{"id":5412,"date":"2014-12-31T09:00:35","date_gmt":"2014-12-31T17:00:35","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=5412"},"modified":"2014-12-22T22:25:48","modified_gmt":"2014-12-23T06:25:48","slug":"using-diagrams-to-solve-gmat-rate-problems-part-1","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/using-diagrams-to-solve-gmat-rate-problems-part-1\/","title":{"rendered":"Using Diagrams to Solve GMAT Rate Problems: Part 1"},"content":{"rendered":"<p>Diagrams are great! Like all types of scratch-work, diagrams can forestall cognitive fatigue because working a problem out on paper is much less demanding than doing all the work in your head.  Diagrams can also help you to visualize relationships, and can make problems more concrete. <\/p>\n<p>Generally though, we use diagrams to generate equations, which we still have to solve.  So good scratch-work doesn\u2019t substitute for other mathematical ability. <\/p>\n<p>Today we\u2019ll use speed problems as examples, since they\u2019re the most common sort of <a href=\"https:\/\/magoosh.com\/gmat\/rates-and-ratios\/\">GMAT rate problem<\/a>, but we\u2019ll talk about work-rate and other problems in a couple of days. <\/p>\n<p>Let\u2019s jump into an example!<br \/>\n&nbsp;<\/p>\n<h2>This is a pretty stock example:<\/h2>\n<ul>\n<li>It involves exactly two travelers.<\/li>\n<li>Those travelers are moving in opposite directions. In this case they\u2019re moving apart, but in other problems they\u2019ll be moving toward one another.<\/li>\n<li>Those travelers are moving simultaneously, at least for the portion of their travels that concern us.<\/li>\n<\/ul>\n<p>Train X and train Y pass one another traveling in opposite directions. Forty minutes later they are 100 miles apart. If train X\u2019s constant speed is 30 miles per hour greater than train Y\u2019s, how far does train X travel during that time?<\/p>\n<p>A. 28<br \/>\nB. 40<br \/>\nC. 60<br \/>\nD. 72<br \/>\nE. 80<\/p>\n<p>The diagram we\u2019ll use for this problem is a RTD Table. It consists simply of three columns (for rate, time, and distance) and some number of rows. For this problem we\u2019ll use three rows, one for each of the trains and one for their combined rate:<\/p>\n<p><center><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0601-225x300.jpg\" alt=\"IMG_0601\" width=\"255\" height=\"300\" class=\"alignnone size-medium wp-image-5413\" \/><\/center><\/p>\n<p>If you like, you can include multiplication and equality signs, to remind yourself how the values in each row are related to one another: RT=D. I\u2019ve included those signs below in red, but in subsequent illustrations of the table I\u2019m going to leave them out.  I will include one more thing though. Circle or otherwise indicate the cell for which we\u2019re solving. In this case, that\u2019s X\u2019s distance.<\/p>\n<p><center><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0602-225x300.jpg\" alt=\"IMG_0602\" width=\"225\" height=\"300\" class=\"alignnone size-medium wp-image-5414\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0602-225x300.jpg 225w, https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0602-768x1024.jpg 768w\" sizes=\"(max-width: 225px) 100vw, 225px\" \/><\/center><\/p>\n<p>Begin by writing in whatever information is given to you. In this case that would include the combined distance traveled by the two trains, and the time traveled.<\/p>\n<p><center><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0603-225x300.jpg\" alt=\"IMG_0603\" width=\"225\" height=\"300\" class=\"alignnone size-medium wp-image-5415\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0603-225x300.jpg 225w, https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0603-768x1024.jpg 768w\" sizes=\"(max-width: 225px) 100vw, 225px\" \/><\/center><\/p>\n<p>A few notes: (1) We didn\u2019t include a distance for either train X or train Y, because we\u2019re not given those. (2) For every story about simultaneous travel, the time value will be the same all the way down the column.  (3) In this case that time value is 2\/3 of an hour. Yes, it was given to us as 40 minutes, but since the speed is given to us in miles per hour, we need to change something to keep the units consistent.  It\u2019s often easier to change from a few large units (like hours) to many small units (like minutes) but in this case I chose to stick with hours. (4) If you\u2019re consistent in your units, and if the units in the table are the same as in the answers, units don\u2019t really matter any more. In math class, you\u2019d probably represent the units, but I\u2019m not going to bother.