chơi xổ số keno trực tuyến

{"appState":{"pageLoadApiCallsStatus":true},"categoryState":{"relatedCategories":{"headers":{"timestamp":"2025-01-31T04:01:08+00:00"},"categoryId":33720,"data":{"title":"Math","slug":"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}],"parentCategory":{"categoryId":33662,"title":"Academics & The Arts","slug":"academics-the-arts","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33662"}},"childCategories":[{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33721"},"image":{"src":"/img/background-image-2.fabfbd5c.png","width":0,"height":0},"hasArticle":true,"hasBook":true,"articleCount":207,"bookCount":13},{"categoryId":33722,"title":"Basic Math","slug":"basic-math","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33722"},"image":{"src":"/img/background-image-1.daf74cf0.png","width":0,"height":0},"hasArticle":true,"hasBook":true,"articleCount":151,"bookCount":4},{"categoryId":33723,"title":"Calculus","slug":"calculus","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33723"},"image":{"src":"/img/background-image-2.fabfbd5c.png","width":0,"height":0},"hasArticle":true,"hasBook":true,"articleCount":258,"bookCount":9},{"categoryId":33724,"title":"Common Core","slug":"common-core","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33724"},"image":{"src":"/img/background-image-1.daf74cf0.png","width":0,"height":0},"hasArticle":true,"hasBook":true,"articleCount":59,"bookCount":2},{"categoryId":33725,"title":"Geometry","slug":"geometry","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33725"},"image":{"src":"/img/background-image-2.fabfbd5c.png","width":0,"height":0},"hasArticle":true,"hasBook":true,"articleCount":199,"bookCount":5},{"categoryId":33726,"title":"Pre-Algebra","slug":"pre-algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33726"},"image":{"src":"/img/background-image-1.daf74cf0.png","width":0,"height":0},"hasArticle":true,"hasBook":true,"articleCount":100,"bookCount":2},{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33727"},"image":{"src":"/img/background-image-2.fabfbd5c.png","width":0,"height":0},"hasArticle":true,"hasBook":true,"articleCount":206,"bookCount":4},{"categoryId":33728,"title":"Statistics","slug":"statistics","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33728"},"image":{"src":"/img/background-image-1.daf74cf0.png","width":0,"height":0},"hasArticle":true,"hasBook":true,"articleCount":219,"bookCount":7},{"categoryId":33729,"title":"Trigonometry","slug":"trigonometry","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33729"},"image":{"src":"/img/background-image-2.fabfbd5c.png","width":0,"height":0},"hasArticle":true,"hasBook":true,"articleCount":180,"bookCount":2}],"description":"Whether you're an apprentice or a fully trained mathmagician, we have clear instruction to help you advance in the craft of math. Start with the basics and work up to calculus, plus everything in between. Yes, you do use this stuff in daily life.","relatedArticles":{"self":"//dummies-api.coursofppt.com/v2/articles?category=33720&offset=0&size=5"},"hasArticle":true,"hasBook":true,"articleCount":1583,"bookCount":49},"_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33720"}},"relatedCategoriesLoadedStatus":"success"},"listState":{"list":{"count":10,"total":1589,"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":"","归类领头羊SEO调整":{"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":"2017-03-26T21:07:46+00:00","modifiedTime":"2024-09-13T18:53:45+00:00","timestamp":"2024-09-13T21:01:03+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":"Pre-Algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33726"},"slug":"pre-algebra","categoryId":33726}],"title":"Use a Pie Chart to Find Percentages and Amounts","strippedTitle":"use a pie chart to find percentages and amounts","slug":"use-a-pie-chart-to-find-percentages-and-amounts","canonicalUrl":"","归类领头羊SEO调整":{"metaDescription":"Pie charts are most often used to show percentages and they provide an easy-to-understand picture of how the whole of something is divided up.","noIndex":0,"noFollow":0},"content":"A pie chart, which looks like a divided circle, shows you how a whole object is cut up into parts. Pie charts are most often used to represent percentages. For example, the following figure is a pie chart representing Eileen’s monthly expenses.\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/216971.image0.jpg\" alt=\"image0.jpg\" width=\"400\" height=\"217\" />\r\n\r\nYou can tell at a glance that Eileen’s largest expense is rent and that her second largest is her car. Unlike a bar graph, the pie chart shows numbers that are dependent upon each other. For example, if Eileen’s rent increases to 30% of her monthly income, she’ll have to decrease her spending in at least one other area.\r\n\r\nHere are a few typical questions you may be asked about a pie chart:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><strong>Individual percentages:</strong> <em>What percentage of her monthly expenses does Eileen spend on food?</em> Find the slice that represents what Eileen spends on food, and notice that she spends 10% of her income there.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><strong>Differences in percentages:</strong> <em>What percentage more does she spend on her car than on entertainment?</em> Eileen spends 20% on her car but only 5% on entertainment, so the difference between these percentages is 15%.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><strong>How much a percent represents in terms of dollars:</strong> <em>If Eileen brings home $2,000 per month, how much does she put away in savings each month?</em> First notice that Eileen puts 15% every month into savings. So you need to figure out 15% of $2,000. Solve this problem by turning 15% into a decimal and multiplying:</p>\r\n<p class=\"child-para\">0.15 2,000 = 300</p>\r\n<p class=\"child-para\">So Eileen saves $300 every month.</p>\r\n</li>\r\n</ul>","description":"A pie chart, which looks like a divided circle, shows you how a whole object is cut up into parts. Pie charts are most often used to represent percentages. For example, the following figure is a pie chart representing Eileen’s monthly expenses.\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/216971.image0.jpg\" alt=\"image0.jpg\" width=\"400\" height=\"217\" />\r\n\r\nYou can tell at a glance that Eileen’s largest expense is rent and that her second largest is her car. Unlike a bar graph, the pie chart shows numbers that are dependent upon each other. For example, if Eileen’s rent increases to 30% of her monthly income, she’ll have to decrease her spending in at least one other area.\r\n\r\nHere are a few typical questions you may be asked about a pie chart:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><strong>Individual percentages:</strong> <em>What percentage of her monthly expenses does Eileen spend on food?</em> Find the slice that represents what Eileen spends on food, and notice that she spends 10% of her income there.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><strong>Differences in percentages:</strong> <em>What percentage more does she spend on her car than on entertainment?</em> Eileen spends 20% on her car but only 5% on entertainment, so the difference between these percentages is 15%.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\"><strong>How much a percent represents in terms of dollars:</strong> <em>If Eileen brings home $2,000 per month, how much does she put away in savings each month?</em> First notice that Eileen puts 15% every month into savings. So you need to figure out 15% of $2,000. Solve this problem by turning 15% into a decimal and multiplying:</p>\r\n<p class=\"child-para\">0.15 2,000 = 300</p>\r\n<p class=\"child-para\">So Eileen saves $300 every month.</p>\r\n</li>\r\n</ul>","blurb":"","authors":[],"primaryCategoryTaxonomy":{"categoryId":33726,"title":"Pre-Algebra","slug":"pre-algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33726"}},"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":291583,"title":"Basic Math and Pre-Algebra All-in-One For Dummies Cheat Sheet","slug":"basic-math-and-pre-algebra-all-in-one-for-dummies","categoryList":["academics-the-arts","math","pre-algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/291583"}},{"articleId":207813,"title":"Basic Math and Pre-Algebra Workbook For Dummies Cheat Sheet","slug":"basic-math-pre-algebra-workbook-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","pre-algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/207813"}},{"articleId":199813,"title":"Multiplying with Scientific Notation","slug":"multiplying-with-scientific-notation","categoryList":["academics-the-arts","math","pre-algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/199813"}},{"articleId":194349,"title":"How to Convert Fractions to Decimals","slug":"how-to-convert-fractions-to-decimals","categoryList":["academics-the-arts","math","pre-algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/194349"}},{"articleId":191352,"title":"How to Determine Likelihoods Using Basic Probability","slug":"how-to-determine-likelihoods-using-basic-probability","categoryList":["academics-the-arts","math","pre-algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/191352"}}]},"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":[{"title":"Pondering the Pi Possibilities","slug":"pondering-the-pi-possibilities","collectionId":297524}],"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;pre-algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-6502230f48b2f\"></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;pre-algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-6502230f49058\"></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":"2024-09-13T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":191187},{"headers":{"creationTime":"2017-03-26T10:58:16+00:00","modifiedTime":"2024-08-28T14:47:37+00:00","timestamp":"2024-08-28T15:01:03+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":"Algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33721"},"slug":"algebra","categoryId":33721}],"title":"Using the Multiplication Property in Sets","strippedTitle":"using the multiplication property in sets","slug":"using-the-multiplication-property-in-sets","canonicalUrl":"","归类领头羊SEO调整":{"metaDescription":"When you want to count up how many things are in a set, you have quite a few options. When the set contains too many elements to count accurately, you look for ","noIndex":0,"noFollow":0},"content":"<p>When you want to count up how many things are in a set, you have quite a few options. When the set contains too many elements to count accurately, you look for some sort of pattern or rule to help out. Here, you practice the multiplication property.