GRE Quant in Focus: Mind-Melting Arithmetic

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Calculator Malfunction!Decimal Planet-cc by 20-Dave Bleasdale

The GRE lets you use a calculator. Hooray! But wait, there's a catch. You can't bring your own calculator. No, you're stuck with the equivalent of a 1987 Windows 2.0 on-screen calculator. Worse still, ETS sometimes seems to go out of its way to make sure that you can't solve the problems using the provided calculator.  

That's the trick. It's a classic bait-and-switch: you want to plug and chug some numbers through the calculator, but you can't. So what is the calculator good for? It's an excellent tool to make sure you don't make arithemetic errors after you have set up a problem correctly. In fact, most GRE problems are constructed such that all calculations could be done without a calculator without too much difficulty, although there are some exceptions. 

In today's post, we will examine a seemingly insuperable arithmetic problem and how to use problem-solving skills to obviate difficult and time-consuming calculations. 

You Cannot Be Serious!

Consider the following problem:


  1. 10-10
  2. 4(10-10)
  3. 4(10-5)
  4. 3(10-5)
  5. 10-5

Wait, what?

"Okay," you're thinking, "Time to cash in my chips. Maybe there's another easier problem to do on this section." You might be surprised, but chances are you are right. It is probably best to tackle a problem like this after you have made at least one pass through a Quant section. That way you can make sure that you have bagged all the easier points before attempting a problem that could suck up valuable time.

However, with some reasoning and observation, you might find that this problem is far less difficult than it appears. 

Analyze the Answer Choices

What do you notice about the answers? They are all expressed as negative exponents. This problem is an excellent reminder of the importance of knowing your math fundamentals cold (and not a bad time to encourage you to register for our free Math Essentials webinar).

As a refresher:

negexp.pngDo you notice a pattern here among the answer choices?

It seems as though we're talking about either some product of either 10-10 or 10-5. Now let's take a second look at the top problem. How can we use what we've observed in the answer choices to help us evaluate the problem?

We might notice that the numerator of the first fraction is a decimal followed by ten 9s. The denominator is 1 plus a decimal, four 0s and then a 1. We could safely assume that the numerator somehow involves 10-10 and the denominator involves 10-5. We should also note that the overall fraction is going to be very close to 1, slightly smaller than 1 in fact. 

With this in mind, examine the second fraction. There appears to be the same kind of 10-10 and 10-5 relationship between the numerator and the denominator but this time we add into the mix 4 and 84. But is 84 really how we should look at this numerator? Keep that 4 in mind. The denominator is more than 1. The numerator is less than one. How much less than one? 84 is 16 less than 100. So let's remember that 16.

It seems as though there's something in common between the two fractions: the numerator seems to be related to the square of the denominator. 

Without doing any more math or analysis, we could note that the difference of 4 and 1 is 3. Take another look at the answers. If I had to make an educated guess, I would go with D. Why?

  1. 10-10. A likely trap answer. This answer completely neglects any kind of 4 and 1 relationship.
  2. 4(10-10). A trap. We have neglected any difference/subtraction operation. Leaving the 4 in there indicates more of a multiplication operation between the multiplicand 1 and the multiplier 4.
  3. 4(10-5). Clever. Also a trap. By repeating the multiplicand 4 in both answer choices B and C, students might think, "Well, it must be either B or C," but again, as with B, this answer neglects the difference/subtraction in the problem.
  4. 3(10-5). This is the correct answer! It involves subtraction: 4 - 1 = 3.
  5. 10-5. A trap answer similar to A. This also completely neglects any kind of 4 and 1 relationship.

Can we solve the problem algebraically?

"That's some cool reasoning, but what if I'm more comfortable doing algebra?" You can definitely solve this problem directly as follows:




















That's some fun algebra, but remember, you don't get any points for showing your work on the GRE!

There's no "right" or "wrong" way to do a Quant problem 

Just as we discussed in our previous two "Quant in Focus" posts (here and here), the right way to do any Quant problem is any way that helps you choose the correct answer as quickly as possible. Success on GRE Quant is a combination of speed and accuracy. Therefore, in your practice, you should experiment with multiple ways to find solutions. Most practice problems offer excellent opportunities to improve your approach and find shortcuts. 

We hope you have found this post helpful. If you have questions or would like to offer your own alternate approach, please comment below! For further free instruction on GRE Quant fundamentals, please consider subscribing to this blog or registering to attend a free GRE webinar. The next webinar will cover GRE Math Essentials and will be held June 14. We hope to see you there!

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