick example:\r\n<blockquote>−2 × 6 ÷ −4</blockquote>\r\nEven though this expression has some negative numbers, the only operations it contains are multiplication and division. So you can evaluate it in two steps from left to right (remembering the rules for multiplying and dividing with negative numbers):\r\n<blockquote>= −2 × 6 ÷ −4\r\n= −12 ÷ −4\r\n= 3</blockquote>\r\nThus,\r\n<blockquote>−2 × 6 ÷ −4 = 3</blockquote>","blurb":"","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> is a math teacher and tutor with degrees in math and English from Rutgers University. He is the author of a dozen books, including <i>Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, and Calculus II For Dummies</i>. Through online tutoring, he teaches multiplication and beyond to preschoolers in a way that sets them up for school success while keeping the natural magic of math alive. Contact Mark at markzegarelli.com. ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"primaryCategoryTaxonomy":{"categoryId":33722,"title":"Basic Math","slug":"basic-math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"}},"secondaryCategoryTaxonomy":{"categoryId":33726,"title":"Pre-Algebra","slug":"pre-algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33726"}},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"The Three Types of Big Four Expressions","target":"#tab1"}],"relatedArticles":{"fromBook":[{"articleId":207780,"title":"Basic Math & Pre-Algebra For Dummies Cheat Sheet","slug":"basic-math-pre-algebra-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/207780"}},{"articleId":194384,"title":"How to Write Numbers in Scientific Notation","slug":"how-to-write-numbers-in-scientific-notation","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/194384"}},{"articleId":158569,"title":"Important Operations that Make Math Problems Easier","slug":"inverse-operations-and-commutative-associative-and-distributive-properties","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/158569"}},{"articleId":158567,"title":"Converting Metric Units to English Units","slug":"converting-metric-units-to-english-units","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/158567"}},{"articleId":158560,"title":"Working with Exponents, Radicals, & Absolute Value","slug":"a-guide-to-working-with-exponents-radicals-and-absolute-value","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/158560"}}],"fromCategory":[{"articleId":291491,"title":"Teaching Your Kids New Math (K-5) For Dummies Cheat Sheet","slug":"teaching-your-kids-new-math-k-5-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/291491"}},{"articleId":253710,"title":"Pre-Algebra: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249996,"title":"Solving Simple Equations in Pre-Algebra Problems","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Rearranging Algebraic Equations to Isolate X","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281978,"slug":"basic-math-pre-algebra-for-dummies-2nd-edition","isbn":"9781119293637","categoryList":["academics-the-arts","math","basic-math"],"amazon":{"default":"//www.amazon.com/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119293634-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/basic-math-and-pre-algebra-for-dummies-2nd-edition-cover-9781119293637-203x255.jpg","width":203,"height":255},"title":"Basic Math & Pre-Algebra For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><p><b><b data-author-id=\"9399\">Mark Zegarelli</b></b> is a math teacher and tutor with degrees in math and English from Rutgers University. He is the author of a dozen books, including <i>Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, and Calculus II For Dummies</i>. Through online tutoring, he teaches multiplication and beyond to preschoolers in a way that sets them up for school success while keeping the natural magic of math alive. Contact Mark at markzegarelli.com.</p>","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> is a math teacher and tutor with degrees in math and English from Rutgers University. He is the author of a dozen books, including <i>Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, and Calculus II For Dummies</i>. Through online tutoring, he teaches multiplication and beyond to preschoolers in a way that sets them up for school success while keeping the natural magic of math alive. Contact Mark at markzegarelli.com. ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293637&quot;]}]\" id=\"du-slot-64ac718f5cb21\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293637&quot;]}]\" id=\"du-slot-64ac718f5ddb5\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2023-10-03T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":150507},{"headers":{"creationTime":"2019-06-25T23:01:36+00:00","modifiedTime":"2024-07-10T18:47:24+00:00","timestamp":"2024-07-10T21:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Basic Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"},"slug":"basic-math","categoryId":33722}],"title":"Pre-Algebra: Comparing Fractions Using Cross-Multiplication","strippedTitle":"pre-algebra: comparing fractions using cross-multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","canonicalUrl":"","寻找网络归类网页引挚改进":{"metaDescription":"Cross-multiplication is a handy tool for finding the common denominator for two fractions, which is important for many operations involving fractions. In the fo","noIndex":0,"noFollow":0},"content":"Cross-multiplication is a handy tool for finding the common denominator for two fractions, which is important for many operations involving fractions. In the following practice questions, you are asked to cross-multiply to compare fractions to find out which is greater or less.\r\n<h2 style=\"padding-left: 30px;\">Practice questions</h2>\r\n<p style=\"padding-left: 30px;\"><strong>1.</strong> Find the lesser fraction:</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253711\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0101.gif\" alt=\"PREALGEBRA_0101\" width=\"65\" height=\"45\" /></p>\r\n<p style=\"padding-left: 30px;\"><strong>2.</strong> Among these three fractions, which is greatest:</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253712\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0102.gif\" alt=\"PREALGEBRA_0102\" width=\"116\" height=\"45\" /></p>\r\n\r\n<h2 id=\"tab2\" >Answers and explanations</h2>\r\n<p style=\"padding-left: 30px;\"><strong>1.</strong> Of the two fractions,</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253713\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0103.gif\" alt=\"PREALGEBRA_0103\" width=\"79\" height=\"45\" /></p>\r\n<p style=\"padding-left: 30px;\">Cross-multiply to compare the two fractions:</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253714\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0104-FINAL.gif\" alt=\"PREALGEBRA_0104\" width=\"59\" height=\"71\" /></p>\r\n<p style=\"padding-left: 30px;\">Because 35 is less than 36,</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253715\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0105.gif\" alt=\"PREALGEBRA_0105\" width=\"131\" height=\"45\" /></p>\r\n<p style=\"padding-left: 30px;\"><strong>2.</strong> Of the three fractions,</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253716\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0106.gif\" alt=\"PREALGEBRA_0106\" width=\"217\" height=\"45\" /></p>\r\n<p style=\"padding-left: 30px;\">Use cross-multiplication to compare the first two fractions.</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253717\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0107-FINAL.gif\" alt=\"PREALGEBRA_0107\" width=\"64\" height=\"71\" /></p>\r\n<p style=\"padding-left: 30px;\">Because 21 is greater than 20, this means that 1/10 is greater than 2/21, so you can rule out 2/21. Next, compare 1/10 and 3/29 by cross-multiplying.</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253718\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0108-FINAL.gif\" alt=\"PREALGEBRA_0108\" width=\"67\" height=\"71\" /></p>\r\n<p style=\"padding-left: 30px;\">Because 30 is greater than 29, 3/29 is greater than 1/10. Therefore, 3/29 is the greatest of the three fractions.</p>","description":"Cross-multiplication is a handy tool for finding the common denominator for two fractions, which is important for many operations involving fractions. In the following practice questions, you are asked to cross-multiply to compare fractions to find out which is greater or less.\r\n<h2 style=\"padding-left: 30px;\">Practice questions</h2>\r\n<p style=\"padding-left: 30px;\"><strong>1.</strong> Find the lesser fraction:</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253711\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0101.gif\" alt=\"PREALGEBRA_0101\" width=\"65\" height=\"45\" /></p>\r\n<p style=\"padding-left: 30px;\"><strong>2.</strong> Among these three fractions, which is greatest:</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253712\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0102.gif\" alt=\"PREALGEBRA_0102\" width=\"116\" height=\"45\" /></p>\r\n\r\n<h2 id=\"tab2\" >Answers and explanations</h2>\r\n<p style=\"padding-left: 30px;\"><strong>1.</strong> Of the two fractions,</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253713\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0103.gif\" alt=\"PREALGEBRA_0103\" width=\"79\" height=\"45\" /></p>\r\n<p style=\"padding-left: 30px;\">Cross-multiply to compare the two fractions:</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253714\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0104-FINAL.gif\" alt=\"PREALGEBRA_0104\" width=\"59\" height=\"71\" /></p>\r\n<p style=\"padding-left: 30px;\">Because 35 is less than 36,</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253715\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0105.gif\" alt=\"PREALGEBRA_0105\" width=\"131\" height=\"45\" /></p>\r\n<p style=\"padding-left: 30px;\"><strong>2.</strong> Of the three fractions,</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253716\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0106.gif\" alt=\"PREALGEBRA_0106\" width=\"217\" height=\"45\" /></p>\r\n<p style=\"padding-left: 30px;\">Use cross-multiplication to compare the first two fractions.</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253717\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0107-FINAL.gif\" alt=\"PREALGEBRA_0107\" width=\"64\" height=\"71\" /></p>\r\n<p style=\"padding-left: 30px;\">Because 21 is greater than 20, this means that 1/10 is greater than 2/21, so you can rule out 2/21. Next, compare 1/10 and 3/29 by cross-multiplying.</p>\r\n<p style=\"padding-left: 30px;\"><img class=\"alignnone size-full wp-image-253718\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_0108-FINAL.gif\" alt=\"PREALGEBRA_0108\" width=\"67\" height=\"71\" /></p>\r\n<p style=\"padding-left: 30px;\">Because 30 is greater than 29, 3/29 is greater than 1/10. Therefore, 3/29 is the greatest of the three fractions.</p>","blurb":"","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> is a math teacher and tutor with degrees in math and English from Rutgers University. He is the author of a dozen books, including <i>Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, and Calculus II For Dummies</i>. Through online tutoring, he teaches multiplication and beyond to preschoolers in a way that sets them up for school success while keeping the natural magic of math alive. Contact Mark at markzegarelli.com. ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"primaryCategoryTaxonomy":{"categoryId":33722,"title":"Basic Math","slug":"basic-math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"}},"secondaryCategoryTaxonomy":{"categoryId":33726,"title":"Pre-Algebra","slug":"pre-algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33726"}},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Practice questions","target":"#tab1"},{"label":"Answers and explanations","target":"#tab2"}],"relatedArticles":{"fromBook":[{"articleId":249996,"title":"Solving Simple Equations in Pre-Algebra Problems","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Rearranging Algebraic Equations to Isolate X","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}},{"articleId":249971,"title":"Pre-Algebra: Cross-Multiply to Solve Equations","slug":"pre-algebra-practice-questions-cross-multiply-solve-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249971"}},{"articleId":249964,"title":"Pre-Algebra Practice Questions: Finding the Volume of Prisms and Cylinders","slug":"pre-algebra-practice-questions-finding-volume-prisms-cylinders","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249964"}}],"fromCategory":[{"articleId":291491,"title":"Teaching Your Kids New Math (K-5) For Dummies Cheat Sheet","slug":"teaching-your-kids-new-math-k-5-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/291491"}},{"articleId":249996,"title":"Solving Simple Equations in Pre-Algebra Problems","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Rearranging Algebraic Equations to Isolate X","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}},{"articleId":249971,"title":"Pre-Algebra: Cross-Multiply to Solve Equations","slug":"pre-algebra-practice-questions-cross-multiply-solve-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249971"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281980,"slug":"basic-math-and-pre-algebra-workbook-for-dummies-with-online-practice-3rd-edition","isbn":"9781119357513","categoryList":["academics-the-arts","math","basic-math"],"amazon":{"default":"//www.amazon.com/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119357519-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/basic-math-and-pre-algebra-workbook-for-dummies-3rd-edition-cover-9781119357513-204x255.jpg","width":204,"height":255},"title":"Basic Math & Pre-Algebra Workbook For Dummies with Online Practice","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"9399\">Mark Zegarelli</b> is a math and test prep teacher who has written a wide variety of basic math and pre-algebra books in the <i>For Dummies</i> series. </p>","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> is a math teacher and tutor with degrees in math and English from Rutgers University. He is the author of a dozen books, including <i>Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, and Calculus II For Dummies</i>. Through online tutoring, he teaches multiplication and beyond to preschoolers in a way that sets them up for school success while keeping the natural magic of math alive. Contact Mark at markzegarelli.com. ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119357513&quot;]}]\" id=\"du-slot-64ac718f54ec3\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119357513&quot;]}]\" id=\"du-slot-64ac718f55642\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2024-07-10T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":253710},{"headers":{"creationTime":"2017-03-26T13:42:37+00:00","modifiedTime":"2024-07-10T18:39:48+00:00","timestamp":"2024-07-10T21:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Basic Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"},"slug":"basic-math","categoryId":33722}],"title":"Important Operations that Make Math Problems Easier","strippedTitle":"important operations that make math problems easier","slug":"inverse-operations-and-commutative-associative-and-distributive-properties","canonicalUrl":"","寻找网络归类网页引挚改进":{"metaDescription":"The Big Four math operations — addition, subtraction, multiplication, and division — let you combine numbers and perform calculations. Certain operations posses","noIndex":0,"noFollow":0},"content":"The Big Four math operations — addition, subtraction, multiplication, and division — let you combine numbers and perform calculations. Certain operations possess properties that enable you to manipulate the numbers in the problem, which comes in handy, especially when you get into higher math like algebra. The important properties you need to know are the commutative property, the associative property, and the distributive property. Understanding what an inverse operation is is also helpful.\r\n<h2 id=\"tab1\" >Inverse operations</h2>\r\n<i>Inverse operations</i> are pairs of operations that you can work \"backward\" to cancel each other out. Two pairs of the Big Four operations — addition, subtraction, multiplication, and division —are inverses of each other:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Addition and subtraction are inverse operations of each other.</b> When you start with any value, then add a number to it and subtract the same number from the result, the value you started with remains unchanged. For example:</p>\r\n\r\n<blockquote>2 + 3 = 5 so 5 – 3 = 2</blockquote>\r\n<blockquote>7 – 1 = 6 so 6 + 1 = 7</blockquote>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Multiplication and division are inverse operations of each other. </b>When you start with any value, then multiply it by a number and divide the result by the same number (except zero), the value you started with remains unchanged. For example:</p>\r\n\r\n<blockquote>3 × 4 = 12 so 12 ÷ 4 = 3</blockquote>\r\n<blockquote>10 ÷ 2 = 5 so 5 × 2 = 10</blockquote>\r\n</li>\r\n</ul>\r\n<h2 id=\"tab2\" >The commutative property</h2>\r\nAn operation is <i>commutative</i> when you apply it to a pair of numbers either forwards or backwards and expect the same result. The two Big Four that are commutative are addition and subtraction.\r\n\r\nAddition is commutative because, for example, 3 + 5 is the same as 5 + 3. In other words\r\n<blockquote>3 + 5 = 5 + 3</blockquote>\r\nMultiplication is <i>commutative</i> because 2 × 7 is the same as 7 × 2. In other words\r\n<blockquote>2 × 7 = 7 × 2</blockquote>\r\n<h2 id=\"tab3\" >The associative property</h2>\r\nAn operation is <i>associative</i> when you can apply it, using parentheses, in different groupings of numbers and still expect the same result. The two Big Four operations that are associative are addition and multiplication.\r\n\r\nAddition is associative because, for example, the problem (2 + 4) + 7 produces the same result as does the problem 2 + (4 + 7). In other words,\r\n<blockquote>(2 + 4) + 7 = 2 + (4 + 7)</blockquote>\r\nNo matter which pair of numbers you add together first, the answer is the same: 13.\r\n\r\nMultiplication is associative because, for example, the problem 3 × (4 × 5) produces the same result as the problem (3 × 4) × 5. In other words,\r\n<blockquote>3 × (4 × 5) = (3 × 4) × 5</blockquote>\r\nAgain, no matter which pair of numbers you multiply first, both problems yield the same answer: 60.\r\n<h2 id=\"tab4\" >The distributive property</h2>\r\nThe <i>distributive property</i> connects the operations of multiplication and addition. When multiplication is described as \"distributive over addition,\" you can split a multiplication problem into two smaller problems and then add the results.\r\n\r\nFor example, suppose you want to multiply 27 × 6. You know that 27 equals 20 + 7, so you can do this multiplication in two steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">First multiply 20 × 6; then multiply 7 × 6.</p>\r\n<p class=\"child-para\">20 × 6 = 1207 × 6 = 42</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Then add the results.</p>\r\n<p class=\"child-para\">120 + 42 = 162</p>\r\n</li>\r\n</ol>\r\nTherefore, 27 × 6 = 162.","description":"The Big Four math operations — addition, subtraction, multiplication, and division — let you combine numbers and perform calculations. Certain operations possess properties that enable you to manipulate the numbers in the problem, which comes in handy, especially when you get into higher math like algebra. The important properties you need to know are the commutative property, the associative property, and the distributive property. Understanding what an inverse operation is is also helpful.