**Substitution**

Substitution is a very useful technique for solving GRE 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.

[Select One or More Answer Choices]

**Example 1:**

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

- n
^{3} - n/4
- $2n+3$
- $n(n+3)$
- $\sqrt{n}$
- 2n - 4

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, *n*^{3} becomes 1^{3}= 1, which is not an even integer. So eliminate (A).

Next, $\frac{n}{4}=\frac{1}{4}$ is not an even integerâ€”eliminate (B).

Next, $2n+3=2\xe2\u20ac\xa21+3=5$ is not an even integerâ€”eliminate (C).

Next, $n(n+3)=1(1+3)=4$ is at least one of the answers.

Next, $\sqrt{n}=\sqrt{1}=1$, which is not evenâ€”eliminate (E). **Finally, 2n - 4 = 2(1) - 4 = -2, which is even and therefore is another answer. Thus, the answer consists of choices (D) and (F).**

**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 answer. 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 of the following CANNOT be an even integer?

- $2n+2$
- $n-5$
- $2n$
- $2n+3$
- $5n+2$

Choose *n* to be 1. Then $2n+2=2\xc3\u20141+2=4$, which is even. So eliminate (A).

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

Next, 2*n* = 2(1) = 2. Eliminate (C).

Next, $2n+3=2\xc3\u20141+3=5$ is not even - it *may* be our answer. However, $5n+2=5\xc3\u20141+2=7$ is not even as well. So we choose another number, say, 2. Then $5n+2=5\xc3\u20142+2=12$ is even, which eliminates (E).

**Thus, choice (D) is the answer.**

**Example 3:**

If $\frac{x}{y}$ is a fraction greater than 1, then which of the following must be less than 1?

- $\frac{3y}{x}$
- $\frac{x}{3y}$
- $\sqrt{\frac{x}{y}}$
- $\frac{y}{x}$
*y*

We must choose *x* and *y* so that $\frac{x}{y}>1$. So choose *x* = 3 and *y* = 2. Now, $\frac{3y}{x}=\frac{3\xe2\u20ac\xa22}{3}=2$ is greater than 1, so eliminate (A).

Next, $\frac{x}{3y}=\frac{3}{3\xc3\u20142}=\frac{1}{2}$, which is less than 1â€”it may be our answer.

Next, $\sqrt{\frac{x}{y}}=\sqrt{\frac{3}{2}}>1$ eliminate (C).

Now, $\frac{y}{x}=\frac{2}{3}<1$. 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 $\frac{x}{3y}=\frac{6}{3\xc3\u20142}=1$, which eliminates (B).

**Therefore, the answer is (D).**

**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 $\frac{1}{2}$ , 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.]

**Column A**: *x*, **Column B**: *x*^{2}, where *x* â‰ 0

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.

If, as above, you get a certain answer when a particular number is substituted and a different answer when another number is substituted (this is called a "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?

**Column A**: <$\sqrt{x}$>, **Column B**: *x*, where *x* â‰¥ 0

If *x* = 0, then <$\sqrt{x}$> = <$\sqrt{0}$> = <0> = 0. In this case, Column A equals Column B. Now, if *x* = 1, then <$\sqrt{x}$> = <$\sqrt{1}$> = 1. In this case, the two columns are again equal. But if *x*= 2, then <$\sqrt{x}$> = <$\sqrt{2}$>, which is greater than 1 but less than 2. Thus, in this case Column B is larger. This is a double case. Hence, the answer is (D) â€” not enough information to decide.

**Substitution (Plugging In)**

Sometimes instead of making up numbers to substitute into the problem, we can use the actual answer-choices. This is called Plugging In. It is a very effective technique but not as common as Substitution.

**Example:**

The digits of a three-digit number add up to 18. If the tenâ€™s digit is twice the hundredâ€™s digit and the hundredâ€™s digit is $\frac{1}{3}$ the unitâ€™s digit, what is the number?

- 246
- 369
- 531
- 855
- 893

First, check to see which of the answer-choices has a sum of digits equal to 18. For choice (A), 2 + 4 + 6 â‰ 18. Eliminate. For choice (B), 3 + 6 + 9 = 18. This may be the answer. For choice (C), 5 + 3 + 1 â‰ 18. Eliminate. For choice (D), 8 + 5 + 5 = 18. This too may be the answer. For choice (E), 8 + 9 + 3 â‰ 18. Eliminate. Now, in choice (D), the tenâ€™s digit is not twice the hundredâ€™s digit, 5 â‰ 2 â€¢ 8. Eliminate. Hence, by process of elimination, the answer is (B). Note that we did not need the fact that the hundredâ€™s digit is $\frac{1}{3}$ the unitâ€™s digit.