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

Reflect
The method for PSAT math questions
PSAT math

Directions: Take a few minutes to recall what you’ve learned and what you’ve been practicing in this chapter. Consider the following questions, jot down your best answer for each one, and then compare your reflections to the expert responses on the following page. Use your level of confidence to determine what to do next.

Think about your current habits when attacking PSAT questions. Are you a strategic test taker? Do you take the time to think through what would be the fastest way to the answer?

Do word problems give you trouble?

What are the steps of the Method for PSAT Math and why is each step important?

Expert Responses

Think about your current habits when attacking PSAT questions. Are you a strategic test taker? Do you take the time to think through what would be the fastest way to the answer?

If yes, good for you! If not, we recommend doing questions more than one way whenever possible as part of your PSAT prep. If you can discover now, while you’re still practicing, that Picking Numbers is faster for you on certain types of questions but not on others, you’ll be that much more efficient on test day.

Do word problems give you trouble?

If word problems are difficult for you, get into the habit of taking an inventory, before you do any math, of whatthe question is asking for and what information you have.

What are the steps of the Method for PSAT Math and why is each step important?

Here are the steps:

Step 1. State what the question is asking

Step 2. Examine the given information

(Taking an inventory is especially important in word problems.)

Step 3. Choose your approach

(Taking a moment to decide what approach will be the fastest way to the answer will ultimately save you time.)

Step 4. Confirm that you answered the right question

(Making sure you solved for the right thing will save you from losing points to speed mistakes on questions that you know how to do and should be getting credit for.)

Next Steps

If you answered most questions correctly in the “How Much Have You Learned?” section, and if your responses to the Reflect questions were similar to those of the PSAT expert, then consider the Method for PSAT Math an area of strength and move on to the next chapter. Do keep using the method as you work on the questions in future chapters.

If you don’t yet feel confident, review those parts of this chapter that you have not yet mastered and try the questions you missed again. As always, be sure to review the explanations closely.

Answers and Explanations

1. B

Difficulty: Medium

Category: Radicals

Strategic Advice: Backsolve by plugging the answer choices in for y to determine which one makes the given equation true.

Getting to the Answer: Simplify the equation by subtracting 4 from both sides to get . Now, check the answer choices, starting with (B) or (C). If the answer you choose is too large or too small, you’ll know which direction to go when testing the next choice.

(B): . This is the correct answer.

If you prefer the algebraic approach, here it is:

Again, (B) is the correct answer.

2. C

Difficulty: Medium

Category: Solving Equations

Strategic Advice: Break apart the question into its mathematical parts; determine what information you have and what value you need to find and then determine how you’ll find that value.

Getting to the Answer: To answer this question, organize the information you know. The capacity of the trailer is 8,000 pounds. However, equipment that is already on the trailer weighs 1,500 pounds, so there is only 8,000− 1,500 = 6,500 pounds of remaining capacity. Each container weighs 300 pounds, so divide 6,500 by 300 to determine the maximum number of containers that can be packed: . Partial containers may not be packed, so round down to 21, which is (C).

3. D

Difficulty: Medium

Category: Solving Equations

Strategic Advice: Begin by determining what you are being asked to find—the difference between the online retailer’s price and the department store’s price. Next, use the information you’re given—the price at the appliance store, as well as fractions that represent the prices at the other two retailers.

Getting to the Answer: To answer this question, determine the price of the vacuum cleaner at each retailer. Online, the vacuum cleaner sells for of the price at the appliance store, or . The department store sells the vacuum cleaner for the price of the appliance store, so the department store’s price is . Now that you know the price of the vacuum cleaner at each store, simply subtract: 540 − 315 = 225, which is (D).

4. C

Difficulty: Easy

Category: Proportions

Strategic Advice: Because the answer choices are widely spaced apart, and the question asks for the answer that is “closest to the number,” estimation will be a better approach than wading into unnecessarily detailed and tedious calculations.

Getting to the Answer: Notice the relationship between the stack of trays in the question and the stack of trays you are asked to solve for: you are given a stack of inches, and you are asked how many plates are needed for a 14-inch stack. The number 14 is very close to twice , so you will need nearly twice the 50 plates given in the question. Thus, 100 is the correct answer, which is (C).

