Lesson: qc05
Quantitative Comparisons Strategies Continued
Choose (D) Only If You Need More Information
Remember: Pick choice (D) only if you can’t make a comparison without having more
information. But if the comparison at hand involves numbers only, you’ll always be
able to calculate specific numerical values for both expressions (assuming you have
time to do the math). You certainly don’t need more information just to compare the
size of two specific numbers.
Example
| Column A |
| Column B |
|---|
| x * y = x(x - y) |
|
| (-1 * -2) * (1 * 2) |
| (1 * -2) * (-1 * 2) |
Explaination
The correct answer is (C). The centered information contains variables—but
calculating the two quantities involves only numbers. Thus, the correct answer
cannot be (D). Just for form, let’s work the problem. For both quantities, first
apply the operation (defined by the symbol * ) to each parenthesized pair, then
apply it again to those results:
Quantity A:
(-1 * -2) = -1(-1 -[-2]) = -1(1) = -1
(1 * 2) = 1(1 - 2) = 1(-1) = -1
Apply the defined operation again, substituting -1 for both x and y:
(-1 * -1) = -1(-1 - [-1]) = -1(0) = 0
Quantity B:
(1 * -2) = 1(1 - [-2]) = 1(3) = 3
(-1 * 2) = -1(-1 - 2) = -1(-3) = 3
Apply the defined operation again, substituting 3 for both x and y:
(3 * 3) = 3(3 - 3) = 3(0) = 0
As you can see, both quantities equal zero (0).
Consider All Possibilities for Variables
When comparing expressions involving variables, unless the centered information
restricts their value, consider positive and negative values, as well as fractions and
the numbers zero (0) and 1. Comparisons often depend on which sort of number is
used. In these cases, the correct answer would be choice (D).
Example
| Column A |
| Column B |
|---|
| P > 0
P ? 1 |
|
| p-3 |
| p4 |
Explaination
The correct answer is (D). Here’s the general rule that applies to this
problem: p
x=1/p
x. Hence, if p > 1, then p
-3 must be a fraction less than 1
while p
4 is greater than 1, and Quantity B is greater. On the other hand, if p < 1, then the opposite is true.
Rewrite a Quantity to Resemble Another
If you have no idea how to analyze a particular problem, try manipulating one or both
of the expressions until they resemble each other more closely. You may be able to
combine numbers or other terms, do some factoring, or restate an expression in a
slightly different form.
Example
| Column A | Column B |
|---|
| 4 - x2 | (2 + x) (2 - x) |
Explaination
The correct answer is (C). Perhaps you recognized that (4 - x
2) is the
difference of two squares (2
2 and x
2) and that the following equation applies: a
2
- b
2 = (a + b)(a - b). If so, then you saw right away that the two quantities
are the same. If you didn’t recognize this, you could tweak Quantity B by
multiplying (2 - x) by (2 + x) using the FOIL method:
(2 + x)(2 - x) = 4 + 2x - 2x - x
2 = 4 - x
2
Rewriting Quantity B,then simplifying it, reveals the comparison. The two
quantities are equal.
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