What Are the Hardest GRE Math Problems?

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Manhattan Prep GRE Blog - What Are the Hardest GRE Math Problems? by Tom Anderson

And what do they tell us about prepping for the GRE?

Students often ask me, “Where can I find the most difficult questions on the GRE?” In this blog entry, I’ll 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’t 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 90th percentile or above without getting any of the very hardest GRE Math problems correct. In case you’re 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.

Start the drum roll.

The Hardest GRE Math Problems: #3

Coming in at #3 is this probability question—85% of test takers missed it. Feel free to try it before you keep reading.

Manhattan Prep GRE Blog - What Are the Hardest GRE Math Problems? by Tom Anderson

What It Teaches Us about GRE Math

This question is a “one-trick pony,” as they say. It’s a relatively simple probability question with a tricky little twist. I’d be willing to bet that it’s that little twist that’s 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’s look first at a simpler version of the same question:

A person rolls a 20-sided die two times. What are the odds that both of the rolls result in 19s or 20s?

To answer this question, you’d first need to write the odds of getting a 19 or 20—that’s 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.

A similar sort of scenario is at work in the “very hard” problem above. At first glance, these problems may seem to be two surface-level flavors that contain the exact same math. (We’re no longer in a basement playing role-playing games; we’ve got a job in a factory checking out lightbulbs. I guess we had to pay the bills.)

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’s left in it. If you’ve got a good bulb in one hand, that’s 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’re pulling them out simultaneously, there are only 17 other good bulbs that might be in your other hand. So the odds change to 17 out of 19.

Multiply (18/20) by (17/19) and you get 153 out of 190—a very ugly fraction that is the correct answer to this tricky little question.

The Hardest GRE Math Problems: #2

Manhattan Prep GRE Blog - What Are the Hardest GRE Math Problems? by Tom Anderson

Here’s #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.

What It Teaches Us about GRE Math

That’s not an excited 25 in there, it’s 25 factorial. 25! Means

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

And it’s this part of the problem that sets up a classic GRE trap. If you were to approach this problem by calculating 25!, you’d either have a calculator with an error screen (too many digits) or you’d be spending 10 minutes doing a long calculation on paper… 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 “can’t calculate” trap, turn your attention instead to the answer choices.

If they’d given us 25 as an answer choice, I bet you’d 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’re 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:

A) 26 = 13 x 2
B) 28 = 14 x 2
C) 36 = 12 x 3
D) 56 = 7 x 8
E) 58 = 29 x 2

4 of the choices are made out of factors on our list. They’ll all divide evenly into 25!. Only one of them contains factors that aren’t on our list for 25! Answer choice E contains a 29, which is a prime number bigger than 25. It won’t be found anywhere between 1 and 25 and it can’t be broken down any further than it is. That makes E the correct choice here.

The Hardest GRE Math Problems: #1

Manhattan Prep GRE Blog - What Are the Hardest GRE Math Problems? by Tom Anderson

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…

What It Teaches Us about GRE Math

If you dealt with this question in an abstract way, it’s a lot to process. Instead, draw out a few variations of lines that don’t go through the origin and look for any patterns.

Manhattan Prep GRE Blog - What Are the Hardest GRE Math Problems? by Tom Anderson

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.

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.

Answer choice C seems strange at first, but rephrase it a little bit: (a – r) refers to the change in x. (b-s) refers to the change in y.

You may have learned to call these—the “run” of the line (a-r) and the “rise” 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’s the very definition of a negative slope. If you’d like to try it with real numbers, try it with the coordinates drawn in on the lines above. You’ll find that answer choice C also proves the line has a negative slope. It’s also correct.

Underneath all of this content, I think, lies the real head-game of this “most difficult” question. You actually check all three of the answer choices. As wild as that might seem, sometimes all three are correct.

This is as Hard as It Gets, Folks

I think these problems have a few big-picture things to teach you about GRE Math:

  1. When faced with a complex or very difficult problem, solve a simpler problem as a step to getting the hard one.
  2. Avoid big calculations. Look for opportunities to work backwards from the answers.
  3. Take any complex information (especially on geometry) and draw it to better understand it.

There’s also definitely some content worth remembering:

  1. With countable objects, probability often changes as you go.
  2. Numbers are divisible by their factors. And you can break big numbers down into factors to check.
  3. The slope of a line is negative when its rise and run have opposite signs—one positive and the other negative.

Perhaps most importantly, remember that this is as hard as it gets. You’ll likely never encounter anything nastier than these. And even these “hardest GRE Math problems” can be cut down to size.


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tom-andersonTom Anderson is a Manhattan Prep instructor based in New York, NY. He has a B.A. in English and a master’s degree in education. Tom has long possessed an understanding of the power of standardized tests in propelling one’s education and career, and he hopes he can help his students see through the intimidating veneer of the GRE. Check out Tom’s upcoming GRE courses here.

  1. Ram October 17, 2018 at 2:56 pm

    Really nice and informative article. However, I have one doubt. At first, you have mentioned:

    “Lines with positive slopes have a positive x-intercept and a negative y-intercept or vice versa.”

    And then you’ve mentioned,

    “The slope of a line is negative when its rise and run have opposite signs—one positive and the other negative.”

    Shouldn’t the latter be, “The slope of the line is positive”? Please correct me if I am mistaken.