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Lesson: qc06

Quantitative Comparisons Strategies

Add or Subtract Across Columns to Simplify Comparison

If both expressions include the same term, you can safely “cancel” that term from each one either by adding or subtracting it. This technique may help simplify one or both of the expressions, thereby revealing the comparison to you. Remember: You don’t change the comparative value of two expressions merely by adding or subtracting the same terms from each one.

Example

Column AColumn B
The sum of all integers from 19 to 50, including 19 and 50 The sum of all integers from 21 to 51, including 21 and 51

Explaination

The correct answer is (B). The two number strings have in common integers 21 through 50. So you can subtract (or cancel) all of these integers from both sides of the comparison. That leaves you to compare (19 1 20) to 51. You can now see clearly that Quantity B is greater.

Avoid Multiplying or Dividing Across Columns

To help simplify the two expressions, you can also multiply or divide across columns, but only if you know for sure that the quantity you’re using is positive. Multiplying or dividing two unequal terms by a negative value changes the inequality; the quantity that was the greater one becomes the smaller one. So think twice before performing either operation on both expressions.

Example

Column A
Column B

x > y
x2y
xy2

Explaination

The correct answer is (D). To simplify this comparison, you may be tempted to divide both columns by x and by y—but that would be wrong. Although you know that x is greater than y, you don’t know whether x and y are positive or negative. You can’t multiply or divide by a quantity unless you’re certain that the quantity is positive. If you do this, here is what could happen: Dividing both sides by xy would leave the comparison between x and y. Given x > y, you’d probably select (A). But if you let x = 1 and y = 0, on this assumption, both quantities = 0 (they’re equal). Or let x = 1 and y = - 1. On this assumption, Quantity A = -1 and Quantity B = 1. Since these results conflict with the previous ones, you’ve proven that the correct answer is choice (D).

Solve Centered Equations for Values

If the centered information includes one or more algebraic equations and you need to know the value of the variable(s) in those equations to make the comparison, then there’s probably no shortcut to avoid doing the algebra.

Example

Column A
Column B

x + y = 6
x - 2y = 3

x
y

Explaination

The correct answer is (A). The centered information presents a system of simultaneous linear equations with two variables, x and y. The quickest way to solve for x is probably by subtracting the second equation from the first:
x + y = 6
-( x - 2y = 3)
3y = 3
y = 1
Substitute this value for y in either equation. Using the first one:
x + 1 = 6
x = 5
Since both equations are linear, you know that each variable has one and only one value. Since x = 5 and y = 1, Quantity A is greater than Quantity B.

Don’t Rely on Geometry Figure Proportions

For Quantitative Comparison questions involving geometry figures, never try to make a comparison by visual estimation or by measuring a figure. Even if you see a note stating that a figure is drawn to scale, you should always make your comparison based on your knowledge of mathematics and whatever nongraphical data the question provides, instead of “eyeballing” it.
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