{"id":5447,"date":"2015-01-14T09:00:57","date_gmt":"2015-01-14T17:00:57","guid":{"rendered":"https:\/\/magoosh.com\/gmat\/?p=5447"},"modified":"2020-01-15T10:48:21","modified_gmt":"2020-01-15T18:48:21","slug":"one-more-rtd-table-problem-average-rates-part-1","status":"publish","type":"post","link":"https:\/\/magoosh.com\/gmat\/one-more-rtd-table-problem-average-rates-part-1\/","title":{"rendered":"One More RTD Table Problem: Average Rates Part 1"},"content":{"rendered":"<p>My last several posts have been devoted to the use of a table to answer rate problems. Today\u2019s post will assume familiarity with that table, so please take a look back at these posts is you\u2019re not already familiar with the RTD table:<\/p>\n<ul>\n<li><a href=\"https:\/\/magoosh.com\/gmat\/using-diagrams-to-solve-gmat-rate-problems-part-1\/\">Using Diagrams to Solve Rate Problems: Part 1<\/a><\/li>\n<li><a href=\"https:\/\/magoosh.com\/gmat\/using-diagrams-to-solve-gmat-rate-problems-part-2\/\">Using Diagrams to Solve Rate Problems: Part 2<\/a><\/li>\n<li><a href=\"https:\/\/magoosh.com\/gmat\/a-different-use-of-the-rtd-table-part-1\/\">A Different Use of the RTD Table: Part 1<\/a><\/li>\n<li><a href=\"https:\/\/magoosh.com\/gmat\/a-different-use-of-the-rtd-table-part-2\/\">A Different Use of the RTD Table: Part 2<\/a><\/li>\n<li><a href=\"https:\/\/magoosh.com\/gmat\/using-the-rtd-table-for-a-complicated-problem\/\">Using the RTD Table for a Complicated Problem<\/a><\/li>\n<\/ul>\n<p>Here\u2019s today\u2019s problem:<\/p>\n<p><em>Div\u2019s bicycle tour consists of three legs of equal length.\u00a0 For the first leg Div averaged 16 kilometers per hour. For the second leg he averaged 24 kilometers per hour. What speed must Div average for the final leg in order to average 24 kilometers per hour for the entire tour? <\/em><\/p>\n<p><em>A) 20 kilometers per hour<br \/>\n<\/em><em>B) 28 kilometers per hour<br \/>\n<\/em><em>C) 32 kilometers per hour<br \/>\n<\/em><em>D) 40 kilometers per hour<br \/>\n<\/em><em>E) 48 kilometers per hour<\/em><\/p>\n<h2>WARNING!<\/h2>\n<p>I usually start with the bare outline of an RTD table, and then fill it in bit-by-bit. Today, though, I\u2019m going to start with a warning.\u00a0 Every problem on average rates has at least one attractive wrong answer that is intuitively appealing and seems promise a quick solution.\u00a0 Can you spot it here?<\/p>\n<p>You might have said 20 kilometers per hour, because that is the average of the given rates, 16 and 24 kilometers per hour. Fair enough. That is wrong and will attract some people who read too quickly. I\u2019m more concerned about a different trap, though.<\/p>\n<p>Average-rate problems encourage you to assume that spending the same <em>distance<\/em> at two (or more) different rates allows you simply average those rates. For instance, in this problem, we might suppose that traveling one leg at 16 kph, one at 24 kph, and one at 32 kph would yield an average speed for the whole tour of 24 kph, because 24 is the simple mean of 16, 24, and 32 . And there\u2019s 32 kilometers per hour waiting for you!<\/p>\n<p>In fact, though, speeds are weighted according to how much <em>time<\/em> you spend at each, not how much <em>distance<\/em> you spend at each. This means that average speeds are generally less than you\u2019d guess based on their component speeds, since it takes you <em>more time<\/em> to travel a given distance when you travel it slowly. (This neat fact deserves a post of its own, and I\u2019ll write one soon.)<\/p>\n<p>As a practical matter, what this means for average speed problems is that you will usually need to approach them <em>not<\/em> as simple averages or even as weighted averages. Instead, you need to view average speed problems through this formula:<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img1.png\"><img decoding=\"async\" class=\"size-full wp-image-5448 aligncenter\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img1.png\" alt=\"ar.ompftrt_img1\" width=\"237\" height=\"64\" \/><\/a><\/p>\n<h2>Back to the RTD table!<\/h2>\n<p>Let\u2019s devote one row of the table to each leg, and a fourth row to the total.<\/p>\n<p><center><a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-151.jpg\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-151-300x225.jpg\" alt=\"Image 15\" width=\"300\" height=\"225\" class=\"alignnone size-medium wp-image-5487\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-151-300x225.jpg 300w, https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-151-1024x768.jpg 1024w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/center><\/p>\n<p>For an average-rate problem, you\u2019ll need a wall between the rates of the various legs and the rate for the whole trip. I\u2019ve represented that wall by drawing a heavy black line above the bottom-left cell. The only way to fill in that cell is from the other information in the bottom row, not directly from the cells above it in the same column.<\/p>\n<p>The red lines below show the direction in you will usually <em>add each of the columns<\/em> and <em>divide the bottom row<\/em>.<\/p>\n<p><center><a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-16.jpg\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-16-300x225.jpg\" alt=\"Image 16\" width=\"300\" height=\"225\" class=\"alignnone size-medium wp-image-5488\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-16-300x225.jpg 300w, https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-16-1024x768.jpg 1024w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/center><\/p>\n<p>Today\u2019s problem is a bit strange though, so we\u2019ll do things in a slightly different way.<\/p>\n<p>&nbsp;<\/p>\n<h2>Rate.<\/h2>\n<p>Okay, let\u2019s get rid of those red lines and add the rate information we\u2019ve been given: the rates for the first two legs and the average rate.<\/p>\n<p><center><a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-17.jpg\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-17-300x225.jpg\" alt=\"Image 17\" width=\"300\" height=\"225\" class=\"alignnone size-medium wp-image-5489\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-17-300x225.jpg 300w, https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-17-1024x768.jpg 1024w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/center><\/p>\n<p>&nbsp;<\/p>\n<h2>Distance.<\/h2>\n<p>Those are the only constant values we\u2019ve been given, but we can add a little more information even so. We don\u2019t know the distance, but we know that it\u2019s the same for every leg, so let\u2019s call that distance <em>d<\/em>, and then sum the legs to get a combined distance.<\/p>\n<p><center><a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-18.jpg\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-18-300x225.jpg\" alt=\"Image 18\" width=\"300\" height=\"225\" class=\"alignnone size-medium wp-image-5490\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-18-300x225.jpg 300w, https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-18-1024x768.jpg 1024w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/center><\/p>\n<h2>Time.<\/h2>\n<p>Now that we\u2019ve represented both rate and distance for three of our four rows, we can determine times for those rows by division;\u00a0<a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img3a.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-5451\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img3a.png\" alt=\"ar.ompftrt_img3a\" width=\"112\" height=\"29\" \/><\/a><\/p>\n<p><center><a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-19.jpg\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-19-300x225.jpg\" alt=\"Image 19\" width=\"300\" height=\"225\" class=\"alignnone size-medium wp-image-5492\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-19-300x225.jpg 300w, https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-19-1024x768.jpg 1024w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/center><\/p>\n<p>&nbsp;<\/p>\n<h2>Back to time.<\/h2>\n<p>Notice that the column for \u201ctime\u201d is almost complete. Since the total time, d\/8, must be the sum of the other times, we can determine the missing time for the third leg.