PSAT/NMSQT Prep 2022 - Eggert M.D., Strelka A. 2022

How to do PSAT math
The method for PSAT math questions
PSAT math

LEARNING OBJECTIVE

After completing this chapter, you will be able to:

· Efficiently apply the Math Method to PSAT Math questions

How to do PSAT math

PSAT Math questions can seem more difficult than they actually are, especially when you are working under time pressure. The method we are about to describe will help you answer PSAT questions, whether you are comfortable with the math content or not. This method is designed to give you the confidence you need to get the right answers on the PSAT by helping you think through each question logically, one piece at a time.

Take a look at this question and take a minute to think about how you would attack it if you saw it on test day:

The Collins Library is one of four public libraries in Madison County. According to data maintained by the county’s public library system, 58 percent of the 15,000 books in Collins Library are fiction titles. If Collins Library is representative of the public libraries in Madison County with regard to the number of fiction vs. nonfiction titles, and the average number of books per public library in Madison County is 15,000, then which of the following is the best estimate of the total number of nonfiction titles held by public libraries in Madison County?

A. 8,700

B. 25,200

C. 34,800

D. 60,000

Many test takers will see a question like this and panic. Others will waste a great deal of time reading and rereading without a clear goal. You want to avoid both of those outcomes.

Start by defining clearly for yourself what the question is actually asking. What do the answer choices represent? In this question, they represent the number of nonfiction books in all the public libraries in Madison County.

Next, examine the information that you have and organize it logically. The question asks about the number of nonfiction books. Okay, then what information do you have about numbers of books? You know that 58% of the 15,000 books in Collins Library are fiction. That’s the opposite of nonfiction. You can deduce that 100% − 58% = 42% of the 15,000 books in Collins Library are nonfiction books.

Now, make a strategic decision about how to proceed. The answer choices are far apart, so you might consider rounding 42% to and estimating. However, this question appears on the calculator section, and it’s a quick calculation. Let’s say that you decide to use your calculator. Plug the numbers into your calculator and jot down what you know so far:

Nonfiction total in Collins Library: 0.42 × 15,000 = 6,300

The question asks for the number of nonfiction books in Madison County, so hunt for information tying Collins Library to Madison County. You’re told that the average number of books per public library in Madison County is 15,000, which is identical to the number of books in Collins Library, and that Collins Library is “representative” of the libraries in Madison County. Translation: what is true for Collins Library is also true for all public libraries in Madison County. You also know that there are four public libraries in Madison County. You can deduce that the number of nonfiction books in Collins Library (6,300) times the total number of public libraries in Madison County (4) will give you the number of nonfiction books in all of Madison County. Plug that into your calculator:

6,300 × 4 = 25,200

Finally, confirm that you answered the right question: you want the number of nonfiction books in all public libraries in Madison County. That’s what you calculated, so you’re done; the correct answer is (B).

Here are the steps of the method we just used:

You can think of these steps as a series of questions to ask yourself: What do they want? What are they giving me to work with? How should I approach this? Did I answer the right question?

Not all PSAT Math questions will require time spent on all of the steps. The question above, because it is a word problem, required a fair amount of analysis in steps 1 and 2, but choosing an approach (step 3) was straightforward; the calculations were quick to do on a calculator, so there was no need to estimate. Other questions will require very little thought in steps 1 and 2, but will benefit from a careful strategy decision in step 3. Step 4 is always quick, but you should always do it: just make sure you answered the question that was actually asked before you bubble in your response. Doing so will save you from speed mistakes on questions that you know how to do and should be getting credit for.

There are several approaches you can choose from in step 3: doing the traditional math, as we did in the question above; Picking Numbers; Backsolving; estimating; or taking a strategic guess. In the next two examples, you’ll see Picking Numbers and Backsolving in action.

Here’s another example. This one is not a word problem, so steps 1 and 2 require negligible mental energy, but pay attention when you get to step 3:

Which of the following expressions is equivalent to ?

A.

B.

C.

D.

Step 1: What do they want? An expression equivalent to .

Step 2: What do they give you? Only the expression .

Step 3: What approach will you use?

Here’s where it gets interesting. The creator of this question may be expecting you to use polynomial long division to solve, and we’ll cover that technique in the online appendix included with this book because you may want to have it in your arsenal. But if you don’t know how to do polynomial long division, there’s no need to panic. You could use an alternate approach called Picking Numbers that will work just as well: choose a number to substitute for x in the question, then substitute the same number in for x in the choices and see which one matches. Like this:

Pick a small number for x, say 2. When x = 2, the original expression becomes the following:

Now, plug x = 2 into the choices:

A.

Not 20, so eliminate (A).

B.

Eliminate (B).

C.

