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QUANTITATIVE COMPARISONS
Generally, quantitative comparison questions require much less calculating than do multiple-choice questions. But they are trickier.

Substitution is very effective with quantitative comparison problems. But you must plug in all five major types of numbers: positives, negatives, fractions, 0, and 1. Test 0, 1, 2, -2, and 1/2, in that order.

General Principles For Solving Quantitative Comparisons

You Can Add or Subtract the Same Term (Number) from Both Sides of a Quantitative Comparison Problem.

You Can Multiply or Divide Both Sides of a Quantitative Comparison Problem by the Same Positive Term (Number). (Caution: This cannot be done if the term can ever be negative or zero.)

You can think of a quantitative comparison problem as an inequality/equation. Your job is to determine whether the correct symbol with which to compare the columns is <, =, >, or that it cannot be determined. Therefore, all the rules that apply to solving inequalities apply to quantitative comparisons. That is, you can always add or subtract the same term to both columns of the problem. If the term is always positive, then you can multiply or divide both columns by it. (The term cannot be negative because it would then invert the inequality. And, of course, it cannot be zero if you are dividing.)

Example:

Don't solve this problem by adding the fractions in each column; that would be too time consuming--the LCD is 120! Instead, merely subtract 1/5 and 1/8 from both columns:

Now 1/3 is larger than 1/4, so Column A is larger than Column B.

Note: If there are only numbers (i.e., no variables) in a quantitative comparison problem, then "not-enough-information" cannot be the answer. Hence (D), not-enough-information, cannot be the answer to Example 1 above.

Caution: You Must Be Certain That the Quantity You Are Multiplying or Dividing by Can Never Be Zero or Negative. (There are no restrictions on adding or subtracting.)

The following example illustrates the false results that can occur if you don't guarantee that the number you are multiplying or dividing by is positive.

Solution (Invalid): Dividing both columns by x yields

We are given that x < 1, so Column B is larger. But this is a false result because when x = 0, the two original columns are equal:

Hence, the answer is actually (D), not-enough-information to decide.

Caution: Don't Cancel Willy-Nilly.

Some people are tempted to cancel the 4x from both columns of the following problem:

You cannot cancel the 4x's from both columns because they do not have the same sign. In Column A, 4x is positive. Whereas in Column B, it is negative.

Substitution (Special Cases)

A. In A Problem with Two Variables, Say, x and y, You Must Check The Case in Which x = y. (This often gives a double case.)

Example:

If x = y = 2, then 2(x + y) = 2(2 + 2) = 8 and 2xy = 2(2)(2) = 8. In this case, the columns are equal. For all other choices of x and y, Column B is greater. (You should check a few cases.) Hence, we have a double case, and therefore the answer is (D).

B. When You Are Given That x < 0, You Must Plug in Negative Whole Numbers, Negative Fractions, and -1. (Choose the numbers -1, -2, and -1/2, in that order.)

C. Sometimes You Have to Plug in The First Three Numbers (But Never More Than Three) From a Class of Numbers.

Example: