{"id":12029,"date":"2020-12-13T01:24:27","date_gmt":"2020-12-13T01:24:27","guid":{"rendered":"https:\/\/www.manhattanprep.com\/gre\/?p=12029"},"modified":"2020-12-22T17:39:22","modified_gmt":"2020-12-22T17:39:22","slug":"hardest-gre-math-problems","status":"publish","type":"post","link":"https:\/\/www.manhattanprep.com\/gre\/blog\/hardest-gre-math-problems\/","title":{"rendered":"What Are the Hardest GRE Math Problems?"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12062\" src=\"https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/what-are-hardest-gre-math-problems-tom-anderson.png\" alt=\"Manhattan Prep GRE Blog - What Are the Hardest GRE Math Problems? by Tom Anderson\" width=\"1200\" height=\"628\" srcset=\"https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/what-are-hardest-gre-math-problems-tom-anderson.png 1200w, https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/what-are-hardest-gre-math-problems-tom-anderson-300x157.png 300w, https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/what-are-hardest-gre-math-problems-tom-anderson-768x402.png 768w, https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/what-are-hardest-gre-math-problems-tom-anderson-1024x536.png 1024w\" sizes=\"auto, (max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><span style=\"font-weight: 400;\">And what do they tell us about prepping for the GRE?<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Students often ask me, \u201cWhere can I find the most difficult questions on the GRE?\u201d In this blog entry, I\u2019ll show you the top three hardest GRE Math problems, ranked by percent of students who got them wrong. Before we get there, I should say: you don\u2019t need to correctly answer questions like these to get a very, very good score on the GRE. This is a test that favors accuracy and consistency on mid-range questions over the ability to get the very hard ones. One can nab a score in the 90<\/span><span style=\"font-weight: 400;\">th<\/span><span style=\"font-weight: 400;\"> percentile or above without getting any of the very hardest GRE Math problems correct. In case you\u2019re curious, though, this is what the hardest GRE Math problems look like. Each of these questions were correctly answered by fewer than 20 percent of GRE test takers.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Start the drum roll.<\/span><\/p>\n<p><!--more--><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/giphy.com\/embed\/GBvkxysAR8Svm\" width=\"480\" height=\"270\" frameBorder=\"0\" class=\"giphy-embed\" allowFullScreen><\/iframe><\/p>\n<p><a href=\"https:\/\/giphy.com\/gifs\/GBvkxysAR8Svm\"><\/a><\/p>\n<h4><b>The Hardest GRE Math Problems: #3<\/b><\/h4>\n<p><span style=\"font-weight: 400;\">Coming in at #3 is this probability question\u201485% of test takers missed it. Feel free to try it before you keep reading.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12031\" src=\"https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/ta-14-image-1.png\" alt=\"Manhattan Prep GRE Blog - What Are the Hardest GRE Math Problems? by Tom Anderson\" width=\"626\" height=\"210\" srcset=\"https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/ta-14-image-1.png 626w, https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/ta-14-image-1-300x101.png 300w\" sizes=\"auto, (max-width: 626px) 100vw, 626px\" \/><\/p>\n<h4><b>What It Teaches Us about GRE Math<\/b><\/h4>\n<p><span style=\"font-weight: 400;\">This question is a \u201cone-trick pony,\u201d as they say. It\u2019s a relatively simple probability question with a tricky little twist. I\u2019d be willing to bet that it\u2019s that little twist that\u2019s making most folks miss this one. Once you figure out the twist, this problem (and others like it) will be a breeze. To illustrate how it works, let\u2019s look first at a simpler version of the same question:<\/span><\/p>\n<p style=\"padding-left: 30px;\"><span style=\"font-weight: 400;\">A person rolls a 20-sided die two times. What are the odds that both of the rolls result in 19s or 20s?<\/span><\/p>\n<p><span style=\"font-weight: 400;\">To answer this question, you\u2019d first need to write the odds of getting a 19 or 20\u2014that\u2019s 2 out of 20. Because you need both rolls to come out with the high numbers, your odds are (2\/20) x (2\/20) or (1\/10) x (1\/10). For this question, our answer is 1\/100. We have a 1% chance of getting such high numbers on both dice.