# Build a Math Tool Kit for the PSAT/NMSQT

One of the first things that every do-it-yourselfer learns is that the proper tool makes all the difference. You don’t need a saw or a screwdriver on the PSAT/NMSQT, but a couple of special techniques help you nail the math questions. What techniques? Read on.

## Plugging in

*Plugging in* is a great technique for solving lots of PSAT/NMSQT problems, especially those involving percents and variables. To plug in, pick a number — almost any number — and work through the problem with that number. Imagine a problem involving percents, such as this one:

A tasteful, orange-and-purple shirt is marked down 40%, but somehow it fails to sell. The store owner lowers the price by an additional 10%. What is the total discount on this fashion-forward item?

(A) 25%

(B) 30%

(C) 35%

(D) 46%

(E) 50%

The answer is Choice (D). The question doesn’t explain how much the shirt cost originally (or who chose the colors). No worries: Just choose a number. For percent problems, 100 is always a good bet. Now work through the problem.

The original price is $100. The first discount is $40, so the new price is $60. The next discount is 10% of $60, or $6. Subtract $6 from $60, and the new price is $54. The original price was $100, so the discount is $100 – $54, or $46. That means that the total discount is 46%, also known as Choice (D).

Here’s another example:

During the hours marked on Jeannie’s calendar as “PSAT/NMSQT Prep,” Jeannie actually spends ½ her time watching reality TV shows. She devotes 2/3 of the remaining prep time to shredding old love letters. During what proportion of the time Jeannie claims to be studying is she actually preparing for the PSAT/NMSQT?

(A) 1/6

(B) 1/3

(C) 1/2

(D) 2/3

(E) 5/6_{}

The answer is Choice (A). You can solve this problem with algebra, naming the time studying as *x. *However, you can also plug in. You don’t know how much time Jeannie *said *she was studying. Her mom checks her calendar, so chances are it’s a respectable amount. Plug in a number.

Because you’re dealing with 1/2 and 2/3, you probably want those denominators to be factors of the number you select. How about 12? Jeannie said she would study for 12 hours, but she watched TV for 6 hours. Subtract 6 from 12, and you have 6 hours left for study. Jeannie shreds her letters for 2/3_{ }of the remaining time, or 4 hours. She has 2 hours for study remaining.

Go back to your plug-in number, 12, and you see that Jeannie spent 2/12, or 1/6, of her time studying. Your answer is Choice (A).

## Backsolving

A variation of plugging in is *backsolving.* This technique is great for simple equations or arithmetic problems. When you backsolve, you plug in the answer choices to see which one works.

Generally, the answer choices are listed in size order — from the smallest to the largest number. Start with Choice (C), which falls in the middle. When you try that answer, you may realize that Choice (C) is too big, and then you know you have to try Choices (A) and (B). Or, you may discover that Choice (C) is too small, and then you can check Choices (D) and (E).

Take a look at these example problems, each answered by backsolving:

A number is tripled, increased by 4, and then halved. If the result is 8, what was the number?

(A) 2

(B) 4

(C) 8

(D) 12

(E) 16

The answer is Choice (B). You *could* solve with algebra, letting *x* represent the original number. However, backsolving works just fine. Try Choice (C), 8, as the original number and see what happens: 8 tripled is 24, which becomes 28 when increased by 4, and then 14 when halved.

Fourteen is too big, so try an answer that’s smaller than Choice (C); Choice (B) is a good next try. If the original number is 4, it becomes 12 when tripled, 16 when increased by 4, and then 8 when halved — the result you want! The correct answer is Choice (B).

Iff(x) =x^{2}– 3x– 2, what value ofxresults inf(x) = 2?

(A) 1

(B) 2

(C) 3

(D) 4

(E) 5

The answer is Choice (D). You can answer this question by creating a quadratic equation and then factoring, but it may be easier for you to backsolve. As usual, start with Choice (C) and go from there. If *x* is 3, you get *f*(3) = (3)^{2} – 3(3) – 2 = 9 – 9 – 2 = –2. Uh-oh, –2 is too small. Try a larger answer, Choice (D). If *x* is 4, you get *f*(4) = (4)^{2} – 3(4) – 2 = 16 – 12– 2= 2, the answer you’re looking for!

## Sketching a diagram

You know those annoying problems where one friend is driving west and the other is on a train heading east, both moving at different speeds? (Why doesn’t everyone just stay home? But back to math.) You may find that a little sketch allows you to “see” the answer or at least the route to the answer. Here’s an example:

Stan and Evan leave school to bicycle home. Both boys ride at a rate of 15 miles per hour. Evan rides directly east for 12 minutes to get home, and Stan rides directly south for 16 minutes to get to his home. How many miles apart are Evan’s and Stan’s homes?

(A) 4

(B) 5

(C) 10

(D) 15

(E) 20

The answer is Choice (B). Diagram time! Make sure you label your diagram so you get a good sense of what’s going on in the problem. But first, determine how far each of the boys live from school.

To get home, Evan rides for 12 minutes, or 1/5 of an hour, meaning that he travels (15 miles per hour) x (1/5 hour) = 3 miles. The formula is (rate) x (time) = distance. Stan rides for 16/60 of an hour, so his distance is (15 miles per hour) x (^{16}/_{60} hour) = 4 miles.

Hopefully you noticed that you have a right triangle, which means that you can use the Pythagorean theorem. Recall that *a*^{2} + *b*^{2} = *c*^{2}, where *a* and *b* are the legs of the triangle and *c* is the hypotenuse. In this case, 3^{2} + 4^{2} = 5^{2}, so Stan and Evan live 5 miles apart, Choice (B).

## Keeping it real

The PSAT/NMSQT doesn’t always give you real-world problems (not counting its role in ruining your life), but sometimes you can use your knowledge of how the world works to help you on the exam. If you’re solving a problem involving decreasing prices, you know that you’re never going to get more than a 100 percent reduction. No store pays you to haul the stuff away!

Nor will you find that 110 students are studying Spanish if the problem tells you that the school has only 50 kids. Keep your eye on reality. If your answer doesn’t fit, go back and try again.

## Using the booklet

Only your answer sheet is graded, but your question booklet is actually a valuable tool for PSAT/NMSQT math. As you read each question, circle key ideas (*integers, largest, less than,* and other such words). The little circles help you focus on the question’s important elements.

Also, use the blank space around each question to jot down the calculations you’re doing to arrive at an answer. If you come up with –12 and none of the answer choices matches that number, you can check your steps to see if you wrote a 2, for example, when you intended to write 4.

If you’ve spent more than a minute on one problem, even if you aren’t done with finding the answer, you should probably move on to the next. If you have time, you can return to that problem. Having the steps written in your booklet helps you jump in where you left off.