</p>\r\n<p>If you can do task one in <i>m</i><sub>1</sub> ways, task two in <i>m</i><sub>2</sub> ways, task three in <i>m</i><sub>3</sub> ways, and so on, then you can perform all the tasks in a total of <i>m</i><sub>1</sub> · <i>m</i><sub>2</sub> · <i>m</i><sub>3 . . . </sub> ways.</p>\r\n<h2 id=\"tab1\" >Sample questions</h2>\r\n<ol class=\"level-one\">\r\n <li><p class=\"first-para\">How many ways can you fly from San Francisco to New York City, stopping in Denver, Chicago, and Buffalo, if the website offers four ways to fly from San Francisco to Denver, six ways to fly from Denver to Chicago, two ways to fly from Chicago to Buffalo, and three ways to fly from Buffalo to New York City?</p>\r\n<p class=\"child-para\"><b>144.</b> Multiply 4 x 6 x 2 x 3 = 144. This method doesn’t tell you what all the routes are; it just tells you how many are possible so you know when you’ve listed all of them. (Better get to work on that.)</p>\r\n </li>\r\n <li><p class=\"first-para\">How many ways can you write a password if the first symbol has to be a digit from 1 to 9; the second, third, and fourth symbols have to be letters of the English alphabet; and the last symbol has to be from the set {!, @, #, $, %, ^, &, *, +}?</p>\r\n<p class=\"child-para\"><b>1,423,656.</b> You multiply 9 x 26 x 26 x 26 x 9 = 1,423,656. This system allows a lot of passwords, but most institutions make you use eight or more characters, which makes the number of possibilities even greater.</p>\r\n </li>\r\n</ol>\r\n<h2 id=\"tab2\" >Practice questions</h2>\r\n<ol class=\"level-one\">\r\n <li><p class=\"first-para\">If you have to take one class in each subject, how many different course loads can you create if you have a choice of four math classes, three history classes, eight English classes, and five science classes?</p>\r\n </li>\r\n <li><p class=\"first-para\">How many different ice-cream sundaes can you create if you have a choice of five ice-cream flavors, three sauces, and five sprinkled toppings if you choose one of each type?</p>\r\n </li>\r\n <li><p class=\"first-para\">How many different automobiles can you order if you have a choice of six colors, four interiors, two trim options, three warranties, and two types of seats?</p>\r\n </li>\r\n <li><p class=\"first-para\">How many different dinners can you order if you have a choice of 12 appetizers, 8 entrees, 5 potatoes, 6 desserts, and a choice of soup or salad?</p>\r\n </li>\r\n</ol>\r\n<p>Following are answers to the practice questions:</p>\r\n<ol class=\"level-one\">\r\n <li><p class=\"first-para\">The answer is 480.</p>\r\n<p class=\"child-para\">Multiply: 4 x 3 x 8 x 5 = 480.</p>\r\n </li>\r\n <li><p class=\"first-para\">The answer is 75.</p>\r\n<p class=\"child-para\">Multiply: 5 x 3 x 5 = 75.</p>\r\n </li>\r\n <li><p class=\"first-para\">The answer is 288.</p>\r\n<p class=\"child-para\">Multiply: 6 x 4 x 2 x 3 x 2 = 288.</p>\r\n </li>\r\n <li><p class=\"first-para\">The answer is 5,760.</p>\r\n<p class=\"child-para\">Multiply: 12 x 8 x 5 x 6 x 2 = 5,760. Don’t forget that <i>soup or salad</i> is two choices for that selection.</p>\r\n </li>\r\n</ol>","description":"<p>When you want to count up how many things are in a set, you have quite a few options. When the set contains too many elements to count accurately, you look for some sort of pattern or rule to help out. Here, you practice the multiplication property.</p>\r\n<p>If you can do task one in <i>m</i><sub>1</sub> ways, task two in <i>m</i><sub>2</sub> ways, task three in <i>m</i><sub>3</sub> ways, and so on, then you can perform all the tasks in a total of <i>m</i><sub>1</sub> · <i>m</i><sub>2</sub> · <i>m</i><sub>3 . . . </sub> ways.</p>\r\n<h2 id=\"tab1\" >Sample questions</h2>\r\n<ol class=\"level-one\">\r\n <li><p class=\"first-para\">How many ways can you fly from San Francisco to New York City, stopping in Denver, Chicago, and Buffalo, if the website offers four ways to fly from San Francisco to Denver, six ways to fly from Denver to Chicago, two ways to fly from Chicago to Buffalo, and three ways to fly from Buffalo to New York City?</p>\r\n<p class=\"child-para\"><b>144.</b> Multiply 4 x 6 x 2 x 3 = 144. This method doesn’t tell you what all the routes are; it just tells you how many are possible so you know when you’ve listed all of them. (Better get to work on that.)</p>\r\n </li>\r\n <li><p class=\"first-para\">How many ways can you write a password if the first symbol has to be a digit from 1 to 9; the second, third, and fourth symbols have to be letters of the English alphabet; and the last symbol has to be from the set {!, @, #, $, %, ^, &, *, +}?</p>\r\n<p class=\"child-para\"><b>1,423,656.</b> You multiply 9 x 26 x 26 x 26 x 9 = 1,423,656. This system allows a lot of passwords, but most institutions make you use eight or more characters, which makes the number of possibilities even greater.</p>\r\n </li>\r\n</ol>\r\n<h2 id=\"tab2\" >Practice questions</h2>\r\n<ol class=\"level-one\">\r\n <li><p class=\"first-para\">If you have to take one class in each subject, how many different course loads can you create if you have a choice of four math classes, three history classes, eight English classes, and five science classes?</p>\r\n </li>\r\n <li><p class=\"first-para\">How many different ice-cream sundaes can you create if you have a choice of five ice-cream flavors, three sauces, and five sprinkled toppings if you choose one of each type?</p>\r\n </li>\r\n <li><p class=\"first-para\">How many different automobiles can you order if you have a choice of six colors, four interiors, two trim options, three warranties, and two types of seats?</p>\r\n </li>\r\n <li><p class=\"first-para\">How many different dinners can you order if you have a choice of 12 appetizers, 8 entrees, 5 potatoes, 6 desserts, and a choice of soup or salad?</p>\r\n </li>\r\n</ol>\r\n<p>Following are answers to the practice questions:</p>\r\n<ol class=\"level-one\">\r\n <li><p class=\"first-para\">The answer is 480.</p>\r\n<p class=\"child-para\">Multiply: 4 x 3 x 8 x 5 = 480.</p>\r\n </li>\r\n <li><p class=\"first-para\">The answer is 75.</p>\r\n<p class=\"child-para\">Multiply: 5 x 3 x 5 = 75.</p>\r\n </li>\r\n <li><p class=\"first-para\">The answer is 288.</p>\r\n<p class=\"child-para\">Multiply: 6 x 4 x 2 x 3 x 2 = 288.</p>\r\n </li>\r\n <li><p class=\"first-para\">The answer is 5,760.</p>\r\n<p class=\"child-para\">Multiply: 12 x 8 x 5 x 6 x 2 = 5,760. Don’t forget that <i>soup or salad</i> is two choices for that selection.</p>\r\n </li>\r\n</ol>","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> taught mathematics for more than 45 years. She was a professor of mathematics at Bradley University for 35 of those years and continues to teach occasional classes either in person or via distance learning. Sterling is the author of several Dummies algebra and higher-level math titles. She is a graduate of the University of New Hampshire with a master's degree in math education.</p>","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":33721,"title":"Algebra","slug":"algebra","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33721"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Sample questions","target":"#tab1"},{"label":"Practice questions","target":"#tab2"}],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":294686,"title":"Algebra II All-in-One For Dummies Cheat Sheet","slug":"algebra-ii-all-in-one-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/294686"}},{"articleId":255800,"title":"Practicing with the Distributive Property in Algebra","slug":"applying-the-distributive-property-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/255800"}},{"articleId":245778,"title":"Algebra: Converting Improper and Mixed Fractions: Algebra Practice Questions","slug":"converting-improper-mixed-fractions-algebra-practice-questions","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/245778"}},{"articleId":210251,"title":"How to Calculate Limits with Algebra","slug":"how-to-calculate-limits-with-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/210251"}},{"articleId":210250,"title":"Understanding the Vocabulary of Algebra","slug":"understanding-the-vocabulary-of-algebra","categoryList":["academics-the-arts","math","algebra"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/210250"}}]},"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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-64ecb6af79955\"></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;algebra&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-64ecb6af7a039\"></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":"2024-08-28T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":149413},{"headers":{"creationTime":"2020-02-20T19:10:31+00:00","modifiedTime":"2024-08-14T19:09:18+00:00","timestamp":"2024-08-14T21:01:03+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":"Pre-Calculus","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33727"},"slug":"pre-calculus","categoryId":33727}],"title":"Pre-Calculus: 10 Habits to Adjust before Calculus","strippedTitle":"pre-calculus: 10 habits to adjust before calculus","slug":"pre-calculus-10-habits-to-adjust-before-calculus","canonicalUrl":"","归类领头羊SEO调整":{"metaDescription":"As you work through pre-calculus, adopting certain tasks as habits can help prepare your brain to tackle your next challenge: calculus. In this article, you fin","noIndex":0,"noFollow":0},"content":"As you work through pre-calculus, adopting certain tasks as habits can help prepare your brain to tackle your next challenge: calculus. In this article, you find ten habits that should be a part of your daily math arsenal. Perhaps you’ve been told to perform some of these tasks since elementary school — such as showing all your work — but other tricks may be new to you. Either way, if you remember these ten pieces of advice, you’ll be even more ready for whatever calculus throws your way.