<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img2.png\"><img decoding=\"async\" class=\"size-full wp-image-5449 aligncenter\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img2.png\" alt=\"ar.ompftrt_img2\" width=\"129\" height=\"55\" \/><\/a><\/p>\n<p>Multiply through by 48, the least common denominator.<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img4.png\"><img decoding=\"async\" class=\"size-full wp-image-5452 aligncenter\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img4.png\" alt=\"ar.ompftrt_img4\" width=\"152\" height=\"26\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img4.png 152w, https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img4-150x26.png 150w\" sizes=\"(max-width: 152px) 100vw, 152px\" \/><\/a><\/p>\n<p>Transpose.<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img5.png\"><img decoding=\"async\" class=\"size-full wp-image-5453 aligncenter\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img5.png\" alt=\"ar.ompftrt_img5\" width=\"64\" height=\"26\" \/><\/a><\/p>\n<p>Divide.<\/p>\n<p><a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img6b.png\"><img decoding=\"async\" class=\"size-full wp-image-5455 aligncenter\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img6b.png\" alt=\"ar.ompftrt_img6b\" width=\"65\" height=\"45\" \/><\/a><\/p>\n<p>Let\u2019s add that to the table.<\/p>\n<p><center><a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-20.jpg\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-20-300x225.jpg\" alt=\"Image 20\" width=\"300\" height=\"225\" class=\"alignnone size-medium wp-image-5491\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-20-300x225.jpg 300w, https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-20-1024x768.jpg 1024w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/center><\/p>\n<p>&nbsp;<\/p>\n<h2>Finally our answer!<\/h2>\n<p>Now that we have the distance and time for the third leg, we can divide to determine the rate;\u00a0<br \/>\n<a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img3.png\"><img decoding=\"async\" class=\"alignnone size-full wp-image-5450\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img3.png\" alt=\"ar.ompftrt_img3\" width=\"111\" height=\"35\" \/><\/a><\/p>\n<p><center><a href=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-21.jpg\"><img decoding=\"async\" src=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-21-300x225.jpg\" alt=\"Image 21\" width=\"300\" height=\"225\" class=\"alignnone size-medium wp-image-5493\" srcset=\"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-21-300x225.jpg 300w, https:\/\/magoosh.com\/gmat\/files\/2015\/01\/Image-21-1024x768.jpg 1024w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/a><\/center><\/p>\n<p>Okay, even using the RTD table, we still had to do an awful lot of work. It turns out, though, that a couple of shortcuts could have saved us most of that work.\u00a0 Come back tomorrow to see how.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>My last several posts have been devoted to the use of a table to answer rate problems. Today\u2019s post will assume familiarity with that table, so please take a look back at these posts is you\u2019re not already familiar with the RTD table: Using Diagrams to Solve Rate Problems: Part 1 Using Diagrams to Solve [&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-5447","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>One More RTD Table Problem: Average Rates 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\/one-more-rtd-table-problem-average-rates-part-1\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"One More RTD Table Problem: Average Rates Part 1\" \/>\n<meta property=\"og:description\" content=\"My last several posts have been devoted to the use of a table to answer rate problems. 