Plugging in - A Basic Math Strategy for the ACT, SAT and GRE

Plugging In Answer Choices

In many math problems on standardized tests such as the ACT, SAT, and GRE, you can get the answer simply by trying each of the answer choices until you find the one that works. Unless you have some intuition as to what the correct answer might be, then you should always start with choice C as your first guess (on the new SAT you can start with choice B or C because there are four choices instead of five). The reason for this is simple. Answers are usually given in increasing or decreasing order. So very often if choice C fails you can eliminate two of the other choices as well.

There are a few exceptions to this rule. If the word least appears in the problem, then start with the smallest number as your first guess. Similarly, if the word greatest appears in the problem, then start with the largest number as your first guess.

Examples

Let’s take a look at two math problems – one where we start with choice C and one where we do not.

Level 4 Geometry Question

When each side of a given square is lengthened by 3 inches, the area is increased by 45 square inches. What is the length, in inches, of a side of the original square?

A) 3
B) 4
C) 5
D) 6
E) 7

Let’s start with choice (C). If the original length of a side of the square is 5, then the length becomes 8 when we increase it by 3. The original square has an area of 5²= 25 and the new square has area 8² = 64. So the area is increased by 64 – 25 = 39 square inches. Thus, we can eliminate choice (C), and most likely (A) and (B) as well.

We next try choice (D). We have 6² = 36, 9² = 81 and 81 – 36 = 45. Thus, the answer is choice (D).

Here is an algebraic solution for those of you that really want to see it.

Let x be the length, in inches, of a side of the original square. The length of a side of the new square is x + 3. The area of the original square is x², and the area of the new square is:

(x + 3)² = (x + 3)(x + 3) = x² + 6x + 9.
(x² + 6x + 9) – x² = 45
6x + 9 = 45
6x = 36
x = 6

Thus, the answer is choice (D).

For this particular question, I prefer the solution by starting with choice C to the more tedious and confusing algebraic solution.

Level 3 Number Theory Question

What is the largest positive integer value of k for which 3^k divides 18⁴?

A) 2
B) 4
C) 6
D) 7
E) 8

Note: This type of question is no longer tested on the SAT, but it could show up on the ACT or GRE.

Pull out your calculator. Since the question has the word “largest” in it, we will start with the largest answer choice which is choice (E), and we will divide 18⁴ by 3⁸. We type 18^4 / 3^8 into our calculator and the output is 16. Since 16 is an integer, the answer is choice (E).

Note that all five answer choices give an integer, but 8 is the largest positive integer that works.

Here is a direct solution for those of you who really want to see it.

The prime factorization of 18 is 18 = 2·3². Therefore:

18⁴ = (2·3²)⁴ = 2⁴(3²)⁴ = 2⁴3⁸.

From this prime factorization it should be clear that 3⁸ divides 18⁴, but 3⁹ does not.

Again, for this particular question, most students will prefer the easier solution of starting with choice E (the largest answer choice) to the more confusing algebraic solution.