# Substitution

Substitution is a very useful technique for solving SAT math problems. It often reduces hard problems to routine ones. In the substitution method, we choose numbers that have the properties given in the problem and plug them into the answer-choices. A few examples will illustrate.

Example-1

If n is an odd integer, which of the following is an even integer?

A.   n3
B.   n/4
C.   2n + 3
D.   n(n + 3)
E.

We are told that n is an odd integer.

So choose an odd integer for n, say, 1 and substitute it into each answer-choice.

Now, n3 becomes 13 = 1, which is not an even integer.
So eliminate (A).

Next, n/4 = 1/4 is not an even integerâ€”eliminate (B).

Next, 2n + 3 = 2 Ã— 1 + 3 = 5 is not an even integerâ€”eliminate (C).

Next, n(n + 3) = 1(1 + 3) = 4 is even and hence the answer is possibly (D).

Finally,  = 1, which is not evenâ€”eliminate (E).

• When using the substitution method, be sure to check every answer-choice because the number you choose may work for more than one answer-choice. If this does occur, then choose another number and plug it in, and so on, until you have eliminated all but the answers. This may sound like a lot of computing, but the calculations can usually be done in a few seconds.
Example-2

If n is an integer, which one of the following CANNOT be an even integer?

A.   2n + 2
B.   n â€“ 5
C.   2n
D.   2n + 3
E.    5n + 2

Choose n to be 1.
Then 2n + 2 = 2(1) + 2 = 4, which is even. So eliminate (A).

Next, n â€“ 5 = 1 â€“ 5 = â€“4. Eliminate (B).

Next, 2n = 2(1) = 2. Eliminate (C).

Next, 2n + 3 = 2(1) + 3 = 5 is not evenâ€”it may be our answer.

However, 5n + 2 = 5(1) + 2 = 7 is not even as well.

So we choose another number, say, 2.
Then 5n + 2 = 5(2) + 2 = 12 is even, which eliminates (E).

Thus, choice (D), 2n + 3, is the answer.

Example-3

If x/y is a fraction greater than 1, then which one of the following must be less than 1?

A.   3y/x
B.   x/3y
C.

D.   y/x
E.    y

We must choose x and y so that x/y > 1.

Let's choose x = 3 and y = 2.

Then  is greater than 1, so eliminate (A).

Next, , which is less than 1â€”it may be our answer.

Next, ; eliminate (C).

Now, , so it too may be our answer.

Next, y = 2 > 1; eliminate (E).

Hence, we must decide between answer-choices (B) and (D). Let x = 6 and y = 2.

Then , which eliminates (B).