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">A similar sort of scenario is at work in the \u201cvery hard\u201d problem above. At first glance, these problems may seem to be two surface-level flavors that contain the exact same math. (We\u2019re no longer in a basement playing role-playing games; we\u2019ve got a job in a factory checking out lightbulbs. I guess we had to pay the bills.)<\/span><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/giphy.com\/embed\/3oriNPdeu2W1aelciY\" width=\"480\" height=\"270\" frameBorder=\"0\" class=\"giphy-embed\" allowFullScreen><\/iframe><\/p>\n<p><a href=\"https:\/\/giphy.com\/gifs\/animation-dragons-dnd-3oriNPdeu2W1aelciY\"><\/a><\/p>\n<p><span style=\"font-weight: 400;\">Underneath that surface-level veneer, though, arises a sneaky little trick. As we pull bulbs out of the box, we change the odds of what\u2019s left in it. If you\u2019ve got a good bulb in one hand, that\u2019s one fewer good bulb that might be in the other hand. To solve this problem, you have 18 good bulbs to choose from (18\/20), but even if you\u2019re pulling them out simultaneously, there are only 17 <\/span><i><span style=\"font-weight: 400;\">other<\/span><\/i><span style=\"font-weight: 400;\"> good bulbs that might be in your other hand. So the odds change to 17 out of 19.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Multiply (18\/20) by (17\/19) and you get 153 out of 190\u2014a very ugly fraction that is the correct answer to this tricky little question.<\/span><\/p>\n<h4><b>The Hardest GRE Math Problems: #2<\/b><\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12032\" src=\"https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/ta-14-image-2.png\" alt=\"Manhattan Prep GRE Blog - What Are the Hardest GRE Math Problems? by Tom Anderson\" width=\"630\" height=\"189\" srcset=\"https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/ta-14-image-2.png 630w, https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/ta-14-image-2-300x90.png 300w\" sizes=\"auto, (max-width: 630px) 100vw, 630px\" \/><\/p>\n<p><span style=\"font-weight: 400;\">Here\u2019s #2 on our list of the hardest GRE Math problems. 89% of test takers missed this one. Before we discuss how to do it, give it a shot on your own.<\/span><\/p>\n<h4><b>What It Teaches Us about GRE Math<\/b><\/h4>\n<p><span style=\"font-weight: 400;\">That\u2019s not an excited 25 in there, it\u2019s 25 factorial. 25! Means<\/span><\/p>\n<p><span style=\"font-weight: 400;\"> 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1<\/span><\/p>\n<p><span style=\"font-weight: 400;\">And it\u2019s this part of the problem that sets up a <\/span><i><span style=\"font-weight: 400;\">classic <\/span><\/i><span style=\"font-weight: 400;\">GRE trap. If you were to approach this problem by calculating 25!, you\u2019d either have a calculator with an error screen (too many digits) or you\u2019d be spending 10 minutes doing a long calculation on paper\u2026 and end up with something totally useless to you. Even typing out all of the numbers (like I did above) takes an annoyingly long amount of time. To beat this \u201ccan\u2019t calculate\u201d trap, turn your attention instead to the answer choices.<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><iframe loading=\"lazy\" src=\"https:\/\/giphy.com\/embed\/UvOaWabxo1enu\" width=\"480\" height=\"367\" frameBorder=\"0\" class=\"giphy-embed\" allowFullScreen><\/iframe><\/p>\n<p><a href=\"https:\/\/giphy.com\/gifs\/commercial-sharp-electronics-UvOaWabxo1enu\"><\/a><\/p>\n<p><\/span><\/p>\n<p><span style=\"font-weight: 400;\">If they\u2019d given us 25 as an answer choice, I bet you\u2019d know immediately that it divided evenly into 25!. Same thing if they gave us 24 or 23 or any of the other smaller numbers listed above. They\u2019re right there in the product, so they could be divided out evenly. The same thing is true about the answer choices they gave us, if you break them down into smaller products like so:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">A) 26 = 13 x 2<br \/>\n<\/span><span style=\"font-weight: 400;\">B) 28 = 14 x 2<\/span><span style=\"font-weight: 400;\"><br \/>\n<\/span><span style=\"font-weight: 400;\">C) 36 = 12 x 3<br \/>\n<\/span><span style=\"font-weight: 400;\">D) 56 = 7 x 8<br \/>\n<\/span><span style=\"font-weight: 400;\">E) 58 = 29 x 2<\/span><\/p>\n<p><span style=\"font-weight: 400;\">4 of the choices are made out of factors on our list. They\u2019ll all divide evenly into 25!. Only one of them contains factors that aren\u2019t on our list for 25! Answer choice E contains a 29, which is a prime number bigger than 25. It won\u2019t be found anywhere between 1 and 25 and it can\u2019t be broken down any further than it is. That makes E the correct choice here.