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Hard Quantitative Comparisons

Most of the time, we have an intuitive feel for whether a problem is hard or easy. But on tricky problems (problems that appear easy but are actually hard) our intuition can fail us.

There are special techniques and strategies that apply to the hard problems only. Do not apply the methods of this section to the easy or medium quantitative comparison problems.

Strategy: On Hard Quantitative Comparison Problems, The Obvious Answer (The Eye-Catcher) Will Almost Always Be Wrong. (If one expression looks at first glance to be larger than another, then it will not be.)


This is so because when people cannot solve a problem, they most often pick the answer-choice that “looks right.” But if that were the answer, most people would answer it correctly and therefore it would not be a “hard” problem.
x ≥ 1
Column A Column B
x10 x100

One would expect x100 to be larger than x10. But this is a hard problem and therefore what we expect will not be the answer (it's the eye-catcher). Now, clearly x100 cannot always be less than x10. And just as clearly x100 cannot always be equal to x10. Hence, the answer is (D)—not-enough-information. (A double case can also be obtained by substituting x = 1 and then x = 2.)

Column A Column B
The number of distinct prime factors of x The number of distinct prime factors of 4x

We expect the number of prime factors of 4x to be larger than the number of prime factors of x. But that is the eye-catcher. Now, the number of prime factors of 4x cannot be less than the number of prime factors of x since 4x contains all the factors of x. So the answer must be that either they are equal or there is not enough information. In fact, there is not enough information, as can be verified by plugging in the numbers x = 2 and then x = 3.


Column A Column B
The area of a square with perimeter 12 The area of a parallelogram with perimeter 16

We expect the area of the parallelogram to be larger. After all, the parallelogram could be a square with perimeter 16, which of course has a larger area than a square with perimeter 12. But that would be too easy.


Hence, there must be a parallelogram whose area is equal to or less than the area of the square. (See whether you can draw it. Hint: Look at the extreme cases.)


Thus, we have a double case, and the answer is (D)—not-enough-information.


Note 1: When plugging in on quantitative comparison problems, be sure to check 0, 1, 2, –2, and 1/2, in that order.


Note 2: If there are only numbers in a quantitative comparison problem, i.e., no variables, then (D), not-enough-information, cannot be the answer.


Note 3: When drawing geometric figures, don’t forget extreme cases.


Strategy: Eliminate Answer-Choices That Are Too Easily Derived or Too Ordinary.


x — y = 3
Column A Column B
x + y 4

The numbers 3 and 1 are solutions to the equation x — y = 3 because 3 — 1 = 3. So for this choice of x and y, Column A equals Column B, since 3 + 1 = 4. But that is too easy: Everyone will notice 1 and 3 as solutions of the equation x — y = 3. Hence, there must be another pair of numbers whose product is 3 and whose sum is not 4. In fact, there are an infinite number of pairs. For example, , but 9 + 1/3 ≠ 4. This is a double case and therefore the answer is (D).


Column A Column B
The greatest number of regions into which two straight lines will divide the shaded region. 4

Most people will draw one or the other of the two drawings below:
In each case, four separate shaded regions are formed. But these drawings are too ordinary, too easy. There must be a way to draw the lines to form more than four regions. Try to draw it before looking at the answer below.
The lines must intersect in the shaded region.

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