<\/p>\n<p>Also capture the rate for each of the trains, using as few variables as possible. In this case, that might mean calling train X\u2019s rate simply r and calling train Y\u2019s rate r-30, since we\u2019re given that train X\u2019s constant speed is 30 miles per hour greater than train Y\u2019s.  (You could also have called train Y\u2019s rate r and train X\u2019s rate r+30.)<\/p>\n<p><center><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0604-225x300.jpg\" alt=\"IMG_0604\" width=\"225\" height=\"300\" class=\"alignnone size-medium wp-image-5416\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0604-225x300.jpg 225w, https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0604-768x1024.jpg 768w\" sizes=\"(max-width: 225px) 100vw, 225px\" \/><\/center><\/p>\n<p>Next add the rates of the two trains to get their combined rate. Every time that you have a story about two travelers moving in opposite directions (whether toward each other or away from each other) and moving simultaneously (starting and stopping at the same time, or both moving for the entire portion of the journey that you\u2019re concerned with) add the rates.<\/p>\n<p><center><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0605-225x300.jpg\" alt=\"IMG_0605\" width=\"225\" height=\"300\" class=\"alignnone size-medium wp-image-5417\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0605-225x300.jpg 225w, https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0605-768x1024.jpg 768w\" sizes=\"(max-width: 225px) 100vw, 225px\" \/><\/center><\/p>\n<p>We now have a complete equation in the bottom row. Since RT=D, we can see that (2r-30)(2\/3)=100. Solve for r:<br \/>\n(2r-30)(2\/3)=100<br \/>\n(4r\/3)-20=100<br \/>\n4r\/3=120<br \/>\n4r=360<br \/>\nr=90<\/p>\n<p>Notice that the diagram merely generated this equation for us; it didn\u2019t solve the equation.  Of course, our ultimate goal is not to solve for r but to solve for the distance traveled by X. Substitute 90 for r and complete the row for X.<\/p>\n<p><center><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0606-225x300.jpg\" alt=\"IMG_0606\" width=\"225\" height=\"300\" class=\"alignnone size-medium wp-image-5418\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0606-225x300.jpg 225w, https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0606-768x1024.jpg 768w\" sizes=\"(max-width: 225px) 100vw, 225px\" \/><\/center><br \/>\n&nbsp;<\/p>\n<h2>Couldn\u2019t we do that without the table?<\/h2>\n<p>Yep. We could have simply noted that for any simultaneous travel problem (combined rate)(time)=(combined distance). We\u2019re given the time (2\/3) and combined distance (100 miles). If we can manage to represent the combined rates (2r-30) we should be able to solve for r. If we can also characterize what we\u2019re solving for (2\/3 of r) then we can answer the question.<\/p>\n<p>Some people can do all of that without the table. <\/p>\n<p>On the other hand, some people who can do all of that without the table in practice find that they need help organizing and translating information once they\u2019re well into the test. So it\u2019s worthwhile learning to use the table.<\/p>\n<p>Another reason to learn to use the table is that some rate problems are a lot more subtle than this one. We\u2019ll look at such problems in a couple of days.<br \/>\n&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Diagrams are great! Like all types of scratch-work, diagrams can forestall cognitive fatigue because working a problem out on paper is much less demanding than doing all the work in your head. Diagrams can also help you to visualize relationships, and can make problems more concrete. Generally though, we use diagrams to generate equations, which [&hellip;]<\/p>\n","protected":false},"author":82,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[193],"tags":[],"ppma_author":[13222],"class_list":["post-5412","post","type-post","status-publish","format-standard","hentry","category-word-problems"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v21.7 (Yoast SEO v21.7) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Using Diagrams to Solve GMAT Rate Problems: Part 1 - Magoosh Blog \u2014 GMAT\u00ae