\r\n<h2 id=\"tab1\" >Figure Out What the Problem Is Asking</h2>\r\nOften, you’ll find that reading comprehension and the ability to work with multiple parts that comprise a whole is an underlying property of a math problem. That’s okay — that’s also what life is all about!! When faced with a math problem, start by reading the whole problem or all the directions to the problem. Look for the question inside the question. Keep your eyes peeled for words like <em>solve, simplify, find,</em> and <em>prove,</em> all of which are common buzz words in any math book. Don’t begin working on a problem until you’re certain of what it wants you to do.\r\n\r\nFor example, take a look at this problem:\r\n<blockquote>The length of a rectangular garden is 24 feet longer than the garden’s width. If you add 2 feet to both the width and the length, the area of the garden is 432 square feet. How long is the new, bigger garden?</blockquote>\r\nIf you miss any of the important information, you may start to solve the problem to figure out how wide the garden is. Or you may find the length but miss the fact that you’re supposed to find out how long it is with 2 feet <em>added</em> to it. Look before you leap!\r\n<p class=\"article-tips tip\">Underlining key words and information in the question is often helpful. This can’t be stressed enough. Highlighting important words and pieces of information solidifies them in your brain so that as you work, you can redirect your focus if it veers off-track. When presented with a word problem, for example, first turn the words into an algebraic equation. If you’re lucky and are given the algebraic equation from the get-go, you can move on to the next step, which is to create a visual image of the situation at hand.</p>\r\nAnd, if you’re wondering what the answer to the example problem is, you’ll find out as you read further.\r\n<h2 id=\"tab2\" >Draw Pictures (the More the Better)</h2>\r\nYour brain is like a movie screen in your skull, and you’ll have an easier time working problems if you project what you see onto a piece of paper. When you visualize math problems, you’re more apt to comprehend them. Draw pictures that correspond to a problem and label all the parts so you have a visual image to follow that allows you to attach mathematical symbols to physical structures. This process works the conceptual part of your brain and helps you remember important concepts. As such, you’ll be less likely to miss steps or get disorganized.\r\n\r\nIf the question is talking about a triangle, for instance, draw a triangle; if it mentions a rectangular garden filled with daffodils for 30 percent of its space, draw that. In fact, every time a problem changes and new information is presented, your picture should change, too.\r\n\r\nIf you were asked to solve the rectangular garden problem from the previous section, you’d start by drawing two rectangles: one for the old, smaller garden and another for the bigger one. Putting pen or pencil to the paper starts you on the way to a solution.\r\n<h2 id=\"tab3\" >Plan Your Attack — Identify Your Targets</h2>\r\nWhen you know and can picture what you must find, you can plan your attack from there, interpreting the problem mathematically and coming up with the equations that you’ll be working with to find the answer:\r\n<ol>\r\n \t<li><strong> Start by writing a </strong><strong> statement. In the garden problem from the last two sections, you’re looking for the length and width of a garden after it has been made bigger. With those in mind, define some variables:</strong>\r\n<ul>\r\n \t<li>Let <em>x</em> = the garden’s width now.</li>\r\n \t<li>Let <em>y</em> = the garden’s length now.</li>\r\n</ul>\r\n</li>\r\n \t<li><strong> Now add those variables to the rectangle you drew of the old garden.</strong></li>\r\n \t<li><strong> But you know that the length is 24 feet greater than the width, so you can rewrite the variable <em>y</em> in terms of the variable <em>x</em> so that <em>y</em> = <em>x</em> + 24.</strong></li>\r\n \t<li><strong> You know that the new garden has had 2 feet added to both its width and length, so you can modify your equations:</strong>\r\n<ul>\r\n \t<li>Let <em>x</em> + 2 = the garden’s new width.</li>\r\n \t<li>Let <em>y</em> + 2 = <em>x</em> + 24 + 2 = <em>x</em> + 26 the garden’s new length.</li>\r\n</ul>\r\n</li>\r\n \t<li><strong> Now add these labels to the picture of the new garden.</strong></li>\r\n</ol>\r\nBy planning your attack, you’ve identified the pieces of the equation that you need to solve.\r\n<h2 id=\"tab4\" >Write Down Any Formulas</h2>\r\nIf you start your attack by writing the formula needed to solve the problem, all you have to do from there is plug in what you know and then solve for the unknown. A problem always makes more sense if the formula is the first thing you write when solving. Before you can do that, though, you need to figure out which formula to use. You can usually find out by taking a close look at the words of the problem.\r\n\r\nIn the case of the garden problem from the previous sections, the formula you need is that for the area of a rectangle. The area of a rectangle is <em>A</em> = <em>lw</em>. You’re told that the area of the new rectangle is 432 square feet, and you have expressions representing the length and width, so you can replace <em>A</em> = <em>lw</em> with 432 = (<em>x</em> + 26)(<em>x</em> + 2).\r\n\r\nAs another example, if you need to solve a right triangle, you may start by writing down the Pythagorean Theorem if you know two sides and are looking for the third. For another right triangle, perhaps you’re given an angle and the hypotenuse and need to find the opposite side; in this situation, you’d start off by writing down the sine ratio.\r\n<h2 id=\"tab5\" >Show Each Step of Your Work</h2>\r\nYes, you’ve been hearing it forever, but your third-grade teacher was right: Showing each step of your work is vital in math. Writing each step on paper minimizes silly mistakes that you can make when you calculate in your head. It’s also a great way to keep a problem organized and clear. And it helps to have your work written down when you get interrupted by a phone call or text message — you can pick up where you left off and not have to start all over again. It may take some time to write every single step down, but it’s well worth your investment.\r\n<h2 id=\"tab6\" >Know When to Quit</h2>\r\nSometimes a problem has no solution. Yes, that can be an answer, too! If you’ve tried all the tricks in your bag and you haven’t found a way, consider that the problem may have no solution at all. Some common problems that may not have a solution include the following:\r\n<ul>\r\n \t<li><strong>Absolute-value equations</strong></li>\r\n</ul>\r\n<p style=\"padding-left: 30px;\">This happens when the absolute value expression is set equal to a negative number. You may not realize the number is negative, at first, if it’s represented by a variable.</p>\r\n\r\n<ul>\r\n \t<li><strong>Equations with the variable under a square-root sign</strong></li>\r\n</ul>\r\n<p style=\"padding-left: 30px;\">If your answer has to be a real number, and complex numbers aren’t an option, then the expression under the radical may represent a negative number. Not allowed.</p>\r\n\r\n<ul>\r\n \t<li><strong>Quadratic equations</strong></li>\r\n</ul>\r\n<p style=\"padding-left: 30px;\">When a quadratic isn’t factorable and you have to resort to the quadratic formula, you may run into a negative under the radical; you can’t use that expression if you’re allowed only real answers.</p>\r\n\r\n<ul>\r\n \t<li><strong>Rational equations</strong></li>\r\n</ul>\r\n<p style=\"padding-left: 30px;\">Rational expressions have numerators and denominators. If there’s a variable in the denominator that ends up creating a zero, then that value isn’t allowed.</p>\r\n\r\n<ul>\r\n \t<li><strong>Trig equations</strong></li>\r\n</ul>\r\n<p style=\"padding-left: 30px;\">Trig functions have restrictions. Sines and cosines have to lie between –1 and 1. Secants and cosecants have to be greater than or equal to 1 or less than or equal to –1. A perfectly nice-looking equation may create an impossible answer.</p>\r\n<p class=\"article-tips warning\">On the other hand, you may get a solution for some problem that just doesn’t make sense. Watch out for the following situations:</p>\r\n\r\n<ul>\r\n \t<li>If you’re solving an equation for a measurement (like length or area) and you get a negative answer, either you made a mistake or no solution exists. Measurement problems include distance, and distance can’t be negative.</li>\r\n \t<li>If you’re solving an equation to find the number of things (like how many books are on a bookshelf) and you get a fraction or decimal answer, then that just doesn’t make any sense. How could you have 13.4 books on a shelf?</li>\r\n</ul>\r\n<h2 id=\"tab7\" >Check Your Answers</h2>\r\nEven the best mathematicians make mistakes. When you hurry through calculations or work in a stressful situation, you may make mistakes more frequently. So, check your work. Usually, this process is very easy: You take your answer and plug it back into the equation or problem description to see if it really works. Making the check takes a little time, but it guarantees you got the question right, so why not do it?\r\n\r\nFor example, if you go back and solve the garden problem from earlier in this chapter by looking at the equation to be solved: 432 = (<em>x</em> + 26)(<em>x</em> + 2), you multiply the two binomials and move the 432 to the other side. Solving <em>x</em><sup>2</sup> + 28<em>x</em> – 380 = 0 you get <em>x</em> = 10 and <em>x</em> = –38. You disregard the <em>x</em> = –38, of course, and find that the original width was 10 feet. So, what was the question? It asks for the length of the new, bigger garden. The length of the new garden is found with <em>y</em> = <em>x</em> + 26. So, the length (and answer) is that the length is 36 feet. Does this check? If you use the new length or 36 and the new width of <em>x</em> + 2 = 12 and multiply 36 times 12 you get 432 square feet. It checks!\r\n<h2 id=\"tab8\" >Practice Plenty of Problems</h2>\r\nYou’re not born with the knowledge of how to ride a bike, play baseball, or even speak. The way you get better at challenging tasks is to practice, practice, practice. And the best way to practice math is to work the problems. You can seek harder or more complicated examples of questions that will stretch your brain and make you better at a concept the next time you see it.\r\n<p class=\"article-tips tip\">Along with working along on the example problems in this book, you can take advantage of the For Dummies workbooks, which include loads of practice exercises. Check out <a href=\"//coursofppt.com/book/academics-the-arts/math/trigonometry/trigonometry-workbook-for-dummies-282641/\"><em>Trigonometry Workbook For Dummies,</em> </a>by Mary Jane Sterling, <a href=\"//coursofppt.com/book/academics-the-arts/math/algebra/algebra-i-workbook-for-dummies-281939/\"><em>Algebra Workbook</em></a> and <a href=\"//coursofppt.com/book/academics-the-arts/math/algebra/algebra-ii-workbook-for-dummies-3rd-edition-281941/\"><em>Algebra II Workbook For Dummies</em></a>, both also by Mary Jane Sterling, and <a href=\"//coursofppt.com/book/academics-the-arts/math/geometry/geometry-workbook-for-dummies-292500/\"><em>Geometry Workbook For Dummies</em></a>, by Mark Ryan (all published by Wiley), to name a few.</p>\r\nEven a math textbook is great for practice. Why not try some (gulp!) problems that weren’t assigned, or maybe go back to an old section to review and make sure you’ve still got it? Typically, textbooks show the answers to the odd problems, so if you stick with those you can always double-check your answers. And if you get a craving for some extra practice, just search the Internet for “practice math problems” to see what you can find! For example, if you search the Internet for “practice systems of equations problems” you’ll find more than a million hits. That’s a lot of practice!\r\n<h2 id=\"tab9\" >Keep Track of the Order of Operations</h2>\r\nDon’t fall for the trap that always is lying there by performing operations in the wrong order. For instance, 2 – 6 × 3 doesn’t become –4 × 3 = –12. You’ll reach those incorrect answers if you forget to do the multiplication first. Focus on following the order of PEMDAS every time, all the time:\r\n<blockquote><strong>P</strong>arentheses (and other grouping devices)\r\n\r\n<strong>E</strong>xponents and roots\r\n\r\n<strong>M</strong>ultiplication and <strong>D</strong>ivision from left to right\r\n\r\n<strong>A</strong>ddition and <strong>S</strong>ubtraction from left to right</blockquote>\r\nDon’t ever go out of order, and that’s an order!\r\n<h2 id=\"tab10\" >Use Caution When Dealing with Fractions</h2>\r\nWorking with denominators can be tricky. It’s okay to write:\r\n\r\n<img class=\"alignnone size-full wp-image-260190\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-denominator.jpg\" alt=\"pre-calculus-denominator\" width=\"272\" height=\"86\" />\r\n\r\nBut, on the other hand: <img class=\"alignnone size-full wp-image-260191\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-denominators2.jpg\" alt=\"pre-calculus-denominators2\" width=\"272\" height=\"78\" />\r\n\r\nAlso, reducing or cancelling in fractions can be performed incorrectly. Every term in the numerator has to be divided by the same factor — the one that divides the denominator. So, it’s true that:\r\n\r\n<img class=\"alignnone size-full wp-image-260192\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-fraction.jpg\" alt=\"pre-calculus-fraction\" width=\"528\" height=\"86\" />because <img class=\"alignnone size-full wp-image-260193\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-fraction2.jpg\" alt=\"pre-calculus-fraction2\" width=\"612\" height=\"135\" />\r\n\r\nBut <img class=\"alignnone size-full wp-image-260194\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-fraction3.jpg\" alt=\"pre-calculus-fraction3\" width=\"510\" height=\"86\" />, because the factor being divided out is just 4, not 4<em>x</em>.\r\n\r\nAnd, again, it has to be the same factor that’s dividing, throughout. Can you spot the error here?\r\n\r\n<img class=\"alignnone size-full wp-image-260195\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-fraction4.jpg\" alt=\"pre-calculus-fraction4\" width=\"319\" height=\"104\" />\r\n\r\nSome poor soul reduced each term and the term directly above it — separately. Nope, doesn’t work that way. The correct process is to factor the trinomials and then divide by the common factor:\r\n\r\n<img class=\"alignnone size-full wp-image-260196\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-fraction5.jpg\" alt=\"pre-calculus-fraction5\" width=\"596\" height=\"112\" />","description":"As you work through pre-calculus, adopting certain tasks as habits can help prepare your brain to tackle your next challenge: calculus. In this article, you find ten habits that should be a part of your daily math arsenal. Perhaps you’ve been told to perform some of these tasks since elementary school — such as showing all your work — but other tricks may be new to you. Either way, if you remember these ten pieces of advice, you’ll be even more ready for whatever calculus throws your way.\r\n<h2 id=\"tab1\" >Figure Out What the Problem Is Asking</h2>\r\nOften, you’ll find that reading comprehension and the ability to work with multiple parts that comprise a whole is an underlying property of a math problem. That’s okay — that’s also what life is all about!! When faced with a math problem, start by reading the whole problem or all the directions to the problem. Look for the question inside the question. Keep your eyes peeled for words like <em>solve, simplify, find,</em> and <em>prove,</em> all of which are common buzz words in any math book. Don’t begin working on a problem until you’re certain of what it wants you to do.\r\n\r\nFor example, take a look at this problem:\r\n<blockquote>The length of a rectangular garden is 24 feet longer than the garden’s width. If you add 2 feet to both the width and the length, the area of the garden is 432 square feet. How long is the new, bigger garden?</blockquote>\r\nIf you miss any of the important information, you may start to solve the problem to figure out how wide the garden is. Or you may find the length but miss the fact that you’re supposed to find out how long it is with 2 feet <em>added</em> to it. Look before you leap!\r\n<p class=\"article-tips tip\">Underlining key words and information in the question is often helpful. This can’t be stressed enough. Highlighting important words and pieces of information solidifies them in your brain so that as you work, you can redirect your focus if it veers off-track. When presented with a word problem, for example, first turn the words into an algebraic equation. If you’re lucky and are given the algebraic equation from the get-go, you can move on to the next step, which is to create a visual image of the situation at hand.</p>\r\nAnd, if you’re wondering what the answer to the example problem is, you’ll find out as you read further.\r\n<h2 id=\"tab2\" >Draw Pictures (the More the Better)</h2>\r\nYour brain is like a movie screen in your skull, and you’ll have an easier time working problems if you project what you see onto a piece of paper. When you visualize math problems, you’re more apt to comprehend them. Draw pictures that correspond to a problem and label all the parts so you have a visual image to follow that allows you to attach mathematical symbols to physical structures. This process works the conceptual part of your brain and helps you remember important concepts. As such, you’ll be less likely to miss steps or get disorganized.\r\n\r\nIf the question is talking about a triangle, for instance, draw a triangle; if it mentions a rectangular garden filled with daffodils for 30 percent of its space, draw that. In fact, every time a problem changes and new information is presented, your picture should change, too.\r\n\r\nIf you were asked to solve the rectangular garden problem from the previous section, you’d start by drawing two rectangles: one for the old, smaller garden and another for the bigger one. Putting pen or pencil to the paper starts you on the way to a solution.\r\n<h2 id=\"tab3\" >Plan Your Attack — Identify Your Targets</h2>\r\nWhen you know and can picture what you must find, you can plan your attack from there, interpreting the problem mathematically and coming up with the equations that you’ll be working with to find the answer:\r\n<ol>\r\n \t<li><strong> Start by writing a </strong><strong> statement. In the garden problem from the last two sections, you’re looking for the length and width of a garden after it has been made bigger. With those in mind, define some variables:</strong>\r\n<ul>\r\n \t<li>Let <em>x</em> = the garden’s width now.</li>\r\n \t<li>Let <em>y</em> = the garden’s length now.</li>\r\n</ul>\r\n</li>\r\n \t<li><strong> Now add those variables to the rectangle you drew of the old garden.</strong></li>\r\n \t<li><strong> But you know that the length is 24 feet greater than the width, so you can rewrite the variable <em>y</em> in terms of the variable <em>x</em> so that <em>y</em> = <em>x</em> + 24.</strong></li>\r\n \t<li><strong> You know that the new garden has had 2 feet added to both its width and length, so you can modify your equations:</strong>\r\n<ul>\r\n \t<li>Let <em>x</em> + 2 = the garden’s new width.</li>\r\n \t<li>Let <em>y</em> + 2 = <em>x</em> + 24 + 2 = <em>x</em> + 26 the garden’s new length.</li>\r\n</ul>\r\n</li>\r\n \t<li><strong> Now add these labels to the picture of the new garden.</strong></li>\r\n</ol>\r\nBy planning your attack, you’ve identified the pieces of the equation that you need to solve.\r\n<h2 id=\"tab4\" >Write Down Any Formulas</h2>\r\nIf you start your attack by writing the formula needed to solve the problem, all you have to do from there is plug in what you know and then solve for the unknown. A problem always makes more sense if the formula is the first thing you write when solving. Before you can do that, though, you need to figure out which formula to use. You can usually find out by taking a close look at the words of the problem.\r\n\r\nIn the case of the garden problem from the previous sections, the formula you need is that for the area of a rectangle. The area of a rectangle is <em>A</em> = <em>lw</em>. You’re told that the area of the new rectangle is 432 square feet, and you have expressions representing the length and width, so you can replace <em>A</em> = <em>lw</em> with 432 = (<em>x</em> + 26)(<em>x</em> + 2).\r\n\r\nAs another example, if you need to solve a right triangle, you may start by writing down the Pythagorean Theorem if you know two sides and are looking for the third. For another right triangle, perhaps you’re given an angle and the hypotenuse and need to find the opposite side; in this situation, you’d start off by writing down the sine ratio.\r\n<h2 id=\"tab5\" >Show Each Step of Your Work</h2>\r\nYes, you’ve been hearing it forever, but your third-grade teacher was right: Showing each step of your work is vital in math. Writing each step on paper minimizes silly mistakes that you can make when you calculate in your head. It’s also a great way to keep a problem organized and clear. And it helps to have your work written down when you get interrupted by a phone call or text message — you can pick up where you left off and not have to start all over again. It may take some time to write every single step down, but it’s well worth your investment.\r\n<h2 id=\"tab6\" >Know When to Quit</h2>\r\nSometimes a problem has no solution. Yes, that can be an answer, too! If you’ve tried all the tricks in your bag and you haven’t found a way, consider that the problem may have no solution at all. Some common problems that may not have a solution include the following:\r\n<ul>\r\n \t<li><strong>Absolute-value equations</strong></li>\r\n</ul>\r\n<p style=\"padding-left: 30px;\">This happens when the absolute value expression is set equal to a negative number. You may not realize the number is negative, at first, if it’s represented by a variable.</p>\r\n\r\n<ul>\r\n \t<li><strong>Equations with the variable under a square-root sign</strong></li>\r\n</ul>\r\n<p style=\"padding-left: 30px;\">If your answer has to be a real number, and complex numbers aren’t an option, then the expression under the radical may represent a negative number. Not allowed.</p>\r\n\r\n<ul>\r\n \t<li><strong>Quadratic equations</strong></li>\r\n</ul>\r\n<p style=\"padding-left: 30px;\">When a quadratic isn’t factorable and you have to resort to the quadratic formula, you may run into a negative under the radical; you can’t use that expression if you’re allowed only real answers.</p>\r\n\r\n<ul>\r\n \t<li><strong>Rational equations</strong></li>\r\n</ul>\r\n<p style=\"padding-left: 30px;\">Rational expressions have numerators and denominators. If there’s a variable in the denominator that ends up creating a zero, then that value isn’t allowed.</p>\r\n\r\n<ul>\r\n \t<li><strong>Trig equations</strong></li>\r\n</ul>\r\n<p style=\"padding-left: 30px;\">Trig functions have restrictions. Sines and cosines have to lie between –1 and 1. Secants and cosecants have to be greater than or equal to 1 or less than or equal to –1. A perfectly nice-looking equation may create an impossible answer.</p>\r\n<p class=\"article-tips warning\">On the other hand, you may get a solution for some problem that just doesn’t make sense. Watch out for the following situations:</p>\r\n\r\n<ul>\r\n \t<li>If you’re solving an equation for a measurement (like length or area) and you get a negative answer, either you made a mistake or no solution exists. Measurement problems include distance, and distance can’t be negative.</li>\r\n \t<li>If you’re solving an equation to find the number of things (like how many books are on a bookshelf) and you get a fraction or decimal answer, then that just doesn’t make any sense. How could you have 13.4 books on a shelf?</li>\r\n</ul>\r\n<h2 id=\"tab7\" >Check Your Answers</h2>\r\nEven the best mathematicians make mistakes. When you hurry through calculations or work in a stressful situation, you may make mistakes more frequently. So, check your work. Usually, this process is very easy: You take your answer and plug it back into the equation or problem description to see if it really works. Making the check takes a little time, but it guarantees you got the question right, so why not do it?\r\n\r\nFor example, if you go back and solve the garden problem from earlier in this chapter by looking at the equation to be solved: 432 = (<em>x</em> + 26)(<em>x</em> + 2), you multiply the two binomials and move the 432 to the other side. Solving <em>x</em><sup>2</sup> + 28<em>x</em> – 380 = 0 you get <em>x</em> = 10 and <em>x</em> = –38. You disregard the <em>x</em> = –38, of course, and find that the original width was 10 feet. So, what was the question? It asks for the length of the new, bigger garden. The length of the new garden is found with <em>y</em> = <em>x</em> + 26. So, the length (and answer) is that the length is 36 feet. Does this check? If you use the new length or 36 and the new width of <em>x</em> + 2 = 12 and multiply 36 times 12 you get 432 square feet. It checks!\r\n<h2 id=\"tab8\" >Practice Plenty of Problems</h2>\r\nYou’re not born with the knowledge of how to ride a bike, play baseball, or even speak. The way you get better at challenging tasks is to practice, practice, practice. And the best way to practice math is to work the problems. You can seek harder or more complicated examples of questions that will stretch your brain and make you better at a concept the next time you see it.\r\n<p class=\"article-tips tip\">Along with working along on the example problems in this book, you can take advantage of the For Dummies workbooks, which include loads of practice exercises. Check out <a href=\"//coursofppt.com/book/academics-the-arts/math/trigonometry/trigonometry-workbook-for-dummies-282641/\"><em>Trigonometry Workbook For Dummies,</em> </a>by Mary Jane Sterling, <a href=\"//coursofppt.com/book/academics-the-arts/math/algebra/algebra-i-workbook-for-dummies-281939/\"><em>Algebra Workbook</em></a> and <a href=\"//coursofppt.com/book/academics-the-arts/math/algebra/algebra-ii-workbook-for-dummies-3rd-edition-281941/\"><em>Algebra II Workbook For Dummies</em></a>, both also by Mary Jane Sterling, and <a href=\"//coursofppt.com/book/academics-the-arts/math/geometry/geometry-workbook-for-dummies-292500/\"><em>Geometry Workbook For Dummies</em></a>, by Mark Ryan (all published by Wiley), to name a few.</p>\r\nEven a math textbook is great for practice. Why not try some (gulp!) problems that weren’t assigned, or maybe go back to an old section to review and make sure you’ve still got it? Typically, textbooks show the answers to the odd problems, so if you stick with those you can always double-check your answers. And if you get a craving for some extra practice, just search the Internet for “practice math problems” to see what you can find! For example, if you search the Internet for “practice systems of equations problems” you’ll find more than a million hits. That’s a lot of practice!\r\n<h2 id=\"tab9\" >Keep Track of the Order of Operations</h2>\r\nDon’t fall for the trap that always is lying there by performing operations in the wrong order. For instance, 2 – 6 × 3 doesn’t become –4 × 3 = –12. You’ll reach those incorrect answers if you forget to do the multiplication first. Focus on following the order of PEMDAS every time, all the time:\r\n<blockquote><strong>P</strong>arentheses (and other grouping devices)\r\n\r\n<strong>E</strong>xponents and roots\r\n\r\n<strong>M</strong>ultiplication and <strong>D</strong>ivision from left to right\r\n\r\n<strong>A</strong>ddition and <strong>S</strong>ubtraction from left to right</blockquote>\r\nDon’t ever go out of order, and that’s an order!