5. D

Difficulty: Medium

Category: Systems of Linear Equations

Strategic Advice: Use the answer choices to your advantage to quickly find Keith’s distance.

Getting to the Answer: The question gives two unknowns and enough information so that a system of equations could be formed. Traditional algebra could be used to solve this system of equations.

However, there is a more efficient way to answer the question: examine the answer choices to see which answers make sense for Keith’s distance. The question states that Keith ran 18 more miles than Mick; thus, Keith must have run more than half of the 76 miles that the two of them ran. Since one-half of 76 is 38, you can eliminate (A) and (B) immediately.

Now, you just have to check either (C) or (D). For (C), if Keith ran 42 miles, then Mick ran 42− 18 = 24 miles, and 42 + 24 = 66 miles, which isn’t correct. Thus, (D) is the correct answer. For the record, if Keith ran 47 miles, then Mick ran 47− 18 = 29 miles, and 47 + 29 = 76, which is correct.

If you are curious about the algebraic approach, let k represent the number of miles Keith ran. Since Keith ran 18 miles more than Mick, Mick ran 18 miles fewer, or k− 18 miles. They ran a combined 76 miles, so the following equation can be created and solved:

Again, this matches (D).

6. A

Difficulty: Medium

Category: Solving Equations

Strategic Advice: You have two variables, but only one equation, so solving for each variable will not be possible. Instead, pick numbers for x and y that will make the equation true.

Getting to the Answer: Pick a simple number for x and solve for y. Hopefully, y will also be easy to work with so you can plug them into the expression you are trying to find. Say x = 4; this gives , or Multiplying both sides by 2 y gives 12 = 12 y, so y = 1. Both numbers are very manageable, so plug them into the expression . This yields , which is (A).

7. A

Difficulty: Medium

Category: Linear Graphs

Strategic Advice: The answer choices are split into two types. The first two are linear equations and the second two are nonlinear. They are quadratic and exponential, respectively. Thus, examine the table to determine whether the relationship between x and y is linear or nonlinear.

Getting to the Answer: Notice that for every increase of 2 in x, y increases by . Thus, the relationship is linear, meaning you can eliminate (C) and (D). To determine whether (A) or (B) is correct, substitution could be used. However, note that if y increases by for every 2 unit increase in x, then dividing both values by 2 shows that y increases by for every 1 unit increase in x. This is the definition of slope, and the only equation that is a line with a slope of is (A).

8. D

Difficulty: Medium

Category: Word Problems

Strategic Advice: Pick a number for c to see how many cups of sugar will be needed, then pick another number for c to see how the number of cups of sugar changes.

Getting to the Answer: Pick a number for c; let’s say c = 2. This means that 3(2) + 5 = 11 cups of sugar are needed for two cakes. Now, try c = 3: 3(3) + 5 = 14 cups of sugar needed for 3 cakes. To go from 2 cakes to 3 cakes, 3 additional cups of sugar were needed, which is (D). You can try c = 4 to confirm: 3(4) + 5 = 17, which is another 3 cups of sugar.

9. B

Difficulty: Medium

Category: Word Problems

Strategic Advice: Backsolve by plugging in answer choices for x to determine which answer matches the cost of four months of membership.

Getting to the Answer: Examine the answer choices, starting with (B) or (C). If the answer you choose is too large or too small, you will know which direction to go. Multiply the answer choice by 4 and add the $25 membership fee.

(B): $7× $4 = $28 ® $28 + $25 = $53. This is a match, so (B) is the correct answer.

Algebra could also be used here:

Algebra also leads to $7, which is (B).

10. B

Difficulty: Medium

Category: Rational Expressions and Equations

Strategic Advice: Pick a number for x to determine the numerical value of the given expression, then plug the same number into the answer choices to find the one that matches.

Getting to the Answer: Pick something easy, like x = 1: . This simplifies to . This fraction can be rewritten as .

Now, plug 1 in for x in each of the answer choices to see which one gives you the same value:

(A):(1 + 5)² = 36. Eliminate.

(B): . Correct! Just to be sure, check the other two answers.

(C): 3(1+ 4) = 3(5) = 15. Eliminate.

(D): . Eliminate.

Since (B) is the only answer that matched the calculated value, it must be the correct answer.