<\/span><\/p>\n<h4><b>The Hardest GRE Math Problems: #1<\/b><\/h4>\n<p><span style=\"font-weight: 400;\"><iframe loading=\"lazy\" src=\"https:\/\/giphy.com\/embed\/RnX4q6yYDoYCI\" width=\"480\" height=\"255\" frameBorder=\"0\" class=\"giphy-embed\" allowFullScreen><\/iframe><\/p>\n<p><a href=\"https:\/\/giphy.com\/gifs\/arrested-development-good-RnX4q6yYDoYCI\"><\/a><\/p>\n<p><\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12033\" src=\"https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/ta-14-image-3.png\" alt=\"Manhattan Prep GRE Blog - What Are the Hardest GRE Math Problems? by Tom Anderson\" width=\"627\" height=\"259\" srcset=\"https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/ta-14-image-3.png 627w, https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/ta-14-image-3-300x124.png 300w\" sizes=\"auto, (max-width: 627px) 100vw, 627px\" \/><\/p>\n<p><span style=\"font-weight: 400;\">There you have it: the hardest GRE Math problem in the book. 90% of test takers missed it. Feel free to give it a go before we discuss\u2026<\/span><\/p>\n<h4><b>What It Teaches Us about GRE Math<\/b><\/h4>\n<p><span style=\"font-weight: 400;\">If you dealt with this question in an abstract way, it\u2019s a lot to process. Instead, draw out a few variations of lines that don\u2019t go through the origin and look for any patterns.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12034\" src=\"https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/ta-14-image-4.png\" alt=\"Manhattan Prep GRE Blog - What Are the Hardest GRE Math Problems? by Tom Anderson\" width=\"639\" height=\"217\" srcset=\"https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/ta-14-image-4.png 639w, https:\/\/www.manhattanprep.com\/gre\/wp-content\/uploads\/sites\/19\/2018\/10\/ta-14-image-4-300x102.png 300w\" sizes=\"auto, (max-width: 639px) 100vw, 639px\" \/><\/p>\n<p><span style=\"font-weight: 400;\">Since the first couple of answer choices ask about x and y intercepts, take a look at our examples and look for patterns. In our negative lines, we hit the axes in two positive spots or two negative spots. On the positive lines, one intercept is in the negative and one is in the positive.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Answer choices A and B are both ways of saying that the x and y intercepts have the same sign. If they have the same sign, their slopes are negative. Lines with positive slopes have a positive x-intercept and a negative y-intercept or vice versa. Both of these choices are correct.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Answer choice C seems strange at first, but rephrase it a little bit: (a \u2013 r) refers to the change in x. (b-s) refers to the change in y.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">You may have learned to call these\u2014the \u201crun\u201d of the line (a-r) and the \u201crise\u201d of the line (b-s). Answer choice C is telling you that if you multiply the rise and run of the line you get a negative. And that\u2019s the very definition of a negative slope. If you\u2019d like to try it with real numbers, try it with the coordinates drawn in on the lines above. You\u2019ll find that answer choice C also proves the line has a negative slope. It\u2019s also correct.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Underneath all of this content, I think, lies the real head-game of this \u201cmost difficult\u201d question. You actually check all three of the answer choices. As wild as that might seem, sometimes all three are correct.<\/span><\/p>\n<h4><b>This is as Hard as It Gets, Folks<\/b><\/h4>\n<p><span style=\"font-weight: 400;\">I think these problems have a few big-picture things to teach you about GRE Math:<\/span><\/p>\n<ol>\n<li><span style=\"font-weight: 400;\"> When faced with a complex or very difficult problem, solve a simpler problem as a step to getting the hard one.