### GRE Math for People Who Hate Math: Backsolving

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You know what I love about GRE Discrete Quant problems? Specifically, multiple-choice Discrete Quant? The answer choices. Think about it: out of the infinite number of numbers in the universe, the GRE has already narrowed it down to just five possibilities. They’ve done almost all of the work for you. And that makes Discrete Quant a huge opportunity for People Who Hate Math.

It’s a lot easier to check whether someone else’s answer is correct than it is to solve a problem from scratch. For example, imagine trying to figure out exactly what you need to eat each day in order to lose a pound per week. It would be difficult and time-consuming to create a plan on your own. However, what if a nutritionist just gave you a meal plan and told you to follow it? Even if you didn’t know how to work out the exact answer on your own, it would be pretty easy to determine whether the nutritionist’s answer was correct. You’d just follow their meal plan for a couple of weeks and see what happened. If it worked, you’d know that you’d found the right answer, without working it out on your own. In fact, you wouldn’t even need to know anything about nutrition at all to use this approach.

There are thousands of different possible diets, but only five possible answers to a multiple-choice DQ problem. And it takes a couple of seconds, not a couple of weeks, to decide whether a particular answer is right. The GRE’s already told you that one of those five answers is the right one. So, why not just test each of them until you find one that works? It’s sometimes much easier than working out the answer on your own – and you can get away with a lot less math.

This style of problem solving is called backsolving. In order to use backsolving on a GRE math problem, a few conditions need to be met. Here they are:

• Multiple-choice problems only! This technique doesn’t work for Quantitative Comparisons, for instance.
• The answer choices have to be numbers. No variables, percents, inequalities, etc.
• The answer choices have to be useful. Useful means that when you look at the numbers, you can tell what math you could do with them. If you look at the answer choices and you aren’t sure where you’d go from there, backsolving might not work.

That’s it. If a problem meets those criteria, you can backsolve!

Backsolving is very similar to ‘checking your work’. When you decide to backsolve, imagine that I’ve just told you what I think the right answer is. Now, you need to determine whether I was right. Let’s try it on an example problem from the 5lb. Book of GRE Practice Problems.

If 150 were increased by 60% and then decreased by y percent, the result would be 192. What is the value of y?

(A) 20
(B) 28
(C) 32
(D) 72
(E) 80

Suppose I told you that the answer was 28, and you wanted to check my work. You’d turn to your calculator. You’d take 150, increase it by 60%, then decrease it by 28%. If the result came out to 192, you’d know that I was right. If it didn’t, that would mean I’d done something wrong.

150, increased by 60%, is 240.

240, decreased by 28%, is 172.8.

That’s not 192, so I was definitely wrong.

You don’t know the right answer, yet – but that doesn’t matter! The right answer must be there among the five answer choices, and you’ve just eliminated one of them. That leaves, at most, four more to check. Should we try an answer choice that’s higher or lower than 28?

If you said ‘lower’, you’re correct. The answer we got was too small, which means we decreased 240 by too large of a percentage. The only answer choice smaller than 28 is (A) 20, so that one must be correct. But if you have the time, and you don’t totally trust yourself, plug those numbers into your calculator to double check:

150, increased by 60%, is 240.

240, decreased by 20%, is 192.

20% is the right answer!

If you’ve ever ‘checked your work’ by plugging an answer back into the problem, you already know how to backsolve. Try it out on these mini problems. They only have two answers each – your job is to figure out which one is right and which one is wrong, without creating equations.

1. A tank that was 40% full of oil was emptied into a 20-gallon bucket. If the oil fills 35% of the bucket’s volume, then what is the total capacity of the tank, in gallons? (15 gallons or 17.5 gallons)
2. The population of antelope in a nature park increases by 10% every two years. If the population at the end of 2007 was 363 antelope, what was the population of antelope in the park at the end of 2003? (290 or 300)
3. After adding a 15% tip, the restaurant bill for a group of 8 people was $386.40. What was the average cost of each person’s meal before the tip was added? ($36 or $42) Think you’ve got it? Open up the 5lb. Book of GRE Practice Problems and try it out on some Discrete Quant problems from the Percents, Word Problems, or Ratios chapters. Remember, you don’t have to do GRE math problems in the same way you solved these problems in middle-school math class. If a solution gets you to the right answer quickly, it’s a good solution – so stay flexible and explore all of your options. ? Answers: 17.5, 300,$42

Want more guidance from our GRE gurus? You can attend the first session of any of our online or in-person GRE courses absolutely free! We’re not kidding. Check out our upcoming courses here.

Chelsey Cooley is a Manhattan Prep instructor based in Seattle, Washington. Chelsey always followed her heart when it came to her education. Luckily, her heart led her straight to the perfect background for GMAT and GRE teaching: she has undergraduate degrees in mathematics and history, a master’s degree in linguistics, a 790 on the GMAT, and a perfect 170/170 on the GRE. Check out Chelsey’s upcoming GRE prep offerings here.