What Makes a Hard GMAT Quant Question: Part 1

Two years ago, GMAC reached out to me with an offer to curate and write explanations for their GMAT Official Advanced Questions. Since the job required selecting the “hardest” of the hard questions, they asked me to explain to them what I thought constituted a difficult GMAT question. I wrote up a pretty lengthy explanation for them, and was ultimately offered the job. Unfortunately, we eventually decided not to pursue it due to contractual issues, so I’m making my write-ups available for the GMAT population. As an independent GMAT tutor, I work with many students who’ve gone through a course or self-study and came to the end of it with scores far below their goal and what they’re capable of. In many cases, the discrepancy is due to misconceptions about what skills the GMAT is actually testing. All too often, students leave a test-prep program with the belief that proficiency on the GMAT is merely a matter of learning and internalizing rules, while all too often neglecting the relevant strategies and reasoning framework required to do well on tougher questions. In this series of blogs, I’m going to use official questions to illustrate what really constitutes difficulty on the GMAT.

Difficulty is, of course, relative, so the best way to illustrate a question’s difficulty is to compare questions that test similar concepts/quantitative abilities but that differ in ways fundamental to what I think constitute difficult questions. Please note that these different characteristics can and often do overlap on one question (indeed, it’s often the overlap of these characteristics that differentiate some of the hardest questions from their counterparts), but, for the purposes of this demonstration, I’ve restricted my analysis of any question to the criterion I consider most relevant.

To make the most of this article, I suggest that you first attempt the questions, then read my analysis below!

Example #1:

Yesterday’s closing prices of 2,420 different stocks listed on a certain stock exchange were all different from today’s closing prices. The number of stocks that closed at a higher price today than yesterday was 20 percent greater than the number that closed at a lower price. How many of the stocks closed at a higher price today than yesterday?

(A) 484
(B) 726
(C) 1,100
(D) 1,320
(E) 1,694

Example #2:

A small, rectangular park has a perimeter of 560 feet and a diagonal measurement of 200 feet. What is its area, in square feet?

A. 19,200
B. 19,600
C. 20,000
D. 20,400
E. 20,800

Question 1 explanation:

Option 1: Algebraic

Let h = the # of stocks closing at a higher price
Let l = the # of stocks closing at a lower price

Since there 2,420 total stocks, we know: h + l = 2,420
We’re further told that the # of stocks that closed higher was 20% greater than the # of stocks that closed lower, so we can create a second equation: h = 1.2l. To solve for h, we can now substitute 1.2l in for h in the first equation, solve for l, and then multiply that value by 1.2 (there are other ways of doing this algebra, but this is the approach I’ll take).

1.2l + l = 2,420
2.2l = 2,420
Multiply both sides by 10:
22l = 24,200
Divide by 10:
l = 24200/22 = 1,100
h = 1.2l = 1,100 (1.2) = 13,200

Answer is D.

Option 2: Use the Choices:

For many test-takers, the algebraic approach on this question can be complicated and time-consuming. Though it might appear that we’re left with no other recourse, we can leverage the dispersion of the choices to our advantage by using a more intuitive approach!

Intuitively, we know two things: a) the total # of stocks is 2,420 and b) there are more stocks that ended up priced higher than priced lower.

Had the # of stocks that ended up higher been the same as the # of stocks that ended up lower, the answer would be 2,420/12 = ~1200. Since more stocks ended up higher than lower, we know the answer must be above ~1,200, so we’re left between D and E. In addressing these two choices, we can use the fact that, proportionately, these numbers are quite far apart. Taking advantage of that fact, we can ask ourselves which choice would make it so that the # of stocks priced higher is 20% greater than the # of stocks priced lower. If we try choice E, we’ll see that if h = ~1600, then l = ~800, yielding a 2:1 ratio. This is greater than the 20% difference we’re looking for, so we know E is too large, leaving us with just choice D.