\r\n<h2 id=\"tab1\" >Inverse operations</h2>\r\n<i>Inverse operations</i> are pairs of operations that you can work \"backward\" to cancel each other out. Two pairs of the Big Four operations — addition, subtraction, multiplication, and division —are inverses of each other:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Addition and subtraction are inverse operations of each other.</b> When you start with any value, then add a number to it and subtract the same number from the result, the value you started with remains unchanged. For example:</p>\r\n\r\n<blockquote>2 + 3 = 5 so 5 – 3 = 2</blockquote>\r\n<blockquote>7 – 1 = 6 so 6 + 1 = 7</blockquote>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Multiplication and division are inverse operations of each other. </b>When you start with any value, then multiply it by a number and divide the result by the same number (except zero), the value you started with remains unchanged. For example:</p>\r\n\r\n<blockquote>3 × 4 = 12 so 12 ÷ 4 = 3</blockquote>\r\n<blockquote>10 ÷ 2 = 5 so 5 × 2 = 10</blockquote>\r\n</li>\r\n</ul>\r\n<h2 id=\"tab2\" >The commutative property</h2>\r\nAn operation is <i>commutative</i> when you apply it to a pair of numbers either forwards or backwards and expect the same result. The two Big Four that are commutative are addition and subtraction.\r\n\r\nAddition is commutative because, for example, 3 + 5 is the same as 5 + 3. In other words\r\n<blockquote>3 + 5 = 5 + 3</blockquote>\r\nMultiplication is <i>commutative</i> because 2 × 7 is the same as 7 × 2. In other words\r\n<blockquote>2 × 7 = 7 × 2</blockquote>\r\n<h2 id=\"tab3\" >The associative property</h2>\r\nAn operation is <i>associative</i> when you can apply it, using parentheses, in different groupings of numbers and still expect the same result. The two Big Four operations that are associative are addition and multiplication.\r\n\r\nAddition is associative because, for example, the problem (2 + 4) + 7 produces the same result as does the problem 2 + (4 + 7). In other words,\r\n<blockquote>(2 + 4) + 7 = 2 + (4 + 7)</blockquote>\r\nNo matter which pair of numbers you add together first, the answer is the same: 13.\r\n\r\nMultiplication is associative because, for example, the problem 3 × (4 × 5) produces the same result as the problem (3 × 4) × 5. In other words,\r\n<blockquote>3 × (4 × 5) = (3 × 4) × 5</blockquote>\r\nAgain, no matter which pair of numbers you multiply first, both problems yield the same answer: 60.\r\n<h2 id=\"tab4\" >The distributive property</h2>\r\nThe <i>distributive property</i> connects the operations of multiplication and addition. When multiplication is described as \"distributive over addition,\" you can split a multiplication problem into two smaller problems and then add the results.\r\n\r\nFor example, suppose you want to multiply 27 × 6. You know that 27 equals 20 + 7, so you can do this multiplication in two steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">First multiply 20 × 6; then multiply 7 × 6.</p>\r\n<p class=\"child-para\">20 × 6 = 1207 × 6 = 42</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Then add the results.</p>\r\n<p class=\"child-para\">120 + 42 = 162</p>\r\n</li>\r\n</ol>\r\nTherefore, 27 × 6 = 162.","blurb":"","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> is a math teacher and tutor with degrees in math and English from Rutgers University. He is the author of a dozen books, including <i>Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, and Calculus II For Dummies</i>. Through online tutoring, he teaches multiplication and beyond to preschoolers in a way that sets them up for school success while keeping the natural magic of math alive. Contact Mark at markzegarelli.com. ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"primaryCategoryTaxonomy":{"categoryId":33722,"title":"Basic Math","slug":"basic-math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"}},"secondaryCategoryTaxonomy":{"categoryId":33726,"title":"Pre-Algebra","slug":"pre-algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33726"}},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Inverse operations","target":"#tab1"},{"label":"The commutative property","target":"#tab2"},{"label":"The associative property","target":"#tab3"},{"label":"The distributive property","target":"#tab4"}],"relatedArticles":{"fromBook":[{"articleId":207780,"title":"Basic Math & Pre-Algebra For Dummies Cheat Sheet","slug":"basic-math-pre-algebra-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/207780"}},{"articleId":194384,"title":"How to Write Numbers in Scientific Notation","slug":"how-to-write-numbers-in-scientific-notation","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/194384"}},{"articleId":158567,"title":"Converting Metric Units to English Units","slug":"converting-metric-units-to-english-units","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/158567"}},{"articleId":158560,"title":"Working with Exponents, Radicals, & Absolute Value","slug":"a-guide-to-working-with-exponents-radicals-and-absolute-value","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/158560"}},{"articleId":158557,"title":"Conversion Guide for Fractions, Decimals, and Percents","slug":"a-quick-conversion-guide-for-fractions-decimals-and-percents","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/158557"}}],"fromCategory":[{"articleId":291491,"title":"Teaching Your Kids New Math (K-5) For Dummies Cheat Sheet","slug":"teaching-your-kids-new-math-k-5-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/291491"}},{"articleId":253710,"title":"Pre-Algebra: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249996,"title":"Solving Simple Equations in Pre-Algebra Problems","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Rearranging Algebraic Equations to Isolate X","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281978,"slug":"basic-math-pre-algebra-for-dummies-2nd-edition","isbn":"9781119293637","categoryList":["academics-the-arts","math","basic-math"],"amazon":{"default":"//www.amazon.com/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119293634-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/basic-math-and-pre-algebra-for-dummies-2nd-edition-cover-9781119293637-203x255.jpg","width":203,"height":255},"title":"Basic Math & Pre-Algebra For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><p><b><b data-author-id=\"9399\">Mark Zegarelli</b></b> is a math teacher and tutor with degrees in math and English from Rutgers University. He is the author of a dozen books, including <i>Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, and Calculus II For Dummies</i>. Through online tutoring, he teaches multiplication and beyond to preschoolers in a way that sets them up for school success while keeping the natural magic of math alive. Contact Mark at markzegarelli.com.</p>","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> is a math teacher and tutor with degrees in math and English from Rutgers University. He is the author of a dozen books, including <i>Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, and Calculus II For Dummies</i>. Through online tutoring, he teaches multiplication and beyond to preschoolers in a way that sets them up for school success while keeping the natural magic of math alive. Contact Mark at markzegarelli.com. ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293637&quot;]}]\" id=\"du-slot-64ac718f08a18\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293637&quot;]}]\" id=\"du-slot-64ac718f091a5\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2024-07-10T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":158569},{"headers":{"creationTime":"2019-02-13T04:21:39+00:00","modifiedTime":"2024-06-28T20:43:33+00:00","timestamp":"2024-06-28T21:01:10+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Basic Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"},"slug":"basic-math","categoryId":33722}],"title":"Solving Simple Equations in Pre-Algebra Problems","strippedTitle":"solving simple equations in pre-algebra problems","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","canonicalUrl":"","寻找网络归类网页引挚改进":{"metaDescription":"When dealing with simple algebraic expressions, you don't always need algebra to solve them. The following practice questions ask you to use three different met","noIndex":0,"noFollow":0},"content":"When dealing with simple algebraic expressions, you don't always need algebra to solve them. The following practice questions ask you to use three different methods: inspecting, rewriting the problem, and guessing and checking.\r\n<h2 id=\"tab1\" >Practice questions</h2>\r\nIn the following questions, solve for <em>x</em> in each case just by looking at the equation.\r\n<p style=\"padding-left: 30px;\"><strong> 1. 18 – <em>x</em> = 12</strong></p>\r\n<p style=\"padding-left: 30px;\"><strong> 2. 4<em>x</em> = 44</strong></p>\r\n<p style=\"padding-left: 30px;\">In the following questions, use the correct inverse operation to rewrite and solve each problem.</p>\r\n<p style=\"padding-left: 30px;\"> <strong>3. 100 – <em>x</em> = 58</strong></p>\r\n<p style=\"padding-left: 30px;\"> <strong>4. 238/<em>x</em> = 17</strong></p>\r\n<p style=\"padding-left: 30px;\">In the following questions, find the value of <em>x</em> in each equation by guessing and checking.</p>\r\n<p style=\"padding-left: 30px;\"><strong> 5. 12<em>x</em> – 17 = 151</strong></p>\r\n<p style=\"padding-left: 30px;\"> <strong>6. 19<em>x</em> – 8 = 600</strong></p>\r\n\r\n<h2 id=\"tab2\" >Answers and explanations</h2>\r\n<ol>\r\n \t<li><em>x</em> = 6You can solve this problem through simple inspection. Because 18 – 6 = 12, <em>x</em> = 6.</li>\r\n \t<li><em>x</em> = 11Again, through simple inspection, because 4(11) = 44, you know that <em>x</em> = 11.