Exam<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/magoosh.com\/gmat\/using-diagrams-to-solve-gmat-rate-problems-part-1\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Using Diagrams to Solve GMAT Rate Problems: Part 1\" \/>\n<meta property=\"og:description\" content=\"Diagrams are great! Like all types of scratch-work, diagrams can forestall cognitive fatigue because working a problem out on paper is much less demanding than doing all the work in your head. Diagrams can also help you to visualize relationships, and can make problems more concrete. Generally though, we use diagrams to generate equations, which [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/magoosh.com\/gmat\/using-diagrams-to-solve-gmat-rate-problems-part-1\/\" \/>\n<meta property=\"og:site_name\" content=\"Magoosh Blog \u2014 GMAT\u00ae Exam\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/MagooshGMAT\/\" \/>\n<meta property=\"article:published_time\" content=\"2014-12-31T17:00:35+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2014-12-23T06:25:48+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/magoosh.com\/gmat\/files\/2014\/12\/IMG_0601-225x300.jpg\" \/>\n<meta name=\"author\" content=\"Michael Schwartz\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@MagooshGMAT\" \/>\n<meta name=\"twitter:site\" content=\"@MagooshGMAT\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Michael Schwartz\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/magoosh.com\/gmat\/using-diagrams-to-solve-gmat-rate-problems-part-1\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/using-diagrams-to-solve-gmat-rate-problems-part-1\/\"},\"author\":{\"name\":\"Michael Schwartz\",\"@id\":\"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/95a95e68f2ccbf70a6bacf5a260eaacf\"},\"headline\":\"Using Diagrams to Solve GMAT Rate Problems: Part 1\",\"datePublished\":\"2014-12-31T17:00:35+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/using-diagrams-to-solve-gmat-rate-problems-part-1\/\"},\"wordCount\":884,\"commentCount\":8,\"publisher\":{\"@id\":\"https:\/\/magoosh.com\/gmat\/#organization\"},\"articleSection\":[\"GMAT Word Problems\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/magoosh.com\/gmat\/using-diagrams-to-solve-gmat-rate-problems-part-1\/\",\"url\":\"https:\/\/magoosh.com\/gmat\/using-diagrams-to-solve-gmat-rate-problems-part-1\/\",\"name\":\"Using Diagrams to Solve GMAT Rate Problems: Part 1 - 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He\u2019s earned multiple perfect scores on the GRE, GMAT, and LSAT. He\u2019d rather have perfect pitch or be able to run low 1:40s for the 800 meters, but you take what you get. He has decades of teaching and curriculum-development experience. One of these days he might finish his dissertation and collect that Ph.D. in philosophy. Might.\",\"url\":\"https:\/\/magoosh.com\/gmat\/author\/michaelschwartz\/\"}]}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Using Diagrams to Solve GMAT Rate Problems: Part 1 - Magoosh Blog \u2014 GMAT\u00ae Exam","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/magoosh.com\/gmat\/using-diagrams-to-solve-gmat-rate-problems-part-1\/","og_locale":"en_US","og_type":"article","og_title":"Using Diagrams to Solve GMAT Rate Problems: Part 1","og_description":"Diagrams are great! 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He\u2019s earned multiple perfect scores on the GRE, GMAT, and LSAT. He\u2019d rather have perfect pitch or be able to run low 1:40s for the 800 meters, but you take what you get. He has decades of teaching and curriculum-development experience. One of these days he might finish his dissertation and collect that Ph.D. in philosophy. Might.","url":"https:\/\/magoosh.com\/gmat\/author\/michaelschwartz\/"}]}},"authors":[{"term_id":13222,"user_id":82,"is_guest":0,"slug":"michaelschwartz","display_name":"Michael Schwartz","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/d93dd6a3ce8134de72c26b5762ae2241005cd9e7537876e27dab68d9c44b2ae8?s=96&d=mm&r=g","user_url":"","last_name":"Schwartz","first_name":"Michael","description":"Michael Schwartz is really good at standardized tests. He\u2019s earned multiple perfect scores on the GRE, GMAT, and LSAT. He\u2019d rather have perfect pitch or be able to run low 1:40s for the 800 meters, but you take what you get. He has decades of teaching and curriculum-development experience. One of these days he might finish his dissertation and collect that Ph.D. in philosophy. 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