Counter-Intuition on the GMAT, Part 3: Problem Solving
In this article, part three of a three-part series, we complete our discussion of counter-intuition on the GMAT. This week, we examine how an unreliable type of test-taker's intuition can disrupt your ability to answer Problem Solving questions. In part one, we covered the Draw a Conclusion question; in part two, we covered Data Sufficiency and Sentence Correction questions.
What exactly constitutes intuition in Problem Solving? The answer to this question will vary quite a bit, depending on the topic matter that is being tested. Intuition on probability questions will mean one thing, whereas intuition on geometry questions with a diagram will mean something else altogether. However, as we saw in earlier editions of this discussion, we will define the intuition that we are speaking of as that which causes the test-taker to hastily decide upon a certain answer choice because it seems to look right. Put differently, this intuition may be thought of as the mechanism that prevents the test-taker from delving into the deepest layer of the question and uncovering the mathematical principles that are being tested. One area where students commonly fall prey to the whims of this type of intuition is in percent problems. Let's take a look at the following example:
One week after its purchase, a stock's price fell to 5/6 of its original purchase price. Two weeks after its purchase, the stock was at 2/3 of its original purchase price. By what percent did the stock fall from week one to week two?
(A) 16 2/3%
(C) 33 1/3%
If we express the price of the stock after the first week as a percent, it equals 83 1/3% of the original price. Likewise the price of the stock after the second week equals 66 2/3% of the original price. A hasty glance at these two numbers reveals a decrease in the percent from week one to week two of 83 1/3 - 66 2/3 or 16 2/3. What's our mistake here? In our hastiness, we have glossed over the concept that is being tested here - percent change. Percent change deals with the change between two values and how that change relates to the original value.
The formula for percent change is: Percent change = (valuenew - valueold)/valueold.
The fact that the values given in the problem are percents themselves makes it easy to overlook that we are dealing with percent change. When we simply subtract the two values we come up with a percent, so it is easy to think that this difference equals the percent change!
The actual percent change should have been calculated in this way, where x equals the original price of the stock:
Notice that we kept the values here in fractions because fractions are typically easier to deal with than percents on the GMAT. As an alternative to using a variable to represent the original price of the stock here, we could have substituted in a value for the original price, making the problem a little less abstract. A good value to use here would have been $6, since taking 5/6 and 2/3 of 6 would leave us with integers:
What is important to notice in this example is that this question is designed to prey on the intuition of the unwary test taker. If the question had merely spoken about the percent change between two stock values given in dollars, it would have been much less effective in stymieing its audience. Most GMAT test takers know the percent change formula. What is out of the ordinary here is the use of the percent change formula to show a change in values that are themselves expressed as percents.
How then can the conscientious test taker prevent herself from falling for this ruse? The best thing to do is to stay alert and always be suspicious. If you find yourself finishing a Problem Solving question with a simple addition, subtraction, or multiplication of the given values, chances are you have jumped the gun. Take a step back and re-evaluate the concept that is being tested in the problem. Every Problem Solving question on the GMAT tests one or more mathematical concepts that must be mastered. If you identify the concept that is relevant for that particular question and apply sound principles to solve, you will adeptly steer clear of the perils of intuition on Problem Solving.
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