\r\n<h2 id=\"tab10\" >Use Caution When Dealing with Fractions</h2>\r\nWorking with denominators can be tricky. It’s okay to write:\r\n\r\n<img class=\"alignnone size-full wp-image-260190\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-denominator.jpg\" alt=\"pre-calculus-denominator\" width=\"272\" height=\"86\" />\r\n\r\nBut, on the other hand: <img class=\"alignnone size-full wp-image-260191\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-denominators2.jpg\" alt=\"pre-calculus-denominators2\" width=\"272\" height=\"78\" />\r\n\r\nAlso, reducing or cancelling in fractions can be performed incorrectly. Every term in the numerator has to be divided by the same factor — the one that divides the denominator. So, it’s true that:\r\n\r\n<img class=\"alignnone size-full wp-image-260192\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-fraction.jpg\" alt=\"pre-calculus-fraction\" width=\"528\" height=\"86\" />because <img class=\"alignnone size-full wp-image-260193\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-fraction2.jpg\" alt=\"pre-calculus-fraction2\" width=\"612\" height=\"135\" />\r\n\r\nBut <img class=\"alignnone size-full wp-image-260194\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-fraction3.jpg\" alt=\"pre-calculus-fraction3\" width=\"510\" height=\"86\" />, because the factor being divided out is just 4, not 4<em>x</em>.\r\n\r\nAnd, again, it has to be the same factor that’s dividing, throughout. Can you spot the error here?\r\n\r\n<img class=\"alignnone size-full wp-image-260195\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-fraction4.jpg\" alt=\"pre-calculus-fraction4\" width=\"319\" height=\"104\" />\r\n\r\nSome poor soul reduced each term and the term directly above it — separately. Nope, doesn’t work that way. The correct process is to factor the trinomials and then divide by the common factor:\r\n\r\n<img class=\"alignnone size-full wp-image-260196\" src=\"//coursofppt.com/wp-content/uploads/pre-calculus-fraction5.jpg\" alt=\"pre-calculus-fraction5\" width=\"596\" height=\"112\" />","blurb":"","authors":[{"authorId":9703,"name":"Yang Kuang","slug":"yang-kuang","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9703"}},{"authorId":9704,"name":"Elleyne Kase","slug":"elleyne-kase","description":"","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/9704"}}],"primaryCategoryTaxonomy":{"categoryId":33727,"title":"Pre-Calculus","slug":"pre-calculus","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33727"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Figure Out What the Problem Is Asking","target":"#tab1"},{"label":"Draw Pictures (the More the Better)","target":"#tab2"},{"label":"Plan Your Attack — Identify Your Targets","target":"#tab3"},{"label":"Write Down Any Formulas","target":"#tab4"},{"label":"Show Each Step of Your Work","target":"#tab5"},{"label":"Know When to Quit","target":"#tab6"},{"label":"Check Your Answers","target":"#tab7"},{"label":"Practice Plenty of Problems","target":"#tab8"},{"label":"Keep Track of the Order of Operations","target":"#tab9"},{"label":"Use Caution When Dealing with Fractions","target":"#tab10"}],"relatedArticles":{"fromBook":[{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/260218"}},{"articleId":260215,"title":"The Differences between Pre-Calculus and Calculus","slug":"the-differences-between-pre-calculus-and-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/260215"}},{"articleId":260207,"title":"10 Polar Graphs","slug":"10-polar-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/260207"}},{"articleId":208308,"title":"Pre-Calculus For Dummies Cheat Sheet","slug":"pre-calculus-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208308"}},{"articleId":192147,"title":"How to Find Local Extrema with the First Derivative Test","slug":"how-to-find-local-extrema-with-the-first-derivative-test","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/192147"}}],"fromCategory":[{"articleId":262884,"title":"10 Pre-Calculus Missteps to Avoid","slug":"10-pre-calculus-missteps-to-avoid","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/262884"}},{"articleId":262851,"title":"Pre-Calculus Review of Real Numbers","slug":"pre-calculus-review-of-real-numbers","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/262851"}},{"articleId":262837,"title":"Fundamentals of Pre-Calculus","slug":"fundamentals-of-pre-calculus","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/262837"}},{"articleId":262652,"title":"Complex Numbers and Polar Coordinates","slug":"complex-numbers-and-polar-coordinates","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/262652"}},{"articleId":260218,"title":"Special Function Types and Their Graphs","slug":"special-function-types-and-their-graphs","categoryList":["academics-the-arts","math","pre-calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/260218"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282496,"slug":"pre-calculus-for-dummies-3rd-edition","isbn":"9781119508779","categoryList":["academics-the-arts","math","pre-calculus"],"amazon":{"default":"//www.amazon.com/gp/product/1119508770/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119508770/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/1119508770-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119508770/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119508770/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/pre-calculus-for-dummies-3rd-edition-cover-9781119508779-203x255.jpg","width":203,"height":255},"title":"Pre-Calculus For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> aught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois for more than 30 years. She is the author of several <i>For Dummies books,</i> including <i>Algebra Workbook For Dummies, Algebra II For Dummies,</i> and <i>Algebra II Workbook For Dummies.</i> </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> taught algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 30 years. She is the author of <i>Trigonometry For Dummies</i> and <i>Finite Math For Dummies.</i> ","hasArticle":false,"_links":{"self":"//dummies-api.coursofppt.com/v2/authors/8985"}}],"_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;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508779&quot;]}]\" id=\"du-slot-64da960fcea07\"></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;pre-calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119508779&quot;]}]\" id=\"du-slot-64da960fcef7d\"></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":"One year","lifeExpectancySetFrom":"2024-08-14T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":260183},{"headers":{"creationTime":"2017-03-26T21:05:45+00:00","modifiedTime":"2024-08-09T16:18:34+00:00","timestamp":"2024-08-09T18:01:03+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":"Calculus","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33723"},"slug":"calculus","categoryId":33723}],"title":"How to Find the Area of a Surface of Revolution","strippedTitle":"how to find the area of a surface of revolution","slug":"how-to-find-the-area-of-a-surface-of-revolution","canonicalUrl":"","归类领头羊SEO调整":{"metaDescription":"The calculus instructions in this article and video show you how to find the area of a surface of revolution.","noIndex":0,"noFollow":0},"content":"A surface of revolution is a three-dimensional surface with circular cross sections, like a vase or a bell, or a wine bottle. This article, and the video, show you how to find its area.\r\n<div class=\"x2 x2-top\">\r\n\r\n<div class=\"video-player-organism\"></div>\r\n\r\n</div>\r\nFor these problems, you divide the surface into narrow circular bands, figure the surface area of a representative band, and then just add up the areas of all the bands to get the total surface area. The following figure shows such a shape with a representative band.\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/219842.image0.jpg\" alt=\"image0.jpg\" width=\"400\" height=\"225\" />\r\n\r\nWhat’s the surface area of a representative band? Well, if you cut the band and unroll it, you get sort of a long, narrow rectangle whose area, of course, is <i>length</i> times <i>width</i>.\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/219843.image1.png\" alt=\"image1.png\" width=\"503\" height=\"221\" />\r\n\r\n<b>Surface of Revolution:</b><b> </b>A surface generated by revolving a function, <i>y</i> = <i>f</i> (<i>x</i>), about an axis has a surface area — between <i>a</i> and <i>b</i> — given by the following integral:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/219844.image2.png\" alt=\"image2.png\" width=\"268\" height=\"59\" />\r\n\r\nBy the way, in the above explanation, you might be wondering why the width of the rectangular band is\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/219845.image3.png\" alt=\"image3.png\" width=\"207\" height=\"56\" />\r\n\r\nIt’s because the little band width is slanted instead of horizontal (in which case it would be just <i>dx</i>). The fact that it’s slanted makes it work like the hypotenuse of a little right triangle. The fancy-looking expression for the width of the band comes from working out the length of this hypotenuse with the Pythagorean Theorem. That should make you feel a lot better!\r\n\r\nIf the axis of revolution is the <i>x</i>-axis, <i>r</i> will equal <i>f</i> (<i>x</i>) — as shown in the above figure. If the axis of revolution is some other line, like <i>y</i> = 5, it’s a bit more complicated — something to look forward to.\r\n\r\nNow try a problem: What’s the surface area — between <i>x</i> = 1 and <i>x</i> = 2 — of the surface generated by revolving\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/219846.image4.png\" alt=\"image4.png\" width=\"45\" height=\"27\" />\r\n\r\nabout the <i>x</i>-axis?\r\n<div class=\"imageBlock\" style=\"width: 314px;\">\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/219847.image5.jpg\" alt=\"A surface of revolution — this one’s shaped sort of like the end of a trumpet.\" width=\"314\" height=\"400\" />\r\n<div class=\"imageCaption\">A surface of revolution — this one’s shaped sort of like the end of a trumpet.</div>\r\n</div>\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Take the derivative of your function.