Backsolving and using answers

When you take any exam, the answer choices can give you useful clues, and the GRE is no exception. There are several ways to use the choices to your advantage, including a few that aren’t obvious at first. In this chapter, you’ll start with the most practical ways to use answer choices to solve GRE questions, and then you’ll finish with a few more subtle tips.

Plug in answer choices and check if they’re correct

Your first instinct might be to read the question, solve it, and then look for a matching answer. But on many GRE Quant questions, it can be faster (and sometimes more natural) to plug in the answer choices and see which one makes the statement true.

When you use this plug-in method, the order you test choices matters.

• GRE Quant answer choices are almost always listed in ascending numerical order.
• Because of that, it’s often best to start with choice B (a middle value).
• After testing B, you’ll usually learn whether your result is too small or too large.
  • If it’s too small, move up to a larger choice (D, then E if needed).
  • If it’s too large, move down to a smaller choice (A, then possibly C).
• Starting in the middle also helps you eliminate choices quickly. For example, if you test D and it’s still too small, then the answer must be E (so you may not need to plug in again).

Try the question below by plugging in the answer choices. You could solve it directly by writing an algebraic equation, but the goal here is to practice the plug-in method.

When twice the value of $z$ is increased by 50%, and 5 is subtracted from that value, the result is 55.

A. 12
B. 15
C. 17
D. 18
E. 20

Start by plugging in answer choice B!

Plugging in choice B means $z=15$.

• Twice 15 is 30.
• Increasing 30 by 50% means multiplying by 1.5: $30\times 1.5=45$.
• Subtract 5: $45-5=40$.

The result should be 55, but 40 is too small. That means you need a larger value of $z$, so move up in the choices and try D.

Plugging in choice D means $z=18$.

• Twice 18 is 36.
• Increase by 50%: $36\times 1.5=54$.
• Subtract 5: $54-5=49$.

This is closer, but it’s still less than 55. Since the choices are in ascending order and E is the only choice larger than D, the answer must be E. 20.

You don’t have to plug in E because it’s the only remaining larger choice. But if you have time, it’s reasonable to double-check.

$$\begin{array}{rl}2z\ast 1.5-5& =55\\ 2\ast 20\ast 1.5-5& =55\\ 40\ast 1.5-5& =55\\ 60-5& =55\\ 55& =55\end{array}$$

It checks out, confirming that E. 20 is the correct answer.

Notice that you didn’t need to write an equation to use the plug-in method, but you did write one to verify your result. Many GRE questions can be solved in more than one way, and using a second method can help you confirm that your answer is correct.

For a complete walkthrough, here’s how you could solve the same question using a direct algebra approach:

$$\begin{array}{rl}2z\ast 1.5-5& =55\\ 3z-5& =55\\ 3z& =60\\ z& =20\end{array}$$

Extra tips to use when backsolving

When exam writers create questions, they’re usually targeting a specific concept. If you can identify what the question is really testing, the answer choices often give you hints about the best path to the solution.

• Examine the format of the answer choices. If the choices share a common structure, try rewriting your work to match that structure.

  For example, suppose the answer choices look like $r=-3/5b-12/5$ (same form, different constants). If the question gives an equation like $5r+3b=12$, isolate $r$ so your expression matches the answer-choice format and is easy to compare.

  Similarly, if all answer choices involve a number raised to an exponent, it’s often fastest to rewrite the prompt’s expression to follow the same exponent pattern.

• Use QB to direct your path. Quantity B can guide your strategy on quantitative comparison questions.

  For example, if QB is 0, you may only need to determine whether QA is positive, negative, or zero, rather than finding QA exactly. Also, the GRE often makes QB very close to QA to increase the chance of small mistakes. If your QA value is just above or just below QB, that’s often a sign you’re on the right track. If your QA value seems far from QB, it may mean you made an error.

• Look out for $\pi$ (pi). If the answer choices all involve $\pi$, you’re probably using a geometry formula for circumference, area, surface area, or volume (often involving circles or cylinders). Don’t replace $\pi$ with 3.14, since the GRE typically expects answers in terms of $\pi$, not decimals.

• Look at the format of QA and QB. Sometimes QA and QB describe the same quantity in different forms, such as a factored expression versus an expanded one. Before comparing, try rewriting both quantities so they look as similar as possible using the simplify and compare strategy.

• Look out for multiple correct answers in ascending order. If more than one answer choice could be correct and the choices are listed in ascending order, it’s often a sign you should use the min-max extremes strategy.