Backsolving and using answers

When taking any exam, you can often get clues from the answer choices, and the GRE is no exception. There are several ways to get hints from the choices, and some aren’t immediately obvious. We’ll begin this chapter with the most practical ways to use answer choices to solve GRE questions, and end with a few more subtle tips.

Plug in answer choices and check if they’re correct

Although our tendency is to read the question, solve it, and then check for corresponding matching answers, sometimes it can be much quick and more natural to just plug in the choices and see if the math works out. When using this plug-in method, the order of testing the choices is important. Since the choices of GRE quantitative questions are almost always in ascending numerical order, it’s often best to test B first. After checking B, and if it wasn’t correct, you’ll have a better understanding of the question and how it relates to your choice, i.e., if you ended up with a result that was too large or too small. Since you started with a choice in the middle, you can move to choice A or choice D (since if D isn’t a match, then you’ll know it’s C or E without plugging in again) depending on the relationship between your result and the question.

Try the question below by plugging in the answer choices. You do have the option of directly solving for the answer by creating an algebraic expression, but the aim of this chapter 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 that $z$ would be 15. Twice 15 is 30, and if we increase 30 by 50%, we get 45. Subtracting 5 from 45 gives us 40. We wanted a result of 55, so 40 is too small. We need a greater starting value, so let’s move up in the choices and try again plugging in D.

Plugging in choice D means that $z$ would be 18. Twice 18 is 36, and 50% more than 36 is 54. Subtract 5 from 54 to get 49. This is closer to our target result of 55, but it’s still less. Therefore, the answer must be the only choice that is greater than D, i.e. E. 20.

We don’t need to plug in E, since it’s the only choice that would end up with a greater result. But if you have time to double-check, it wouldn’t hurt.

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

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

We never wrote the equation before, but we did when double-checking. Most GRE questions can be solved multiple ways, and using a different method to solve them can give you even more confidence that your choice is correct.

For the sake of a thorough walkthrough, here’s how we would have solved the question using a top-down mathematical 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 the exam writers are creating questions, they have a goal in mind of testing a specific topic. By understanding what a question is really asking, you can get some hints about the correct answer.

• Examine the format of the answer choices. For instance, if all the answer choices might be written as similar equations, your goal should be your goal to create an equation from the question and rewrite it in the format of the answer choices. As a concrete example, maybe the question has answer choices that look like $r=-3/5b-12/5$, using different constants for each, but all in the same form. If the question gives an equation like $5r+3b=12$, you should try to isolate the $r$, just like in the answer choices, so your equation can easily be compared to them. Another situation you might see is if all answer choices are written in terms of some number raised to an exponent. The simplest path forward would probably be to adjust the equation in the prompt to follow the same exponent pattern.

• Use QB to direct your path. Quantity B can be very helpful in determining how to solve quantitative comparison questions. For example, if QB is 0, it is very likely that you only need to understand if QA is positive or negative. This can be much easier than finding the exact value of QA. Furthermore, the GRE often makes QB very close to QA, since they want to make the problem susceptible to mistakes. If you find that QA is just above or below QB, it probably means you’re on track! Finding a value for QA that seems far off from QB is sometimes a sign that you didn’t solve the question correctly.

• Look out for $\pi$ (pi). If the answer choices all involve pi, you’re probably going to be working with a geometric equation for the circumference, area, surface area, or volume of a circle or cylinder. Don’t replace the pi symbol with $3.14$, since you’ll want your final answer to be expressed in terms of $\pi$ rather than a decimal.

• Look at the format of QA and QB. Sometimes QA and QB are different ways of describing the same thing, like the factored and unfactored version of an expression. Before comparing, try making both quantities as similar as possible using the simplify and compare strategy.

• Look out for multiple correct answers in ascending order. If multiple choices can be correct, and they are written in ascending order. It is very likely that you should use the min-max extremes strategy.