How to Approach Difficult GMAT Problems

SAT/ACTMy students have a hard time understanding what makes a difficult GMAT question difficult. They assume that the tougher questions are either testing something they don’t know, or that these problems involve a dizzying level of complexity that requires an algebraic proficiency that’s simply beyond them.

One of my main goals in teaching a class is to persuade everyone that this is not, in fact, how hard questions work on this test. Hard questions don’t ask you do to something you don’t know how to do. Rather, they’re cleverly designed to provoke an anxiety response that makes it difficult to realize that you do know exactly how to solve the problem.

Take this official question, for example:

Let a, b, c and d be nonzero real numbers. If the quadratic equation ax(cx + d) = -b(cx +d) is solved for x, which of the following is a possible ratio of the 2 solutions?

A) –ab/cd
B) –ac/bd
C) –ad/bc
D) ab/cd
E) ad/bc

Most students see this and panic. Often, they’ll start by multiplying out the left side of the equation, see that the expression is horrible (acx^2 + adx), and take this as evidence that this question is beyond their skill level. And, of course, the question was designed to elicit precisely this response. So when I do this problem in class, I always start by telling my students, much to their surprise, that every one of them already knows how to do this. They’ve just succumbed to the question writer’s attempt to convince them otherwise.

So let’s start simple. I’ll write the following on the board: xy = 0. Then I’ll ask what we know about x or y. And my students shrug and say x or y (or both) is equal to 0. They’ll also wonder what on earth such a simple identity has to do with the algebraic mess of the question they’d been struggling with.

I’ll then write this: zx + zy = 0. Again, I’ll ask what we know about the variables. Most will quickly see that we can factor out a “z” and get z(x+y) = 0. And again, applying the same logic, we see that one of the two components of the product must equal zero – either z = 0 or x + y = 0.

Next, I’ll ask if they would approach the problem any differently if I’d given them zx = -zy – they wouldn’t.

Now it clicks. We can take our initial equation in the aforementioned problem: ax(cx +d) = -b(cx+d), and see that we have a ‘cx + d’ on both sides of the equation, just as we’d had a “z” on both sides of the previous example. If I’m able to get everything on one side of the equation, I can factor out the common term.

Now ax(cx +d) = -b(cx+d) becomes ax(cx +d) + b(cx+d) = 0.

Just as we factored out a “z” in the previous example, we can factor out “cx + d” in this one.

Now we have (cx + d)(ax + b) = 0.

Again, if we multiply two expressions to get a product of zero, we know that at least one of those expressions must equal 0. Either cx + d = 0 or ax + b = 0.

If cx + d = 0, then x = -d/c.

If ax + b = 0, then x = -b/a.

Therefore, our two possible solutions for x are –d/c and –b/a. So, the ratio of the two would simply be (-d/c)/(-b/a). Recall that dividing by a fraction is the equivalent of multiplying by the reciprocal, so we’re ultimately solving for (-d/c)(-a/b). Multiplying two negatives gives us a positive, and we end up with da/cb, which is equivalent to answer choice E.

Takeaway: Anytime you see something on the GMAT that you think you don’t know how to do, remind yourself that the question was designed to create this false impression. You know how to do it – don’t hesitate to dive in and search for how to apply this knowledge.

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By David Goldstein, a Veritas Prep GMAT instructor based in Boston. You can find more articles written by him here.

GMAT Tip of the Week

Fractional Focus, Exponential Improvement

(This is one of a series of GMAT tips that we offer on our blog.)

The very fact that the GMAT does not permit you to use a calculator should give you some insight in to the nature of the quantitative section’s emphasis – the GMAT is much less concerned with your ability to “crunch numbers” than it is with your logical skills, and the way in which you can make yourself more efficient while assessing and manipulating numbers.

Perhaps the best application of this insight is for you to focus on using fractions whenever possible – in lieu of either decimals or division problems. The GMAT almost exclusively features fractions that will ultimately reduce through calculation, whereas long division and/or the use of decimals will often waste quite a bit of your time and leave you prone to calculation errors simply because of the mass of digits you will need to carry.

Say, for example, you arrive at a calculation that requires you to divide 12 by 5. Rather than calculating that the answer will be 2.4, you’re best served to simply leave the number at 12/5, and wait to determine which step will be required next. The problem will likely take the form of the following:

A group of five children finds $12 on the playground, and decides to split it evenly between them. How much will each child have if the playground bully demands a 16.67% commission from each for allowing them to take the money inside?

While 12/5 is, indeed, 2.4, this problem is best calculated by keeping your fractions until the end. In this case, each child will receive 12/5, and be allowed to keep all but 1/6 of their amount (.1667 = 1/6). So, each child will keep 5/6 of 12/5.

This problem is relatively easy to solve: 5/6 * 12/5 allows you to cancel the fives in the numerator and denominator, and then divide the 12 by 6 to arrive cleanly at $2. By keeping your calculations in fraction form as long as you can, you’ll typically save yourself time and hassle, and can often perform the calculations mentally.

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