</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/219848.image6.png\" alt=\"image6.png\" width=\"63\" height=\"67\" />\r\n<p class=\"child-para\">Now you can finish the problem by just plugging everything into the formula, but you should do it step by step to reinforce the idea that whenever you integrate, you write down a representative little bit of something — that’s the integrand — then you add up all the little bits by integrating.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Figure the surface area of a representative narrow band.</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/219849.image7.png\" alt=\"image7.png\" width=\"500\" height=\"128\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Add up the areas of all the bands from 1 to 2 by integrating.</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/219850.image8.png\" alt=\"image8.png\" width=\"473\" height=\"461\" /></li>\r\n</ol>","description":"A surface of revolution is a three-dimensional surface with circular cross sections, like a vase or a bell, or a wine bottle. This article, and the video, show you how to find its area.\r\n<div class=\"x2 x2-top\">\r\n\r\n<div class=\"video-player-organism\"></div>\r\n\r\n</div>\r\nFor these problems, you divide the surface into narrow circular bands, figure the surface area of a representative band, and then just add up the areas of all the bands to get the total surface area. The following figure shows such a shape with a representative band.\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/219842.image0.jpg\" alt=\"image0.jpg\" width=\"400\" height=\"225\" />\r\n\r\nWhat’s the surface area of a representative band? Well, if you cut the band and unroll it, you get sort of a long, narrow rectangle whose area, of course, is <i>length</i> times <i>width</i>.\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/219843.image1.png\" alt=\"image1.png\" width=\"503\" height=\"221\" />\r\n\r\n<b>Surface of Revolution:</b><b> </b>A surface generated by revolving a function, <i>y</i> = <i>f</i> (<i>x</i>), about an axis has a surface area — between <i>a</i> and <i>b</i> — given by the following integral:\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/219844.image2.png\" alt=\"image2.png\" width=\"268\" height=\"59\" />\r\n\r\nBy the way, in the above explanation, you might be wondering why the width of the rectangular band is\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/219845.image3.png\" alt=\"image3.png\" width=\"207\" height=\"56\" />\r\n\r\nIt’s because the little band width is slanted instead of horizontal (in which case it would be just <i>dx</i>). The fact that it’s slanted makes it work like the hypotenuse of a little right triangle. The fancy-looking expression for the width of the band comes from working out the length of this hypotenuse with the Pythagorean Theorem. That should make you feel a lot better!\r\n\r\nIf the axis of revolution is the <i>x</i>-axis, <i>r</i> will equal <i>f</i> (<i>x</i>) — as shown in the above figure. If the axis of revolution is some other line, like <i>y</i> = 5, it’s a bit more complicated — something to look forward to.\r\n\r\nNow try a problem: What’s the surface area — between <i>x</i> = 1 and <i>x</i> = 2 — of the surface generated by revolving\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/219846.image4.png\" alt=\"image4.png\" width=\"45\" height=\"27\" />\r\n\r\nabout the <i>x</i>-axis?\r\n<div class=\"imageBlock\" style=\"width: 314px;\">\r\n\r\n<img src=\"//coursofppt.com/wp-content/uploads/219847.image5.jpg\" alt=\"A surface of revolution — this one’s shaped sort of like the end of a trumpet.\" width=\"314\" height=\"400\" />\r\n<div class=\"imageCaption\">A surface of revolution — this one’s shaped sort of like the end of a trumpet.</div>\r\n</div>\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Take the derivative of your function.</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/219848.image6.png\" alt=\"image6.png\" width=\"63\" height=\"67\" />\r\n<p class=\"child-para\">Now you can finish the problem by just plugging everything into the formula, but you should do it step by step to reinforce the idea that whenever you integrate, you write down a representative little bit of something — that’s the integrand — then you add up all the little bits by integrating.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Figure the surface area of a representative narrow band.</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/219849.image7.png\" alt=\"image7.png\" width=\"500\" height=\"128\" /></li>\r\n \t<li>\r\n<p class=\"first-para\">Add up the areas of all the bands from 1 to 2 by integrating.</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/219850.image8.png\" alt=\"image8.png\" width=\"473\" height=\"461\" /></li>\r\n</ol>","blurb":"","authors":[],"primaryCategoryTaxonomy":{"categoryId":33723,"title":"Calculus","slug":"calculus","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33723"}},"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":256336,"title":"Solve a Difficult Limit Problem Using the Sandwich Method","slug":"solve-a-difficult-limit-problem-using-the-sandwich-method","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/256336"}},{"articleId":255765,"title":"Solve Limit Problems on a Calculator Using Graphing Mode","slug":"solve-limit-problems-on-a-calculator-using-graphing-mode","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/255765"}},{"articleId":255755,"title":"Solve Limit Problems on a Calculator Using the Arrow-Number","slug":"solve-limit-problems-on-a-calculator-using-the-arrow-number","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/255755"}},{"articleId":255261,"title":"Limit and Continuity Graphs: Practice Questions","slug":"limit-and-continuity-graphs-practice-questions","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/255261"}},{"articleId":255255,"title":"Use the Vertical Line Test to Identify a Function","slug":"use-the-vertical-line-test-to-identify-a-function","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/255255"}}]},"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;calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-64d3d45f0a144\"></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;calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[null]}]\" id=\"du-slot-64d3d45f0a6b6\"></div></div>"},"articleType":{"articleType":"Videos","articleList":null,"content":null,"videoInfo":{"videoId":"694655204001","name":"How to Find the Surface Area of a Surface Revolution","accountId":"622696558001","playerId":"default","thumbnailUrl":"//cf-images.us-east-1.prod.boltdns.net/v1/static/622696558001/76f0595c-da6a-42d9-ac74-6f5d34e4edc9/a0a533ae-7b93-47e8-a41c-befd77598689/160x90/match/image.jpg","description":"The surface area of a surface of revolution applies to many three-dimensional, radially symmetrical shapes. Formulas in this calculus video tutorial reveal how to estimate, measure, and solve for the surface area of a three-dimensional object like a","uploadDate":"2023-07-15T08:18:07.281Z"}},"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":"2024-08-09T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":190931},{"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":"","归类领头羊SEO调整":{"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":"2017-03-26T18:17:12+00:00","modifiedTime":"2024-07-14T13:05:10+00:00","timestamp":"2024-07-14T15:01:02+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":"Calculus","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33723"},"slug":"calculus","categoryId":33723}],"title":"Finding the Area of a Surface of Revolution","strippedTitle":"finding the area of a surface of revolution","slug":"finding-the-area-of-a-surface-of-revolution","canonicalUrl":"","归类领头羊SEO调整":{"metaDescription":"The nice thing about finding the area of a surface of revolution is that there’s a formula you can use. Memorize it and you’re halfway done. To find the area of","noIndex":0,"noFollow":0},"content":"<p>The nice thing about finding the area of a surface of revolution is that there’s a formula you can use. Memorize it and you’re halfway done.</p>\r\n<p>To find the area of a surface of revolution between <i>a</i> and <i>b</i><i>,</i> watch this video tutorial or follow the steps below:</p>\r\n<div class='x2 x2-top'><div class=\"video-player-organism\"></div></div>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314778.image0.png\" width=\"153\" height=\"57\" alt=\"image0.png\"/>\r\n<p>This formula looks long and complicated, but it makes more sense when you spend a minute thinking about it. The integral is made from two pieces:</p>\r\n<ul class=\"level-one\">\r\n <li><p class=\"first-para\">The arc-length formula, which measures the length along the surface </p>\r\n </li>\r\n <li><p class=\"first-para\">The formula for the circumference of a circle, which measures the length around the surface</p>\r\n </li>\r\n</ul>\r\n<p>So multiplying these two pieces together is similar to multiplying length and width to find the area of a rectangle. In effect, the formula allows you to measure surface area as an infinite number of little rectangles.</p>\r\n<p>When you’re measuring the surface of revolution of a function <i>f</i>(<i>x</i>) around the <i>x</i>-axis, substitute <i>r</i> = <i>f</i>(<i>x</i>) into the formula:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314779.image1.png\" width=\"188\" height=\"53\" alt=\"image1.png\"/>\r\n<p>For example, suppose that you want to find the area of revolution that’s shown in this figure.</p>\r\n<div class=\"imageBlock\" style=\"width:286px;\"><img src=\"//coursofppt.com/wp-content/uploads/314780.image2.jpg\" width=\"286\" height=\"400\" alt=\"Measuring the surface of revolution of <i>y</i> = <i>x</i><sup>3</sup> between <i>x</i> = 0 and <i>\"/><div class=\"imageCaption\">Measuring the surface of revolution of <i>y</i> = <i>x</i><sup>3</sup> between <i>x</i> = 0 and <i>x</i> = 1.