Substitution (Quantitative Comparisons)
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

x^{2}

A. Column A
B. Column B
C. Both are Equal
D. Cannot be Determined
If x = 2, then x^{2} = 4. In this case, Column B is larger.
However, if x equals 1, then x^{2} = 1. In this case, the two columns are equal.
Hence, the answer is (D)â€”not enough information to decide.
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

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.