<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> Avoid big calculations. Look for opportunities to work backwards from the answers.<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> Take any complex information (especially on geometry) and draw it to better understand it. <\/span><\/li>\n<\/ol>\n<p><span style=\"font-weight: 400;\">There\u2019s also definitely some content worth remembering:<\/span><\/p>\n<ol>\n<li><span style=\"font-weight: 400;\"> With countable objects, probability often changes as you go.<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> Numbers are divisible by their factors. And you can break big numbers down into factors to check.<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> The slope of a line is negative when its rise and run have opposite signs\u2014one positive and the other negative.<\/span><\/li>\n<\/ol>\n<p><span style=\"font-weight: 400;\">Perhaps most importantly, remember that this is as hard as it gets. You\u2019ll likely never encounter anything nastier than these. And even these \u201chardest GRE Math problems\u201d can be cut down to size. <\/span><\/p>\n<hr \/>\n<p><b><i>Want more guidance from our GRE gurus? You can attend the first session of any of our online or in-person GRE courses absolutely free! We\u2019re not kidding.\u00a0<\/i><\/b><a id=\"bloglink\" href=\"https:\/\/www.manhattanprep.com\/gre\/classes\/\" target=\"_blank\" rel=\"noopener noreferrer\"><b><i>Check out our upcoming courses here<\/i><\/b><\/a><b><i>.<\/i><\/b><\/p>\n<hr \/>\n<p><em><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignleft wp-image-10555 size-thumbnail\" src=\"https:\/\/d27gmszdzgfpo3.cloudfront.net\/gre\/wp-content\/uploads\/sites\/19\/2017\/08\/tom-anderson-150x150.png\" alt=\"tom-anderson\" width=\"150\" height=\"150\" \/><a id=\"bloglink\" href=\"https:\/\/www.manhattanprep.com\/instructors\/tom-anderson\/\" target=\"_blank\" rel=\"noopener noreferrer\">Tom Anderson<\/a>\u00a0is a Manhattan Prep instructor based in New York, NY.<\/strong>\u00a0He has a B.A. in English and a master\u2019s degree in education. Tom has long possessed an understanding of the power of standardized tests in propelling one\u2019s education and career, and he hopes he can help his students see through the intimidating veneer of the GRE.\u00a0<a id=\"bloglink\" href=\"https:\/\/www.manhattanprep.com\/gre\/classes\/#instructor\/53\" target=\"_blank\" rel=\"noopener noreferrer\">Check out Tom\u2019s upcoming GRE courses here<\/a>.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>And what do they tell us about prepping for the GRE? Students often ask me, \u201cWhere can I find the most difficult questions on the GRE?\u201d In this blog entry, I\u2019ll show you the top three hardest GRE Math problems, ranked by percent of students who got them wrong. Before we get there, I should [&hellip;]<\/p>\n","protected":false},"author":173,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2,474284,921840,421,7,9,733445],"tags":[133,151,1362594,229],"yst_prominent_words":[1362788,1362782,1362778,1362781,1362790,1362627,1362701,1362700,1362779,1362702,1362699,1362787,1362703,1362785,1362786,1362789,1362783,1362704,1362784,1362780],"class_list":["post-12029","post","type-post","status-publish","format-standard","hentry","category-challenge-problems","category-current-studiers","category-gre-prep-2","category-gre-quant-2","category-how-to-study","category-math-gre-strategies","category-study-tips-2","tag-gre","tag-gre-math","tag-hardest-gre-math-problems","tag-math"],"_links":{"self":[{"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/posts\/12029","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/users\/173"}],"replies":[{"embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/comments?post=12029"}],"version-history":[{"count":6,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/posts\/12029\/revisions"}],"predecessor-version":[{"id":12941,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/posts\/12029\/revisions\/12941"}],"wp:attachment":[{"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/media?parent=12029"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/categories?post=12029"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/tags?post=12029"},{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.manhattanprep.com\/gre\/wp-json\/wp\/v2\/yst_prominent_words?post=12029"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}