</li>\r\n \t<li><em>x </em>= 42Turn the problem around by changing the subtraction to addition: 100 – <em>x = </em>58 means the same thing as 100 – 58 =<em> x</em>, so<em> x = </em>42.</li>\r\n \t<li><em>x </em>= 14Turn the problem around by switching around the division:\r\n\r\n<img class=\"alignnone size-full wp-image-249997\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3001.gif\" alt=\"PREALGEBRA_3001\" width=\"183\" height=\"45\" />\r\n\r\nso <em>x </em>= 14.</li>\r\n \t<li><em>x</em> = 14Guess what you think the answer may be. For example, perhaps it's <em>x</em> = 10:\r\n\r\n12(10) – 17 = 120 – 17 = 103\r\n\r\n103 is less than 151, so this guess is too low. Try <em>x</em> = 20:\r\n\r\n12(20) – 17 = 240 – 17 = 223\r\n\r\n223 is greater than 151, so this guess is too high. Therefore, <em>x</em> is between 10 and 20. Try <em>x</em> = 15:\r\n\r\n12(15) – 17 = 180 – 17 = 163\r\n\r\n163 is a little greater than 151, so this guess is a little too high. Try <em>x </em>= 14:\r\n\r\n12(14) – 17 = 168 – 17 = 151\r\n\r\n151 is correct, so <em>x</em> = 14.</li>\r\n \t<li><em>x</em> = 32Again, start by guessing. First, try <em>x</em> = 10:\r\n\r\n19(10) – 8 = 190 – 8 = 182\r\n\r\n182 is much less than 600, so this guess is much too low. Try <em>x</em> = 30:\r\n\r\n19(30) – 8 = 570 – 8 = 562\r\n\r\n562 is still less than 600, so this guess is still too low. Try <em>x</em> = 35:\r\n\r\n19(35) – 8 = 665 – 8 = 657\r\n\r\n657 is greater than 600, so this guess is too high. Therefore, <em>x</em> is between 30 and 35. Try <em>x</em> = 32:\r\n\r\n19(32) – 8 = 608 – 8 = 600\r\n\r\n600 is correct, so <em>x</em> = 32.</li>\r\n</ol>\r\n ","description":"When dealing with simple algebraic expressions, you don't always need algebra to solve them. The following practice questions ask you to use three different methods: inspecting, rewriting the problem, and guessing and checking.\r\n<h2 id=\"tab1\" >Practice questions</h2>\r\nIn the following questions, solve for <em>x</em> in each case just by looking at the equation.\r\n<p style=\"padding-left: 30px;\"><strong> 1. 18 – <em>x</em> = 12</strong></p>\r\n<p style=\"padding-left: 30px;\"><strong> 2. 4<em>x</em> = 44</strong></p>\r\n<p style=\"padding-left: 30px;\">In the following questions, use the correct inverse operation to rewrite and solve each problem.</p>\r\n<p style=\"padding-left: 30px;\"> <strong>3. 100 – <em>x</em> = 58</strong></p>\r\n<p style=\"padding-left: 30px;\"> <strong>4. 238/<em>x</em> = 17</strong></p>\r\n<p style=\"padding-left: 30px;\">In the following questions, find the value of <em>x</em> in each equation by guessing and checking.</p>\r\n<p style=\"padding-left: 30px;\"><strong> 5. 12<em>x</em> – 17 = 151</strong></p>\r\n<p style=\"padding-left: 30px;\"> <strong>6. 19<em>x</em> – 8 = 600</strong></p>\r\n\r\n<h2 id=\"tab2\" >Answers and explanations</h2>\r\n<ol>\r\n \t<li><em>x</em> = 6You can solve this problem through simple inspection. Because 18 – 6 = 12, <em>x</em> = 6.</li>\r\n \t<li><em>x</em> = 11Again, through simple inspection, because 4(11) = 44, you know that <em>x</em> = 11.</li>\r\n \t<li><em>x </em>= 42Turn the problem around by changing the subtraction to addition: 100 – <em>x = </em>58 means the same thing as 100 – 58 =<em> x</em>, so<em> x = </em>42.</li>\r\n \t<li><em>x </em>= 14Turn the problem around by switching around the division:\r\n\r\n<img class=\"alignnone size-full wp-image-249997\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3001.gif\" alt=\"PREALGEBRA_3001\" width=\"183\" height=\"45\" />\r\n\r\nso <em>x </em>= 14.</li>\r\n \t<li><em>x</em> = 14Guess what you think the answer may be. For example, perhaps it's <em>x</em> = 10:\r\n\r\n12(10) – 17 = 120 – 17 = 103\r\n\r\n103 is less than 151, so this guess is too low. Try <em>x</em> = 20:\r\n\r\n12(20) – 17 = 240 – 17 = 223\r\n\r\n223 is greater than 151, so this guess is too high. Therefore, <em>x</em> is between 10 and 20. Try <em>x</em> = 15:\r\n\r\n12(15) – 17 = 180 – 17 = 163\r\n\r\n163 is a little greater than 151, so this guess is a little too high. Try <em>x </em>= 14:\r\n\r\n12(14) – 17 = 168 – 17 = 151\r\n\r\n151 is correct, so <em>x</em> = 14.</li>\r\n \t<li><em>x</em> = 32Again, start by guessing. First, try <em>x</em> = 10:\r\n\r\n19(10) – 8 = 190 – 8 = 182\r\n\r\n182 is much less than 600, so this guess is much too low. Try <em>x</em> = 30:\r\n\r\n19(30) – 8 = 570 – 8 = 562\r\n\r\n562 is still less than 600, so this guess is still too low. Try <em>x</em> = 35:\r\n\r\n19(35) – 8 = 665 – 8 = 657\r\n\r\n657 is greater than 600, so this guess is too high. Therefore, <em>x</em> is between 30 and 35. Try <em>x</em> = 32:\r\n\r\n19(32) – 8 = 608 – 8 = 600\r\n\r\n600 is correct, so <em>x</em> = 32.</li>\r\n</ol>\r\n ","blurb":"","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> earned degrees in mathematics and English from Rutgers University. He is the founder of SimpleStep Learning, an online educational platform that teaches courses in basic concepts in ten minutes or less, keeping students engaged and learning in every moment. Mark is also author of several other successful <i>For Dummies</i> books.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"primaryCategoryTaxonomy":{"categoryId":33722,"title":"Basic Math","slug":"basic-math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"}},"secondaryCategoryTaxonomy":{"categoryId":33726,"title":"Pre-Algebra","slug":"pre-algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33726"}},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Practice questions","target":"#tab1"},{"label":"Answers and explanations","target":"#tab2"}],"relatedArticles":{"fromBook":[{"articleId":253710,"title":"Pre-Algebra Practice Questions: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Rearranging Algebraic Equations to Isolate X","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}},{"articleId":249971,"title":"Pre-Algebra: Cross-Multiply to Solve Equations","slug":"pre-algebra-practice-questions-cross-multiply-solve-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249971"}},{"articleId":249964,"title":"Pre-Algebra Practice Questions: Finding the Volume of Prisms and Cylinders","slug":"pre-algebra-practice-questions-finding-volume-prisms-cylinders","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249964"}}],"fromCategory":[{"articleId":291491,"title":"Teaching Your Kids New Math (K-5) For Dummies Cheat Sheet","slug":"teaching-your-kids-new-math-k-5-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/291491"}},{"articleId":253710,"title":"Pre-Algebra Practice Questions: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Rearranging Algebraic Equations to Isolate X","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}},{"articleId":249971,"title":"Pre-Algebra: Cross-Multiply to Solve Equations","slug":"pre-algebra-practice-questions-cross-multiply-solve-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249971"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281980,"slug":"basic-math-and-pre-algebra-workbook-for-dummies-with-online-practice-3rd-edition","isbn":"9781119357513","categoryList":["academics-the-arts","math","basic-math"],"amazon":{"default":"//www.amazon.com/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119357519-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/basic-math-and-pre-algebra-workbook-for-dummies-3rd-edition-cover-9781119357513-204x255.jpg","width":204,"height":255},"title":"Basic Math & Pre-Algebra Workbook For Dummies with Online Practice","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"9399\">Mark Zegarelli</b> is a math and test prep teacher who has written a wide variety of basic math and pre-algebra books in the <i>For Dummies</i> series. </p>","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> earned degrees in mathematics and English from Rutgers University. He is the founder of SimpleStep Learning, an online educational platform that teaches courses in basic concepts in ten minutes or less, keeping students engaged and learning in every moment. Mark is also author of several other successful <i>For Dummies</i> books.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119357513&quot;]}]\" id=\"du-slot-649c9f961784f\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119357513&quot;]}]\" id=\"du-slot-649c9f9618136\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2024-06-28T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":249996},{"headers":{"creationTime":"2017-03-26T11:06:26+00:00","modifiedTime":"2024-06-22T13:57:07+00:00","timestamp":"2024-06-22T15:01:04+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Basic Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"},"slug":"basic-math","categoryId":33722}],"title":"Applying Order of Operations in Expressions with Exponents","strippedTitle":"applying order of operations in expressions with exponents","slug":"applying-order-of-operations-in-expressions-with-exponents","canonicalUrl":"","寻找网络归类网页引挚改进":{"metaDescription":"Evaluate exponents from left to right before you begin evaluating Big Four operations (adding, subtracting, multiplying, and dividing). The trick here is to tur","noIndex":0,"noFollow":0},"content":"<p>Evaluate exponents from left to right <i>before</i><i> </i>you begin evaluating Big Four operations (adding, subtracting, multiplying, and dividing). The trick here is to turn the expression into a Big Four expression and then use the other basic order of operations rules.</p>\r\n<h2 id=\"tab1\" >The Three Types of Big Four Expressions</h2>\r\n<table>\r\n<tr>\r\n<th>Expression</th>\r\n<th>Example</th>\r\n<th>Rule</th>\r\n</tr>\r\n<tr>\r\n<td>Contains only addition and subtraction</td>\r\n<td>12 + 7 - 6 - 3 + 8</td>\r\n<td>Evaluate left to right.</td>\r\n</tr>\r\n<tr>\r\n<td>Contains only multiplication and division</td>\r\n<td>18 ÷ 3 x 7 ÷ 14</td>\r\n<td>Evaluate left to right.</td>\r\n</tr>\r\n<tr>\r\n<td>Mixed-operator expression: contains a combination of\r\naddition/subtraction and multiplication/division</td>\r\n<td>9 + 6 ÷ 3</td>\r\n<td>1. Evaluate multiplication and division left to right.