This is a match. When using Picking Numbers, it is possible that another answer choice can produce the same result, so check (D) to be sure there isn’t another match when x = 2. (If there is, go back and pick another number to distinguish between the choices that match.)

D.

Eliminate (D).

Step 4: Did you solve for the right thing? You found the equivalent expression, so yes. Only (C) is a match, and therefore it is correct.

When picking numbers, use numbers that are permissible and manageable. That is, use numbers that are allowed by the stipulations of the question and that are easy to work with. In this question, you could have picked any real number because x was not defined as positive, negative, odd, even, a fraction, etc. A small positive integer is usually the best choice in this situation. In other questions, other kinds of numbers may be more manageable. For example, in percents questions, 100 is typically a smart number to pick.

Try one more:

A child is arranging plates of apples to serve at a party. If the child places 6 apples on each plate, there will be 5 apples left over. In order to place 7 apples on each plate, with no apples left over, 5 more apples are needed. How many apples does the child have to arrange?

A. 32

B. 41

C. 56

D. 65

Step 1: What do they want? The number of apples.

Step 2: What do they give you? Two unknowns (the number of plates and the number of apples) and sufficient information to set up a system of equations.

Step 3: What approach will you use? You could set up the system of equations, but it might be faster to use a technique called Backsolving: plug the answer choices in for the unknown and see which one works. Here, you need an answer choice that will leave a remainder of 5 when divided by 6. Choices (A) and (C) don’t meet this condition, so the answer must be (B) or (D).

Check (B). If there are 41 apples, and they are distributed 6 to a plate, there will indeed be 5 apples left over since 41 ÷ 6 = 6 R5. Now, what happens in the other situation? With an extra 5 apples, there should be enough to distribute 7 to a plate with none left over. But 41 + 5 = 46, which is not evenly divisible by 7. There would be 4 apples left over. Eliminate (B).

You’ve now eliminated every choice but (D), so it must be correct—you don’t even need to test it! For the record:

(D) If there are 65 apples and they are distributed 6 to a plate, there would indeed be 5 left over since 65 ÷ 6 = 10 R5. With an extra 5 apples, it should be possible to distribute them evenly to 7 plates, and this is in fact what happens: 65 + 5 = 70, which is evenly divisible by 7.

Step 4: Did you solve for the right thing? The question asked for the number of apples. You found that 65 apples satisfies all conditions of the question. Choose (D) and move on.

Although it wasn’t the case in this question, when backsolving it often makes sense to start with (B) or (C) in case you can tell from the context whether you’ll need a larger or smaller answer choice if the one you’re testing fails.

Now, it’s your turn. Be deliberate with these questions. If there is analysis to do up front, do it. If there is more than one way to do a question, consider carefully before choosing your approach. And be sure to check whether you answered the right question. Forming good habits now, in slow and careful practice, will build your confidence for test day.

Try on Your Own

Directions: Take as much time as you need on these questions. Work carefully and methodically. There will be opportunities for timed practice in future chapters.

1. In the equation above, what is the value of y?

A. 3

B.7

C.9

D. 11

2. A tractor trailer has a maximum capacity of 8,000 pounds. The equipment needed to load and unload the trailer must travel with the trailer and weighs a combined 1,500 pounds. The trailer will be loaded with x containers, each of which weighs 300 pounds. What is the largest value of x such that the trailer’s capacity is not exceeded?

A. 5

B.15

C.21

D. 26

3. A certain vacuum cleaner is priced at $450 at a local appliance store. The same model of vacuum cleaner sells online for of the price at the appliance store. At a department store, the same model vacuum cleaner sells for of the appliance store’s price. How many dollars more is the price of the vacuum cleaner at the department store than at the online retailer?

A. 90

B.135

C.180

D. 225

4. A stack of 50 kitchen serving trays forms a column that is approximately inches tall. What is closest to the number of kitchen trays that would be needed to form a column that is 14 inches tall?

A. 70

B.83

C.100

D. 113

5. Last month, Keith ran 18 more miles than Mick ran. If they ran a total of 76 miles, how many miles did Keith run?

A. 29

B.38

C.42

D. 47

6. If , what is the value of ?

A.

B.

C.

D. 1

7. Which of the following equations relates y to x according to the values in the table above?

A.

B.

C.

D.

8. In a restaurant’s kitchen, c cakes are made by adding s cups of sugar to a mix of eggs and butter. If s = 3 c + 5, how many more cups of sugar are needed to make one additional cake?

A. 0

B.

C.1

D. 3

9. A bowling league charges a one-time membership fee of $25, plus x dollars each month. If a bowler has paid $53 for the first 4 months, including the membership fee, what is the value of x?

A. 4

B.7

C.10

D. 13

10. If x > 0, which of the following is equivalent to ?

A. (x + 5)²

B.

C.3(x + 4)

D.