</div></div>\r\n<p>To solve this problem, first note that for </p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314781.image3.png\" width=\"149\" height=\"29\" alt=\"image3.png\"/>\r\n<p>So set up the problem as follows:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314782.image4.png\" width=\"168\" height=\"52\" alt=\"image4.png\"/>\r\n<p>To start off, simplify the problem a bit:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314783.image5.png\" width=\"135\" height=\"52\" alt=\"image5.png\"/>\r\n<p>You can solve this problem by using the following variable substitution:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314784.image6.png\" width=\"100\" height=\"51\" alt=\"image6.png\"/>\r\n<p>Now substitute <i>u</i> for 1+ 9<i>x</i><sup>4</sup> and </p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314785.image7.png\" width=\"41\" height=\"37\" alt=\"image7.png\"/>\r\n<p>for <i>x</i><sup>3</sup> <i>dx </i>into the equation:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314786.image8.png\" width=\"112\" height=\"52\" alt=\"image8.png\"/>\r\n<p>Notice that you change the limits of integration: When <i>x</i> = 0, <i>u</i> = 1. And when <i>x</i> = 1, <i>u</i> = 10.</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314787.image9.png\" width=\"96\" height=\"52\" alt=\"image9.png\"/>\r\n<p>Now you can perform the integration:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314788.image10.png\" width=\"117\" height=\"119\" alt=\"image10.png\"/>\r\n<p>Finally, evaluate the definite integral:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314789.image11.png\" width=\"136\" height=\"115\" alt=\"image11.png\"/>","description":"<p>The nice thing about finding the area of a surface of revolution is that there’s a formula you can use. Memorize it and you’re halfway done.</p>\r\n<p>To find the area of a surface of revolution between <i>a</i> and <i>b</i><i>,</i> watch this video tutorial or follow the steps below:</p>\r\n<div class='x2 x2-top'><div class=\"video-player-organism\"></div></div>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314778.image0.png\" width=\"153\" height=\"57\" alt=\"image0.png\"/>\r\n<p>This formula looks long and complicated, but it makes more sense when you spend a minute thinking about it. The integral is made from two pieces:</p>\r\n<ul class=\"level-one\">\r\n <li><p class=\"first-para\">The arc-length formula, which measures the length along the surface </p>\r\n </li>\r\n <li><p class=\"first-para\">The formula for the circumference of a circle, which measures the length around the surface</p>\r\n </li>\r\n</ul>\r\n<p>So multiplying these two pieces together is similar to multiplying length and width to find the area of a rectangle. In effect, the formula allows you to measure surface area as an infinite number of little rectangles.</p>\r\n<p>When you’re measuring the surface of revolution of a function <i>f</i>(<i>x</i>) around the <i>x</i>-axis, substitute <i>r</i> = <i>f</i>(<i>x</i>) into the formula:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314779.image1.png\" width=\"188\" height=\"53\" alt=\"image1.png\"/>\r\n<p>For example, suppose that you want to find the area of revolution that’s shown in this figure.</p>\r\n<div class=\"imageBlock\" style=\"width:286px;\"><img src=\"//coursofppt.com/wp-content/uploads/314780.image2.jpg\" width=\"286\" height=\"400\" alt=\"Measuring the surface of revolution of <i>y</i> = <i>x</i><sup>3</sup> between <i>x</i> = 0 and <i>\"/><div class=\"imageCaption\">Measuring the surface of revolution of <i>y</i> = <i>x</i><sup>3</sup> between <i>x</i> = 0 and <i>x</i> = 1.</div></div>\r\n<p>To solve this problem, first note that for </p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314781.image3.png\" width=\"149\" height=\"29\" alt=\"image3.png\"/>\r\n<p>So set up the problem as follows:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314782.image4.png\" width=\"168\" height=\"52\" alt=\"image4.png\"/>\r\n<p>To start off, simplify the problem a bit:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314783.image5.png\" width=\"135\" height=\"52\" alt=\"image5.png\"/>\r\n<p>You can solve this problem by using the following variable substitution:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314784.image6.png\" width=\"100\" height=\"51\" alt=\"image6.png\"/>\r\n<p>Now substitute <i>u</i> for 1+ 9<i>x</i><sup>4</sup> and </p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314785.image7.png\" width=\"41\" height=\"37\" alt=\"image7.png\"/>\r\n<p>for <i>x</i><sup>3</sup> <i>dx </i>into the equation:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314786.image8.png\" width=\"112\" height=\"52\" alt=\"image8.png\"/>\r\n<p>Notice that you change the limits of integration: When <i>x</i> = 0, <i>u</i> = 1. And when <i>x</i> = 1, <i>u</i> = 10.</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314787.image9.png\" width=\"96\" height=\"52\" alt=\"image9.png\"/>\r\n<p>Now you can perform the integration:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314788.image10.png\" width=\"117\" height=\"119\" alt=\"image10.png\"/>\r\n<p>Finally, evaluate the definite integral:</p>\r\n<img src=\"//coursofppt.com/wp-content/uploads/314789.image11.png\" width=\"136\" height=\"115\" alt=\"image11.png\"/>","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":33723,"title":"Calculus","slug":"calculus","_links":{"self":"//dummies-api.coursofppt.com/v2/categories/33723"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":208670,"title":"Calculus II For Dummies Cheat Sheet","slug":"calculus-ii-for-dummies-cheat-sheet","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/208670"}},{"articleId":179236,"title":"Computing Integrals and Representing Integrals as Functions","slug":"computing-integrals-and-representing-integrals-as-functions","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/179236"}},{"articleId":179235,"title":"Drawing with 3-D Cartesian Coordinates","slug":"drawing-with-3-d-cartesian-coordinates","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/179235"}},{"articleId":179234,"title":"Evaluating Triple Integrals","slug":"evaluating-triple-integrals","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/179234"}},{"articleId":179233,"title":"Find the Area Between Two Functions","slug":"find-the-area-between-two-functions","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/179233"}}],"fromCategory":[{"articleId":256336,"title":"Solve a Difficult Limit Problem Using the Sandwich Method","slug":"solve-a-difficult-limit-problem-using-the-sandwich-method","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/256336"}},{"articleId":255765,"title":"Solve Limit Problems on a Calculator Using Graphing Mode","slug":"solve-limit-problems-on-a-calculator-using-graphing-mode","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/255765"}},{"articleId":255755,"title":"Solve Limit Problems on a Calculator Using the Arrow-Number","slug":"solve-limit-problems-on-a-calculator-using-the-arrow-number","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/255755"}},{"articleId":255261,"title":"Limit and Continuity Graphs: Practice Questions","slug":"limit-and-continuity-graphs-practice-questions","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/255261"}},{"articleId":255255,"title":"Use the Vertical Line Test to Identify a Function","slug":"use-the-vertical-line-test-to-identify-a-function","categoryList":["academics-the-arts","math","calculus"],"_links":{"self":"//dummies-api.coursofppt.com/v2/articles/255255"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282046,"slug":"calculus-ii-for-dummies-2nd-edition","isbn":"9781119986614","categoryList":["academics-the-arts","math","calculus"],"amazon":{"default":"//www.amazon.com/gp/product/1119986613/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119986613/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/1119986613-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119986613/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119986613/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//coursofppt.com/wp-content/uploads/calculus-ii-for-dummies-3rd-edition-cover-1119986613-203x255.jpg","width":203,"height":255},"title":"Calculus II 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;calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119986614&quot;]}]\" id=\"du-slot-64b1632edf82d\"></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;calculus&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119986614&quot;]}]\" id=\"du-slot-64b1632ee0108\"></div></div>"},"articleType":{"articleType":"Videos","articleList":null,"content":null,"videoInfo":{"videoId":"694655204001","name":"How to Find the Surface Area of a Surface Revolution","accountId":"622696558001","playerId":"default","thumbnailUrl":"//cf-images.us-east-1.prod.boltdns.net/v1/static/622696558001/76f0595c-da6a-42d9-ac74-6f5d34e4edc9/a0a533ae-7b93-47e8-a41c-befd77598689/160x90/match/image.jpg","description":"The surface area of a surface of revolution applies to many three-dimensional, radially symmetrical shapes. Formulas in this calculus video tutorial reveal how to estimate, measure, and solve for the surface area of a three-dimensional object like a","uploadDate":"2023-07-15T08:18:07.281Z"}},"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":"2024-07-13T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":178407},{"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":"","归类领头羊SEO调整":{"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":"","归类领头羊SEO调整":{"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":"","归类领头羊SEO调整":{"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. 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