<br />\r\n2. Evaluate addition and subtraction left to right.</td>\r\n</tr>\r\n</table>\r\n<p class=\"Remember\">For example, suppose you want to evaluate the following:</p>\r\n<blockquote>3 + 5² - 6</blockquote>\r\n<p>First, evaluate the exponent:</p>\r\n<blockquote>3 + 25 - 6</blockquote>\r\n<p>At this point, the expression contains only addition and subtraction, so you can evaluate it from left to right in two steps:</p>\r\n<blockquote>= 28 - 6</blockquote>\r\n<blockquote>= 22</blockquote>\r\n<p>So</p>\r\n<blockquote>3 + 5² - 6 = 22</blockquote>","description":"<p>Evaluate exponents from left to right <i>before</i><i> </i>you begin evaluating Big Four operations (adding, subtracting, multiplying, and dividing). The trick here is to turn the expression into a Big Four expression and then use the other basic order of operations rules.</p>\r\n<h2 id=\"tab1\" >The Three Types of Big Four Expressions</h2>\r\n<table>\r\n<tr>\r\n<th>Expression</th>\r\n<th>Example</th>\r\n<th>Rule</th>\r\n</tr>\r\n<tr>\r\n<td>Contains only addition and subtraction</td>\r\n<td>12 + 7 - 6 - 3 + 8</td>\r\n<td>Evaluate left to right.</td>\r\n</tr>\r\n<tr>\r\n<td>Contains only multiplication and division</td>\r\n<td>18 ÷ 3 x 7 ÷ 14</td>\r\n<td>Evaluate left to right.</td>\r\n</tr>\r\n<tr>\r\n<td>Mixed-operator expression: contains a combination of\r\naddition/subtraction and multiplication/division</td>\r\n<td>9 + 6 ÷ 3</td>\r\n<td>1. Evaluate multiplication and division left to right.<br />\r\n2. Evaluate addition and subtraction left to right.</td>\r\n</tr>\r\n</table>\r\n<p class=\"Remember\">For example, suppose you want to evaluate the following:</p>\r\n<blockquote>3 + 5² - 6</blockquote>\r\n<p>First, evaluate the exponent:</p>\r\n<blockquote>3 + 25 - 6</blockquote>\r\n<p>At this point, the expression contains only addition and subtraction, so you can evaluate it from left to right in two steps:</p>\r\n<blockquote>= 28 - 6</blockquote>\r\n<blockquote>= 22</blockquote>\r\n<p>So</p>\r\n<blockquote>3 + 5² - 6 = 22</blockquote>","blurb":"","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> earned degrees in mathematics and English from Rutgers University. He is the founder of SimpleStep Learning, an online educational platform that teaches courses in basic concepts in ten minutes or less, keeping students engaged and learning in every moment. Mark is also author of several other successful <i>For Dummies</i> books.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"primaryCategoryTaxonomy":{"categoryId":33722,"title":"Basic Math","slug":"basic-math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"}},"secondaryCategoryTaxonomy":{"categoryId":33726,"title":"Pre-Algebra","slug":"pre-algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33726"}},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"The Three Types of Big Four Expressions","target":"#tab1"}],"relatedArticles":{"fromBook":[{"articleId":207780,"title":"Basic Math & Pre-Algebra For Dummies Cheat Sheet","slug":"basic-math-pre-algebra-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/207780"}},{"articleId":194384,"title":"How to Write Numbers in Scientific Notation","slug":"how-to-write-numbers-in-scientific-notation","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/194384"}},{"articleId":158569,"title":"Inverse Operations and Commutative, Associative, and Distributive Properties","slug":"inverse-operations-and-commutative-associative-and-distributive-properties","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/158569"}},{"articleId":158567,"title":"Converting Metric Units to English Units","slug":"converting-metric-units-to-english-units","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/158567"}},{"articleId":158560,"title":"Working with Exponents, Radicals, & Absolute Value","slug":"a-guide-to-working-with-exponents-radicals-and-absolute-value","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/158560"}}],"fromCategory":[{"articleId":291491,"title":"Teaching Your Kids New Math (K-5) For Dummies Cheat Sheet","slug":"teaching-your-kids-new-math-k-5-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/291491"}},{"articleId":253710,"title":"Pre-Algebra Practice Questions: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249996,"title":"Pre-Algebra Practice Questions: Solving Simple Algebraic Equations","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Rearranging Algebraic Equations to Isolate X","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281978,"slug":"basic-math-pre-algebra-for-dummies-2nd-edition","isbn":"9781119293637","categoryList":["academics-the-arts","math","basic-math"],"amazon":{"default":"//www.amazon.com/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119293634-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/basic-math-and-pre-algebra-for-dummies-2nd-edition-cover-9781119293637-203x255.jpg","width":203,"height":255},"title":"Basic Math & Pre-Algebra For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><b><b data-author-id=\"9399\">Mark Zegarelli</b></b> earned degrees in mathematics and English from Rutgers University. He is the founder of SimpleStep Learning, an online educational platform that teaches courses in basic concepts in ten minutes or less, keeping students engaged and learning in every moment. Mark is also author of several other successful <i>For Dummies</i> books.</p>","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> earned degrees in mathematics and English from Rutgers University. He is the founder of SimpleStep Learning, an online educational platform that teaches courses in basic concepts in ten minutes or less, keeping students engaged and learning in every moment. Mark is also author of several other successful <i>For Dummies</i> books.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293637&quot;]}]\" id=\"du-slot-64946230164be\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293637&quot;]}]\" id=\"du-slot-64946230169f1\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2024-06-22T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":150546},{"headers":{"creationTime":"2019-02-13T02:47:47+00:00","modifiedTime":"2024-06-20T20:14:39+00:00","timestamp":"2024-06-20T21:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Basic Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"},"slug":"basic-math","categoryId":33722}],"title":"Rearranging Algebraic Equations to Isolate X","strippedTitle":"rearranging algebraic equations to isolate x","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","canonicalUrl":"","寻找网络归类网页引挚改进":{"metaDescription":"A quick method for solving algebra problems is to re-arrange the equation by placing all x terms on one side of the equal sign and all constants (non- x terms) ","noIndex":0,"noFollow":0},"content":"A quick method for solving algebra problems is to re-arrange the equation by placing all <em>x</em> terms on one side of the equal sign and all constants (non-<em>x</em> terms) on the other side. Essentially, you're doing the addition and subtraction without showing it. You can then isolate <em>x</em>.\r\n<h2 id=\"tab1\" >Practice questions</h2>\r\n<ol>\r\n \t<li>Rearrange the equation 10<em>x</em> + 5 = 3<em>x</em> + 19 to solve for <em>x</em>.</li>\r\n \t<li>Solve –[2(<em>x</em> + 7) + 1] = <em>x</em> – 12 for <em>x</em>.</li>\r\n</ol>\r\n<h2 id=\"tab2\" >Answers and explanations</h2>\r\n<ol>\r\n \t<li><em>x</em> = 2First, rearrange the terms of the equation so that the <em>x</em> terms are on one side and the constants are on the other. In this case, you can do this in two steps:\r\n\r\n<img class=\"alignnone size-full wp-image-249981\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3201.gif\" alt=\"PREALGEBRA_3201\" width=\"149\" height=\"71\" />\r\n\r\nSecond, combine like terms on both sides:\r\n\r\n7<em>x</em> = 14\r\n\r\nThe third and final step is to divide (in this case, by 7) to isolate <em>x</em>:\r\n\r\n<img class=\"alignnone size-full wp-image-249982\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3202.gif\" alt=\"PREALGEBRA_3202\" width=\"96\" height=\"71\" /></li>\r\n \t<li><em>x</em> = –1Before you can begin rearranging terms, remove the parentheses on the left side of the equation. Start with the inner parentheses, multiplying 2 by every term inside that set:\r\n\r\n–[2(<em>x</em> + 7) + 1] = <em>x</em> – 12\r\n–[2<em>x</em> + 14 + 1] = <em>x</em> – 12\r\n\r\nNext, remove the remaining parentheses, switching the sign of every term within that set:\r\n\r\n–2<em>x</em> – 14 – 1 = <em>x</em> – 12\r\n\r\nNow you can solve for <em>x</em> by, first, rearranging the terms of the equation:\r\n\r\n–2<em>x</em> – 14 – 1 + 12 = <em>x</em>\r\n–14 – 1 + 12 = <em>x</em> + 2<em>x</em>\r\n\r\nThen you combine like terms on both sides:\r\n\r\n–3 = 3<em>x</em>\r\n\r\nFinally, you divide (in this example, by 3) to isolate <em>x</em>:\r\n\r\n<img class=\"alignnone size-full wp-image-249983\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3203.gif\" alt=\"PREALGEBRA_3203\" width=\"85\" height=\"65\" /></li>\r\n</ol>","description":"A quick method for solving algebra problems is to re-arrange the equation by placing all <em>x</em> terms on one side of the equal sign and all constants (non-<em>x</em> terms) on the other side. Essentially, you're doing the addition and subtraction without showing it. You can then isolate <em>x</em>.\r\n<h2 id=\"tab1\" >Practice questions</h2>\r\n<ol>\r\n \t<li>Rearrange the equation 10<em>x</em> + 5 = 3<em>x</em> + 19 to solve for <em>x</em>.</li>\r\n \t<li>Solve –[2(<em>x</em> + 7) + 1] = <em>x</em> – 12 for <em>x</em>.