GRE Solutions Manual, Problem 5.11

Exam Prep, GRE, Quantitative Reasoning

This page is part of my unofficial solutions manual to the GRE Paper Practice Book (2e), a free resource available on the ETS website. They publish the questions; I explain the answers. If you haven’t worked through the Practice Book, give Section 5 a shot before reading this!

5.11: The Variable That Wasn’t

This problem illustrates another GRE scare tactic: “variables” that are actually constants. The initial expression looks like an equation in two variables, a and b, which could be a real pain to work with. In the very next equation, however, we’re told that b = 1; in other words, b is a constant. Consequently, we can go right back to that first equation and plug in 1 everywhere we see a b:

GRE 5.11, Eqn. 1

At this point, as in many of the QC problems we saw earlier, there are two distinct ways to proceed: solve algebraically or try plugging in answers. Because the algebra here is relatively simple, the first method is the one I’d recommend in most cases.


Solving Algebraically

We begin by multiplying out the denominator:

GRE 5.11, Eqn. 2

Next, we combine like terms:

GRE 5.11, Eqn. 3

Divide out the coefficient (-1), and we get our solution:

GRE 5.11, Eqn. 4

Answer (E) is correct.


Plugging In

But suppose for a moment that you have only 30 seconds on the clock. This is the last problem you’ll have time for, and you won’t be able to finish it in its entirety. In this situation, plugging in may actually be more useful than attempting to solve algebraically, because plugging in lets you eliminate wrong answers along the way. Here’s how the process might work out for this problem:

  • If a = 1, the numerator (and thus the fraction) equals 0, because 1 – 1 = 0. So answer (A) can’t be right.
  • If a = 0, the fraction evaluates to (-1)/(1) = -1. Cross (B) off the list.
  • If a = -1, the denominator is (-1 + 1) = 0, which results in an undefined number. Get rid of answer (C).
  • If a = -2, the fraction evaluates to (-3)/(-1) = 3. There goes answer (D).
  • Since (A), (B), (C), and (D) don’t work, the answer must be (E).

When I teach problem 5.11 in class, most of my students find the algebraic route to be faster and more straightforward than plugging in. In an end-of-section time crunch, though, it makes sense to focus on eliminating wrong answers as quickly as possible. Although there’s no partial credit on the GRE, it can still be helpful to think in terms of “statistical partial credit,” in the sense that every wrong answer you eliminate increases your expected score by a fraction of a point.


Math Review Reference

For more on this topic, see the following section of the GRE Math Review:

  • 2.1: Operations With Algebraic Expressions (pp. 17-18)