# Substitution (Quantitative Comparisons)

When substituting in quantitative comparison problems, donâ€™t rely on only positive whole numbers. You must also check negative numbers, fractions, 0, and 1 because they often give results different from those of positive whole numbers. Plug in the numbers 0, 1, 2, â€“2, and 1/2, in that order.

Example-1

Determine which of the two expressions below is larger, whether they are equal, or whether there is not enough information to decide. [The answer is (A) if Column A is larger, (B) if Column B is larger, (C) if the columns are equal, and (D) if there is not enough information to decide.]

x â‰  0

 Column A Column B x x2
Which column is greater?
A. Column A
B. Column B
C. Both are Equal
D. Cannot be
Determined

If x = 2, then x2 = 4. In this case, Column B is larger.

However, if x equals 1, then x2 = 1. In this case, the two columns are equal.

Hence, the answer is (D)â€”not enough information to decide.

Note: If, as above, you get a certain answer when a particular number is substituted and a different answer when another number is substituted (Double Case), then the answer is (D)â€”not enough information to decide.

Example-2

Let <x> denote the greatest integer less than or equal to x.  For example:  <5.5> = 5 and <3> = 3.  Now, which column below is larger?

x â‰¥ 0

 Column A Column B x
Which column is greater?
A. Column A
B. Column B
C. Both are Equal
D. Cannot be Determined

If x = 0, then .

In this case, Column A equals Column B. Now, if x = 1, then .

In this case, the two columns are again equal. But if x = 2, then .

Thus, in this case Column B is larger. This is a double case.

Hence, the answer is (D)â€”not enough information to decide.