</li>\r\n</ol>\r\n<h2 id=\"tab2\" >Answers and explanations</h2>\r\n<ol>\r\n \t<li><em>x</em> = 2First, rearrange the terms of the equation so that the <em>x</em> terms are on one side and the constants are on the other. In this case, you can do this in two steps:\r\n\r\n<img class=\"alignnone size-full wp-image-249981\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3201.gif\" alt=\"PREALGEBRA_3201\" width=\"149\" height=\"71\" />\r\n\r\nSecond, combine like terms on both sides:\r\n\r\n7<em>x</em> = 14\r\n\r\nThe third and final step is to divide (in this case, by 7) to isolate <em>x</em>:\r\n\r\n<img class=\"alignnone size-full wp-image-249982\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3202.gif\" alt=\"PREALGEBRA_3202\" width=\"96\" height=\"71\" /></li>\r\n \t<li><em>x</em> = –1Before you can begin rearranging terms, remove the parentheses on the left side of the equation. Start with the inner parentheses, multiplying 2 by every term inside that set:\r\n\r\n–[2(<em>x</em> + 7) + 1] = <em>x</em> – 12\r\n–[2<em>x</em> + 14 + 1] = <em>x</em> – 12\r\n\r\nNext, remove the remaining parentheses, switching the sign of every term within that set:\r\n\r\n–2<em>x</em> – 14 – 1 = <em>x</em> – 12\r\n\r\nNow you can solve for <em>x</em> by, first, rearranging the terms of the equation:\r\n\r\n–2<em>x</em> – 14 – 1 + 12 = <em>x</em>\r\n–14 – 1 + 12 = <em>x</em> + 2<em>x</em>\r\n\r\nThen you combine like terms on both sides:\r\n\r\n–3 = 3<em>x</em>\r\n\r\nFinally, you divide (in this example, by 3) to isolate <em>x</em>:\r\n\r\n<img class=\"alignnone size-full wp-image-249983\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3203.gif\" alt=\"PREALGEBRA_3203\" width=\"85\" height=\"65\" /></li>\r\n</ol>","blurb":"","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> earned degrees in mathematics and English from Rutgers University. He is the founder of SimpleStep Learning, an online educational platform that teaches courses in basic concepts in ten minutes or less, keeping students engaged and learning in every moment. Mark is also author of several other successful <i>For Dummies</i> books.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"primaryCategoryTaxonomy":{"categoryId":33722,"title":"Basic Math","slug":"basic-math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"}},"secondaryCategoryTaxonomy":{"categoryId":33726,"title":"Pre-Algebra","slug":"pre-algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33726"}},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Practice questions","target":"#tab1"},{"label":"Answers and explanations","target":"#tab2"}],"relatedArticles":{"fromBook":[{"articleId":253710,"title":"Pre-Algebra Practice Questions: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249996,"title":"Pre-Algebra Practice Questions: Solving Simple Algebraic Equations","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249971,"title":"Pre-Algebra: Cross-Multiply to Solve Equations","slug":"pre-algebra-practice-questions-cross-multiply-solve-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249971"}},{"articleId":249964,"title":"Pre-Algebra Practice Questions: Finding the Volume of Prisms and Cylinders","slug":"pre-algebra-practice-questions-finding-volume-prisms-cylinders","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249964"}}],"fromCategory":[{"articleId":291491,"title":"Teaching Your Kids New Math (K-5) For Dummies Cheat Sheet","slug":"teaching-your-kids-new-math-k-5-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/291491"}},{"articleId":253710,"title":"Pre-Algebra Practice Questions: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249996,"title":"Pre-Algebra Practice Questions: Solving Simple Algebraic Equations","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249971,"title":"Pre-Algebra: Cross-Multiply to Solve Equations","slug":"pre-algebra-practice-questions-cross-multiply-solve-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249971"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281980,"slug":"basic-math-and-pre-algebra-workbook-for-dummies-with-online-practice-3rd-edition","isbn":"9781119357513","categoryList":["academics-the-arts","math","basic-math"],"amazon":{"default":"//www.amazon.com/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119357519-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/basic-math-and-pre-algebra-workbook-for-dummies-3rd-edition-cover-9781119357513-204x255.jpg","width":204,"height":255},"title":"Basic Math & Pre-Algebra Workbook For Dummies with Online Practice","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"9399\">Mark Zegarelli</b> is a math and test prep teacher who has written a wide variety of basic math and pre-algebra books in the <i>For Dummies</i> series. </p>","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> earned degrees in mathematics and English from Rutgers University. He is the founder of SimpleStep Learning, an online educational platform that teaches courses in basic concepts in ten minutes or less, keeping students engaged and learning in every moment. Mark is also author of several other successful <i>For Dummies</i> books.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119357513&quot;]}]\" id=\"du-slot-6492138f2c0ce\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119357513&quot;]}]\" id=\"du-slot-6492138f2cbc0\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2024-06-20T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":249980},{"headers":{"creationTime":"2019-02-13T02:22:14+00:00","modifiedTime":"2024-06-06T18:16:37+00:00","timestamp":"2024-06-06T21:01:02+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Basic Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"},"slug":"basic-math","categoryId":33722}],"title":"Pre-Algebra: Cross-Multiply to Solve Equations","strippedTitle":"pre-algebra: cross-multiply to solve equations","slug":"pre-algebra-practice-questions-cross-multiply-solve-equations","canonicalUrl":"","寻找网络归类网页引挚改进":{"metaDescription":"When an algebraic equation contains fractions, you can use cross-multiplication to solve the equation. The following practice questions contain two equal fracti","noIndex":0,"noFollow":0},"content":"When an algebraic equation contains fractions, you can use cross-multiplication to solve the equation. The following practice questions contain two equal fractions, where you need to cross-multiply to solve them.\r\n<h2 id=\"tab1\" >Practice questions</h2>\r\n<ol>\r\n \t<li>Rearrange the equation<img class=\"alignnone size-full wp-image-249972\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3301.gif\" alt=\"PREALGEBRA_3301\" width=\"80\" height=\"45\" />\r\n\r\nto solve for <em>x</em>.</li>\r\n \t<li>Solve the equation<img class=\"alignnone size-full wp-image-249973\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3302.gif\" alt=\"PREALGEBRA_3302\" width=\"108\" height=\"45\" /></li>\r\n</ol>\r\n<h2 id=\"tab2\" >Answers and explanations</h2>\r\n<ol>\r\n \t<li><em>x</em> = –4Remove the fraction from the equation by cross-multiplying:\r\n\r\n<img class=\"alignnone size-full wp-image-249974\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3303.gif\" alt=\"PREALGEBRA_3303\" width=\"101\" height=\"79\" />\r\n\r\nMultiply to remove the parentheses from the left side of the equation:\r\n\r\n8<em>x</em> + 40 = –2<em>x</em>\r\n\r\nAt this point, you can solve for <em>x</em>:\r\n\r\n<img class=\"alignnone size-full wp-image-249975\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3304.gif\" alt=\"PREALGEBRA_3304\" width=\"108\" height=\"124\" /></li>\r\n \t<li><em>x</em> = 5Remove the fractions from the equation by cross-multiplying:\r\n\r\n<img class=\"alignnone size-full wp-image-249976\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3305.gif\" alt=\"PREALGEBRA_3305\" width=\"189\" height=\"79\" />\r\n\r\nRemove the parentheses from the left side of the equation by multiplying through by <em>x</em>; remove parentheses from the right side of the equation by FOILing:\r\n\r\n4<em>x</em><sup>2</sup> – 7<em>x</em> = 4<em>x</em><sup>2</sup> – 10<em>x</em> + 6<em>x</em> – 15\r\n\r\nRearrange the equation:\r\n\r\n4<em>x</em><sup>2</sup> – 7<em>x</em> – 4<em>x</em><sup>2</sup> + 10<em>x</em> – 6<em>x</em> = –15\r\n\r\nNotice that the two <em>x</em><sup>2</sup> terms cancel each other out:\r\n\r\n<img class=\"alignnone size-full wp-image-249977\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3306.gif\" alt=\"PREALGEBRA_3306\" width=\"147\" height=\"124\" /></li>\r\n</ol>","description":"When an algebraic equation contains fractions, you can use cross-multiplication to solve the equation. The following practice questions contain two equal fractions, where you need to cross-multiply to solve them.\r\n<h2 id=\"tab1\" >Practice questions</h2>\r\n<ol>\r\n \t<li>Rearrange the equation<img class=\"alignnone size-full wp-image-249972\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3301.gif\" alt=\"PREALGEBRA_3301\" width=\"80\" height=\"45\" />\r\n\r\nto solve for <em>x</em>.</li>\r\n \t<li>Solve the equation<img class=\"alignnone size-full wp-image-249973\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3302.gif\" alt=\"PREALGEBRA_3302\" width=\"108\" height=\"45\" /></li>\r\n</ol>\r\n<h2 id=\"tab2\" >Answers and explanations</h2>\r\n<ol>\r\n \t<li><em>x</em> = –4Remove the fraction from the equation by cross-multiplying:\r\n\r\n<img class=\"alignnone size-full wp-image-249974\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3303.gif\" alt=\"PREALGEBRA_3303\" width=\"101\" height=\"79\" />\r\n\r\nMultiply to remove the parentheses from the left side of the equation:\r\n\r\n8<em>x</em> + 40 = –2<em>x</em>\r\n\r\nAt this point, you can solve for <em>x</em>:\r\n\r\n<img class=\"alignnone size-full wp-image-249975\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3304.gif\" alt=\"PREALGEBRA_3304\" width=\"108\" height=\"124\" /></li>\r\n \t<li><em>x</em> = 5Remove the fractions from the equation by cross-multiplying:\r\n\r\n<img class=\"alignnone size-full wp-image-249976\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3305.gif\" alt=\"PREALGEBRA_3305\" width=\"189\" height=\"79\" />\r\n\r\nRemove the parentheses from the left side of the equation by multiplying through by <em>x</em>; remove parentheses from the right side of the equation by FOILing:\r\n\r\n4<em>x</em><sup>2</sup> – 7<em>x</em> = 4<em>x</em><sup>2</sup> – 10<em>x</em> + 6<em>x</em> – 15\r\n\r\nRearrange the equation:\r\n\r\n4<em>x</em><sup>2</sup> – 7<em>x</em> – 4<em>x</em><sup>2</sup> + 10<em>x</em> – 6<em>x</em> = –15\r\n\r\nNotice that the two <em>x</em><sup>2</sup> terms cancel each other out:\r\n\r\n<img class=\"alignnone size-full wp-image-249977\" src=\"//coursofppt.com/wp-content/uploads/PREALGEBRA_3306.gif\" alt=\"PREALGEBRA_3306\" width=\"147\" height=\"124\" /></li>\r\n</ol>","blurb":"","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> earned degrees in mathematics and English from Rutgers University. He is the founder of SimpleStep Learning, an online educational platform that teaches courses in basic concepts in ten minutes or less, keeping students engaged and learning in every moment. Mark is also author of several other successful <i>For Dummies</i> books.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"primaryCategoryTaxonomy":{"categoryId":33722,"title":"Basic Math","slug":"basic-math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"}},"secondaryCategoryTaxonomy":{"categoryId":33726,"title":"Pre-Algebra","slug":"pre-algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33726"}},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Practice questions","target":"#tab1"},{"label":"Answers and explanations","target":"#tab2"}],"relatedArticles":{"fromBook":[{"articleId":253710,"title":"Pre-Algebra Practice Questions: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249996,"title":"Pre-Algebra Practice Questions: Solving Simple Algebraic Equations","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Pre-Algebra Practice Questions: Rearranging Equations to Isolate x","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}},{"articleId":249964,"title":"Pre-Algebra Practice Questions: Finding the Volume of Prisms and Cylinders","slug":"pre-algebra-practice-questions-finding-volume-prisms-cylinders","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249964"}}],"fromCategory":[{"articleId":291491,"title":"Teaching Your Kids New Math (K-5) For Dummies Cheat Sheet","slug":"teaching-your-kids-new-math-k-5-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/291491"}},{"articleId":253710,"title":"Pre-Algebra Practice Questions: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249996,"title":"Pre-Algebra Practice Questions: Solving Simple Algebraic Equations","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Pre-Algebra Practice Questions: Rearranging Equations to Isolate x","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281980,"slug":"basic-math-and-pre-algebra-workbook-for-dummies-with-online-practice-3rd-edition","isbn":"9781119357513","categoryList":["academics-the-arts","math","basic-math"],"amazon":{"default":"//www.amazon.com/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119357519-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119357519/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/basic-math-and-pre-algebra-workbook-for-dummies-3rd-edition-cover-9781119357513-204x255.jpg","width":204,"height":255},"title":"Basic Math & Pre-Algebra Workbook For Dummies with Online Practice","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"9399\">Mark Zegarelli</b> is a math and test prep teacher who has written a wide variety of basic math and pre-algebra books in the <i>For Dummies</i> series. </p>","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> earned degrees in mathematics and English from Rutgers University. He is the founder of SimpleStep Learning, an online educational platform that teaches courses in basic concepts in ten minutes or less, keeping students engaged and learning in every moment. Mark is also author of several other successful <i>For Dummies</i> books.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119357513&quot;]}]\" id=\"du-slot-647f9e8eab4c8\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119357513&quot;]}]\" id=\"du-slot-647f9e8eac4ca\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2024-06-06T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":249971},{"headers":{"creationTime":"2017-03-26T13:42:35+00:00","modifiedTime":"2024-06-06T16:48:33+00:00","timestamp":"2024-06-06T18:01:04+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Basic Math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"},"slug":"basic-math","categoryId":33722}],"title":"Working with Exponents, Radicals, & Absolute Value","strippedTitle":"working with exponents, radicals, & absolute value","slug":"a-guide-to-working-with-exponents-radicals-and-absolute-value","canonicalUrl":"","寻找网络归类网页引挚改进":{"metaDescription":"Exponents, radicals, and absolute value are mathematical operations that go beyond addition, subtraction, multiplication, and division. They are useful in more ","noIndex":0,"noFollow":0},"content":"Exponents, radicals, and absolute value are mathematical operations that go beyond addition, subtraction, multiplication, and division. They are useful in more advanced math, such as algebra, but they also have real-world applications, especially in geometry and measurement.\r\n\r\n<i>Exponents </i>(powers) are repeated multiplication: When you raise a number to the power of an exponent, you multiply that number by itself the number of times indicated by the exponent. For example:\r\n<blockquote>7<sup>2</sup> = 7 × 7 = 49</blockquote>\r\n<blockquote>2<sup>5</sup> = 2 × 2 × 2 × 2 × 2 = 32</blockquote>\r\n<i>Square roots </i>(radicals) are the inverse of exponent 2 — that is, the number that, when multiplied by itself, gives you the indicated value.\r\n\r\n<i>Absolute value </i>is the positive value of a number — that is, the value of a negative number when you drop the minus sign. For example:\r\n\r\nAbsolute value is used to describe numbers that are always positive, such as the distance between two points or the area inside a polygon.","description":"Exponents, radicals, and absolute value are mathematical operations that go beyond addition, subtraction, multiplication, and division. They are useful in more advanced math, such as algebra, but they also have real-world applications, especially in geometry and measurement.\r\n\r\n<i>Exponents </i>(powers) are repeated multiplication: When you raise a number to the power of an exponent, you multiply that number by itself the number of times indicated by the exponent. For example:\r\n<blockquote>7<sup>2</sup> = 7 × 7 = 49</blockquote>\r\n<blockquote>2<sup>5</sup> = 2 × 2 × 2 × 2 × 2 = 32</blockquote>\r\n<i>Square roots </i>(radicals) are the inverse of exponent 2 — that is, the number that, when multiplied by itself, gives you the indicated value.\r\n\r\n<i>Absolute value </i>is the positive value of a number — that is, the value of a negative number when you drop the minus sign. For example:\r\n\r\nAbsolute value is used to describe numbers that are always positive, such as the distance between two points or the area inside a polygon.","blurb":"","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> earned degrees in mathematics and English from Rutgers University. He is the founder of SimpleStep Learning, an online educational platform that teaches courses in basic concepts in ten minutes or less, keeping students engaged and learning in every moment. Mark is also author of several other successful <i>For Dummies</i> books.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"primaryCategoryTaxonomy":{"categoryId":33722,"title":"Basic Math","slug":"basic-math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"}},"secondaryCategoryTaxonomy":{"categoryId":33726,"title":"Pre-Algebra","slug":"pre-algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33726"}},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":207780,"title":"Basic Math & Pre-Algebra For Dummies Cheat Sheet","slug":"basic-math-pre-algebra-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/207780"}},{"articleId":194384,"title":"How to Write Numbers in Scientific Notation","slug":"how-to-write-numbers-in-scientific-notation","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/194384"}},{"articleId":158569,"title":"Inverse Operations and Commutative, Associative, and Distributive Properties","slug":"inverse-operations-and-commutative-associative-and-distributive-properties","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/158569"}},{"articleId":158567,"title":"Converting Metric Units to English Units","slug":"converting-metric-units-to-english-units","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/158567"}},{"articleId":158557,"title":"Conversion Guide for Fractions, Decimals, and Percents","slug":"a-quick-conversion-guide-for-fractions-decimals-and-percents","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/158557"}}],"fromCategory":[{"articleId":291491,"title":"Teaching Your Kids New Math (K-5) For Dummies Cheat Sheet","slug":"teaching-your-kids-new-math-k-5-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/291491"}},{"articleId":253710,"title":"Pre-Algebra Practice Questions: Comparing Fractions Using Cross-Multiplication","slug":"pre-algebra-practice-questions-comparing-fractions-using-cross-multiplication","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/253710"}},{"articleId":249996,"title":"Pre-Algebra Practice Questions: Solving Simple Algebraic Equations","slug":"pre-algebra-practice-questions-solving-simple-algebraic-equations","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249996"}},{"articleId":249986,"title":"Pre-Algebra: Isolating x in an Equation","slug":"pre-algebra-practice-questions-isolating-x-equation","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249986"}},{"articleId":249980,"title":"Pre-Algebra Practice Questions: Rearranging Equations to Isolate x","slug":"pre-algebra-practice-questions-rearranging-equations-isolate-x","categoryList":["academics-the-arts","math","basic-math"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/249980"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":281978,"slug":"basic-math-pre-algebra-for-dummies-2nd-edition","isbn":"9781119293637","categoryList":["academics-the-arts","math","basic-math"],"amazon":{"default":"//www.amazon.com/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119293634-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119293634/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/basic-math-and-pre-algebra-for-dummies-2nd-edition-cover-9781119293637-203x255.jpg","width":203,"height":255},"title":"Basic Math & Pre-Algebra For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><b><b data-author-id=\"9399\">Mark Zegarelli</b></b> earned degrees in mathematics and English from Rutgers University. He is the founder of SimpleStep Learning, an online educational platform that teaches courses in basic concepts in ten minutes or less, keeping students engaged and learning in every moment. Mark is also author of several other successful <i>For Dummies</i> books.