Question 2 explanation:

Option 1: Algebraic:

Let l = length of the rectangle and w = width of the rectangle.

Since the perimeter is 560, we know that 2l + 2w = 560 –> l + w = 280
The diagonal of the rectangle forms a right triangle with the sides of the rectangle as its legs and the diagonal as the hypotenuse, so we can use the pythagorean theorem to say: l^2 + w^2 = 280^2.

We now have two equations:
a) l + w = 280
b) l^2 + w^2 = 200^2

This is where things get interesting/messy. How will we combine these two to solve for lw? We can of course substitute to solve for each individual variable, but that will be time-consuming and error prone. A faster approach is to use quadratic templates to our advantage:

square both sides of equation 1: (l+w)^2 = 280^2 –> l^2 + 2lw + w^2 = 280^2
Using equation 2, substitute 200^2 for (l^2 + w^2) in equation 1 to arrive at: 2lw + 200^2 = 280^2.
Subtract 200^2 from both sides: 2lw = 280^2 – 200^2
Notice that the right side is a difference of squares that can be re-written as (280 + 200)(280 – 200). Therefore:
2lw = (280 + 200)(280 – 200) = 480(80)
Divide by 2:
lw = 480(40) = 19,200.

Answer is D.

Option 2: not available! There’s no other way to solve this question other than to implement the algebraic steps above.

Analysis: Many official GMAT questions allow multiple pathways to arrive at the correct answer. Generally, the availability of these pathways makes questions “easier” in that the pool of test-takers likely to arrive at a viable solution-path within two minutes increases. Conversely, one characteristic of difficult questions is the narrowing of options test-takers have either from the beginning of the question or at some intermediate stage. In the above examples, both questions require setting up two algebraic relationships and solving for an unknown. However, in the second example (the harder one), after step #1, the only pathway to an answer is a relatively complex algebraic approach, meaning that test-takers weaker in algebra/quadratic equations have no way to arrive at answer! Thus, most of that subset of test-takers will miss this question, making it a harder question.

In contrast, in the first example, test-takers can solve for the relevant variable by either taking an algebraic approach or by using the choices. The point is that the spacing of the choices gives test-takers weak in algebra/arithmetic an alternate pathway to the answer. Thus, both algebraic and “strategic/intuitive” test-takers can answer this question correctly, meaning, relative to scenario #1, a higher proportion of test-takers will get this answer correct.

What’s important to take away from this is that the difficulty of a question is fundamentally based on the percentage of test-takers who get that question correct! So if you’re accustomed to taking only the algebraic approach and miss question #1, you’ll be missing a question that other members of the test-taking population with a more intuitive approach answer correctly, meaning that you’re inherently putting a ceiling on your score. Throughout the years, I’ve seen countless students who’ve learned only the algebraic approach toward questions and who have consequently short-changed themselves. While it’s of course essential to have the core fundamentals of algebra in your arsenal, the reality is that if you limit yourself to just this approach and don’t leverage the other opportunities GMAC makes available on certain questions, you’ll miss more questions than you should and ultimately put a cap on your score. This is also why I always recommend focusing on official GMAT questions in your preparation. GMAC spends thousands of dollars on each question to ensure that they’re measuring student’s reasoning skills, not their math skills. Reasoning skills require flexible thinking, and GMAC intentionally constructs questions that reward this. Test prep companies, on the other hand, will often construct questions that reward one way of thinking (usually how they teach the exam), and students who focus only on those materials will be pigeonholed into looking at questions from just one perspective. Moving forward, how can you implement what I’ve discussed here? Mindful studying! When reviewing questions that you missed or had difficulty on, ask yourself whether there were other approaches available. Were the choices far apart? Could you have taken advantage of that? Could you have plugged in numbers? Could you have backsolved? One of the key skills of high scorers is the ability to see numerous pathways before settling on a plan of attack, so the more time you spend developing this muscle, the better the results you’ll see on test day.