SAT Math: Inventing Numbers
Here’s a question from the May 2007 exam to demonstrate the usefulness of “inventing numbers” when it comes to math problems with variables in the answer choices.
19. The toll for trucks to cross a certain bridge is a total of $2 for the first two axles plus $2 for each additional axle. Which of the following functions gives the toll T(n), in dollars, for a truck with n axles?
(A) T(n) = 2n – 2
(B) T(n) = 2n – 1
(C) T(n) = 2n
(D) T(n) = 2n + 1
(E) T(n) = 2n + 2
Don’t be intimidated by the use of the word “function” in this problem. A function is just an equation that, when x (or in this case n) is plugged in, will yield only one y (or in this case, T(n)). In other words, this problem asks you to find an equation to determine the total toll paid by a truck, which varies based upon the number of axles. N is the number of axles. T(n) is the total amount paid in tolls.
Start by inventing a number of axles (n). Make it a number that’s easy to work with. Let’s say that our truck has 3 axles (n = 3). How much money would we pay in tolls? Well, the first two axles cost $2, and the third axle costs another $2, so that’s $4 in total. Circle that answer.
Now plug your variable into the answer choices (n = 3) until you find the one that yields a value of $4. We can eliminate B, D, and E. But how do we determine whether A or C is correct?
Invent a different number. Say the truck has only two axles (n = 2). That truck would pay only $2 in tolls. Plugging n = 2 into the answer choices now eliminates choice C because 2n = 2(2) = $4 in tolls.
Therefore, the answer is choice A: T(n) = 2n – 2.
Make sure to always test all answer choices when you are using the “invent numbers” strategy. Do not stop just because you found one right answer, especially if the question appears toward the end of the section (as this one did). If you find duplicate correct answers, invent another number and try again.