</p>","authors":[{"authorId":9399,"name":"Mark Zegarelli","slug":"mark-zegarelli","description":" <p><b>Mark Zegarelli</b> earned degrees in mathematics and English from Rutgers University. He is the founder of SimpleStep Learning, an online educational platform that teaches courses in basic concepts in ten minutes or less, keeping students engaged and learning in every moment. Mark is also author of several other successful <i>For Dummies</i> books.</p> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9399"}}],"_links":{"self":"//dummies-api.coursofppt.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293637&quot;]}]\" id=\"du-slot-647f746066aa8\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;academics-the-arts&quot;,&quot;math&quot;,&quot;basic-math&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119293637&quot;]}]\" id=\"du-slot-647f74606732b\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2024-06-06T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":158560}],"_links":{"self":{"self":"//dummies-api.coursofppt.com/v2/categories/33722/categoryArticles?sortField=time&sortOrder=1&size=10&offset=0"},"next":{"self":"//dummies-api.coursofppt.com/v2/categories/33722/categoryArticles?sortField=time&sortOrder=1&size=10&offset=10"},"last":{"self":"//dummies-api.coursofppt.com/v2/categories/33722/categoryArticles?sortField=time&sortOrder=1&size=10&offset=144"}}},"objectTitle":"","status":"success","pageType":"article-category","objectId":"33722","page":1,"sortField":"time","sortOrder":1,"categoriesIds":[],"articleTypes":[],"filterData":{"categoriesFilter":[{"itemId":0,"itemName":"All Categories","count":151}],"articleTypeFilter":[{"articleType":"All Types","count":151},{"articleType":"Articles","count":144},{"articleType":"Cheat Sheet","count":7}]},"filterDataLoadedStatus":"success","pageSize":10},"adsState":{"pageScripts":{"headers":{"timestamp":"2025-03-04T05:50:01+00:00"},"adsId":0,"data":{"scripts":[{"pages":["all"],"location":"header","script":"<!--Optimizely Script-->\r\n<script src=\"//cdn.optimizely.com/js/10563184655.js\"></script>","enabled":false},{"pages":["all"],"location":"header","script":"<!-- comScore Tag -->\r\n<script>var _comscore = _comscore || [];_comscore.push({ c1: \"2\", c2: \"15097263\" });(function() {var s = document.createElement(\"script\"), el = document.getElementsByTagName(\"script\")[0]; s.async = true;s.src = (document.location.protocol == \"https:\" ? \"//sb\" : \"//b\") + \".scorecardresearch.com/beacon.js\";el.parentNode.insertBefore(s, el);})();</script><noscript><img src=\"//sb.scorecardresearch.com/p?c1=2&c2=15097263&cv=2.0&cj=1\" /></noscript>\r\n<!-- / comScore Tag -->","enabled":true},{"pages":["all"],"location":"footer","script":"<!--BEGIN QUALTRICS WEBSITE FEEDBACK SNIPPET-->\r\n<script type='text/javascript'>\r\n(function(){var g=function(e,h,f,g){\r\nthis.get=function(a){for(var a=a+\"=\",c=document.cookie.split(\";\"),b=0,e=c.length;b<e;b++){for(var d=c[b];\" \"==d.charAt(0);)d=d.substring(1,d.length);if(0==d.indexOf(a))return d.substring(a.length,d.length)}return null};\r\nthis.set=function(a,c){var b=\"\",b=new Date;b.setTime(b.getTime()+6048E5);b=\"; expires=\"+b.toGMTString();document.cookie=a+\"=\"+c+b+\"; path=/; \"};\r\nthis.check=function(){var a=this.get(f);if(a)a=a.split(\":\");else if(100!=e)\"v\"==h&&(e=Math.random()>=e/100?0:100),a=[h,e,0],this.set(f,a.join(\":\"));else return!0;var c=a[1];if(100==c)return!0;switch(a[0]){case \"v\":return!1;case \"r\":return c=a[2]%Math.floor(100/c),a[2]++,this.set(f,a.join(\":\")),!c}return!0};\r\nthis.go=function(){if(this.check()){var a=document.createElement(\"script\");a.type=\"text/javascript\";a.src=g;document.body&&document.body.appendChild(a)}};\r\nthis.start=function(){var t=this;\"complete\"!==document.readyState?window.addEventListener?window.addEventListener(\"load\",function(){t.go()},!1):window.attachEvent&&window.attachEvent(\"onload\",function(){t.go()}):t.go()};};\r\ntry{(new g(100,\"r\",\"QSI_S_ZN_5o5yqpvMVjgDOuN\",\"//zn5o5yqpvmvjgdoun-wiley.siteintercept.qualtrics.com/SIE/?Q_ZID=ZN_5o5yqpvMVjgDOuN\")).start()}catch(i){}})();\r\n</script><div id='ZN_5o5yqpvMVjgDOuN'><!--DO NOT REMOVE-CONTENTS PLACED HERE--></div>\r\n<!--END WEBSITE FEEDBACK SNIPPET-->","enabled":false},{"pages":["all"],"location":"header","script":"<!-- Hotjar Tracking Code for //coursofppt.com -->\r\n<script>\r\n (function(h,o,t,j,a,r){\r\n h.hj=h.hj||function(){(h.hj.q=h.hj.q||[]).push(arguments)};\r\n h._hjSettings={hjid:257151,hjsv:6};\r\n a=o.getElementsByTagName('head')[0];\r\n r=o.createElement('script');r.async=1;\r\n r.src=t+h._hjSettings.hjid+j+h._hjSettings.hjsv;\r\n a.appendChild(r);\r\n })(window,document,'//static.hotjar.com/c/hotjar-','.js?sv=');\r\n</script>","enabled":false},{"pages":["article"],"location":"header","script":"<!-- //Connect Container: dummies --> <script src=\"//get.s-onetag.com/bffe21a1-6bb8-4928-9449-7beadb468dae/tag.min.js\" async defer></script>","enabled":true},{"pages":["homepage"],"location":"header","script":"<meta name=\"facebook-domain-verification\" content=\"irk8y0irxf718trg3uwwuexg6xpva0\" />","enabled":true},{"pages":["homepage","article","category","search"],"location":"footer","script":"<!-- Facebook Pixel Code -->\r\n<noscript>\r\n<img height=\"1\" width=\"1\" src=\"//www.facebook.com/tr?id=256338321977984&ev=PageView&noscript=1\"/>\r\n</noscript>\r\n<!-- End Facebook Pixel Code -->","enabled":true}]}},"pageScriptsLoadedStatus":"success"},"navigationState":{"navigationCollections":[{"collectionId":287568,"title":"BYOB (Be Your Own Boss)","hasSubCategories":false,"url":"/collection/for-the-entry-level-entrepreneur-287568"},{"collectionId":293237,"title":"Be a Rad Dad","hasSubCategories":false,"url":"/collection/be-the-best-dad-293237"},{"collectionId":295890,"title":"Career Shifting","hasSubCategories":false,"url":"/collection/career-shifting-295890"},{"collectionId":294090,"title":"Contemplating the Cosmos","hasSubCategories":false,"url":"/collection/theres-something-about-space-294090"},{"collectionId":287563,"title":"For Those Seeking Peace of Mind","hasSubCategories":false,"url":"/collection/for-those-seeking-peace-of-mind-287563"},{"collectionId":287570,"title":"For the Aspiring Aficionado","hasSubCategories":false,"url":"/collection/for-the-bougielicious-287570"},{"collectionId":291903,"title":"For the Budding Cannabis Enthusiast","hasSubCategories":false,"url":"/collection/for-the-budding-cannabis-enthusiast-291903"},{"collectionId":299891,"title":"For the College Bound","hasSubCategories":false,"url":"/collection/for-the-college-bound-299891"},{"collectionId":291934,"title":"For the Exam-Season Crammer","hasSubCategories":false,"url":"/collection/for-the-exam-season-crammer-291934"},{"collectionId":301547,"title":"For the Game Day Prepper","hasSubCategories":false,"url":"/collection/big-game-day-prep-made-easy-301547"}],"navigationCollectionsLoadedStatus":"success","navigationCategories":{"books":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/books/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/books/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/books/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/books/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/books/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/books/level-0-category-0"}},"articles":{"0":{"data":[{"categoryId":33512,"title":"Technology","hasSubCategories":true,"url":"/category/articles/technology-33512"},{"categoryId":33662,"title":"Academics & The Arts","hasSubCategories":true,"url":"/category/articles/academics-the-arts-33662"},{"categoryId":33809,"title":"Home, Auto, & Hobbies","hasSubCategories":true,"url":"/category/articles/home-auto-hobbies-33809"},{"categoryId":34038,"title":"Body, Mind, & Spirit","hasSubCategories":true,"url":"/category/articles/body-mind-spirit-34038"},{"categoryId":34224,"title":"Business, Careers, & Money","hasSubCategories":true,"url":"/category/articles/business-careers-money-34224"}],"breadcrumbs":[],"categoryTitle":"Level 0 Category","mainCategoryUrl":"/category/articles/level-0-category-0"}}},"navigationCategoriesLoadedStatus":"success"},"searchState":{"searchList":[],"searchStatus":"initial","relatedArticlesList":[],"relatedArticlesStatus":"initial"},"routeState":{"name":"ArticleCategory","path":"/category/articles/basic-math-33722/","hash":"","query":{},"params":{"category":"basic-math-33722"},"fullPath":"/category/articles/basic-math-33722/","meta":{"routeType":"category","breadcrumbInfo":{"suffix":"Articles","baseRoute":"/category/articles"},"prerenderWithAsyncData":true},"from":{"name":null,"path":"/","hash":"","query":{},"params":{},"fullPath":"/","meta":{}}},"profileState":{"auth":{},"userOptions":{},"status":"success"}}
fun88 casino net cách chơi keno trực tuyến game đánh bài baccarat baccarat quốc tế sòng bài trực tuyến