Are you making this mistake?

To better understand a mistake I see all the time while tutoring, put on your game face, set a timer, and try answering the following question from the Official Guide (no peeking at the solution below): 

A store currently charges the same price for each towel that it sells. If the current price of each towel were to be increased by $1, 10 fewer of the towels could be bought for $120, excluding sales tax. What is the current price of each towel?

(A) $ 1

(B) $ 2

(C) $ 3

(D) $ 4

(E) $12

Let’s examine two very different approaches:

 

The approach that would please your former math teachers and probably earn you a gold star for showing all of your work

  • Assign variable expressions to current price per towel (x) and increased price per towel (x+1).
  • Produce an equation something like this: 120/x = 120/(x+1) + 10
  • Algebra, algebra, algebra ….voila: x² + x - 12 = 0
  • Factor: (x + 4)(x - 3) = 0, which means either x = -4 or x = 3
  • Since towel prices can’t be negative, the correct answer is 3.
  • Locate 3 among the answer choices…..select C

The next approach uses the fact that, with GMAT math questions, the correct answer is hiding among 5 answer choices. This wonderful feature should be exploited at every opportunity.

 

Recognizing that this question is only as difficult as you want it to be

As I’m reading a GMAT math question for the first time, I’m determining whether it’s possible to solve it by testing the answer choices. If it is possible, then this becomes Plan A. I then give myself 20 seconds to identify a faster approach. If I don’t spot a faster approach, I start testing answer choices. That said, for this particular question, I’d probably start testing the answer choices immediately, since they work so nicely with $120.

A) $1. At this price, I can buy 120 towels. At the increased price of $2, I can buy 60 towels.

That’s 60 fewer towels. The question says 10 fewer towels. ELIMINATE A.

B) $2. At this price, I get 60 towels. At $3 per towel, I get 40 towels.

That’s 20 fewer towels. ELIMINATE B.

C) $3. This gets me 40 towels. $4 gets me 30 towels.

That’s 10 fewer towels. DONE!

Aside: In most cases, I’d start by testing the middle value (C) and then, if that’s not correct, use the outcome to determine which numbers to test next. However, since the answer choices are so easy to work with, I‘d probably just start with A.

It’s worth noting that a lot of students resist any strategy that doesn’t match the strategies they learned in school. This resistance is apparent in GMAT Club’s statistics for the towel question, where it has a difficulty rating in the 600-700 range. This inflated difficulty level is largely due to the fact that the average student took 2 minutes and 21 seconds to answer the question correctly, and 2 minutes and 37 seconds to answer it incorrectly. So, even if you answered it correctly, you may have lost some time in the process.   

The funny thing is, for this question, testing the answer choices is crazy fast (30 seconds tops), AND it requires a low-tech mathematical operation you learned in elementary school (division). Conversely, the algebraic approach requires you to create and solve a quadratic equation. This approach is like renting a backhoe to plant a sunflower seed.

 

One final point about testing the answer choices

Even if you use algebra to create the equation 120/x = 120/(x+1) + 10, you’re under no obligation to solve the equation algebraically. You can still use your fancy equation to conveniently test the answer choices. Doing so might actually be faster, and you’re less likely to make a mistake testing the answer choices than using algebra.   

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