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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":"One More RTD Table Problem: Average Rates 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\/one-more-rtd-table-problem-average-rates-part-1\/","og_locale":"en_US","og_type":"article","og_title":"One More RTD Table Problem: Average Rates Part 1","og_description":"My last several posts have been devoted to the use of a table to answer rate problems. Today\u2019s post will assume familiarity with that table, so please take a look back at these posts is you\u2019re not already familiar with the RTD table: Using Diagrams to Solve Rate Problems: Part 1 Using Diagrams to Solve [&hellip;]","og_url":"https:\/\/magoosh.com\/gmat\/one-more-rtd-table-problem-average-rates-part-1\/","og_site_name":"Magoosh Blog \u2014 GMAT\u00ae Exam","article_publisher":"https:\/\/www.facebook.com\/MagooshGMAT\/","article_published_time":"2015-01-14T17:00:57+00:00","article_modified_time":"2020-01-15T18:48:21+00:00","og_image":[{"url":"https:\/\/magoosh.com\/gmat\/files\/2015\/01\/ar.ompftrt_img1.png"}],"author":"Michael Schwartz","twitter_card":"summary_large_image","twitter_creator":"@MagooshGMAT","twitter_site":"@MagooshGMAT","twitter_misc":{"Written by":"Michael Schwartz","Est. reading time":"7 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/magoosh.com\/gmat\/one-more-rtd-table-problem-average-rates-part-1\/#article","isPartOf":{"@id":"https:\/\/magoosh.com\/gmat\/one-more-rtd-table-problem-average-rates-part-1\/"},"author":{"name":"Michael Schwartz","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/95a95e68f2ccbf70a6bacf5a260eaacf"},"headline":"One More RTD Table Problem: Average Rates Part 1","datePublished":"2015-01-14T17:00:57+00:00","mainEntityOfPage":{"@id":"https:\/\/magoosh.com\/gmat\/one-more-rtd-table-problem-average-rates-part-1\/"},"wordCount":800,"commentCount":0,"publisher":{"@id":"https:\/\/magoosh.com\/gmat\/#organization"},"articleSection":["GMAT Word Problems"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/magoosh.com\/gmat\/one-more-rtd-table-problem-average-rates-part-1\/","url":"https:\/\/magoosh.com\/gmat\/one-more-rtd-table-problem-average-rates-part-1\/","name":"One More RTD Table Problem: Average Rates Part 1 - Magoosh Blog \u2014 GMAT\u00ae Exam","isPartOf":{"@id":"https:\/\/magoosh.com\/gmat\/#website"},"datePublished":"2015-01-14T17:00:57+00:00","breadcrumb":{"@id":"https:\/\/magoosh.com\/gmat\/one-more-rtd-table-problem-average-rates-part-1\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/magoosh.com\/gmat\/one-more-rtd-table-problem-average-rates-part-1\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/magoosh.com\/gmat\/one-more-rtd-table-problem-average-rates-part-1\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/magoosh.com\/gmat\/"},{"@type":"ListItem","position":2,"name":"One More RTD Table Problem: Average Rates Part 1"}]},{"@type":"WebSite","@id":"https:\/\/magoosh.com\/gmat\/#website","url":"https:\/\/magoosh.com\/gmat\/","name":"Magoosh Blog \u2014 GMAT\u00ae Exam","description":"Everything you need to know about the GMAT","publisher":{"@id":"https:\/\/magoosh.com\/gmat\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/magoosh.com\/gmat\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/magoosh.com\/gmat\/#organization","name":"Magoosh","url":"https:\/\/magoosh.com\/gmat\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/logo\/image\/","url":"https:\/\/magoosh.com\/gmat\/files\/2019\/04\/Magoosh-logo-purple-60h.png","contentUrl":"https:\/\/magoosh.com\/gmat\/files\/2019\/04\/Magoosh-logo-purple-60h.png","width":265,"height":60,"caption":"Magoosh"},"image":{"@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/MagooshGMAT\/","https:\/\/twitter.com\/MagooshGMAT"]},{"@type":"Person","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/95a95e68f2ccbf70a6bacf5a260eaacf","name":"Michael Schwartz","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/magoosh.com\/gmat\/#\/schema\/person\/image\/e394ca0b8f7a8ce5d11ab69a2756a9cf","url":"https:\/\/secure.gravatar.com\/avatar\/d93dd6a3ce8134de72c26b5762ae2241005cd9e7537876e27dab68d9c44b2ae8?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/d93dd6a3ce8134de72c26b5762ae2241005cd9e7537876e27dab68d9c44b2ae8?s=96&d=mm&r=g","caption":"Michael Schwartz"},"description":"Michael Schwartz is really good at standardized tests. 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