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

Word problems
Linear equations and graphs
The heart of algebra

Learning objective

After this lesson, you will be able to:

· Translate word problems into equations

To answer a question like this:

A local gym sells two types of memberships. One package costs $325 for one year of membership with an unlimited number of visits. The second package has a $185 enrollment fee, includes five free visits, and costs an additional $4 per visit after the first five. How many visits over a one-year period would a person who purchases the second package need to make for the cost to be the same as the one-year unlimited membership?

A. 30

B. 35

C. 40

D. 45

You need to know this:

The PSAT likes to test your understanding of how to describe real-world situations using math equations. For some questions, it will be up to you to extract and solve an equation; for others, you’ll have to interpret an equation in a real-life context. The following table shows some of the most common phrases and mathematical equivalents you’re likely to see on the PSAT.

You need to do this:

When translating from English to math, start by defining the variables, choosing letters that make sense. Then, break the question down into small pieces, writing down the translation for one phrase at a time.

Explanation:

The phrase “how many visits” indicates an unknown, so you need a variable. Use an intuitive letter to represent the number of visits; call it v. The question asks when the two memberships will cost the “same amount,” so write an equation that sets the total membership costs equal to each other.

The first membership type costs $325 for unlimited visits, so write 325 on one side of the equal sign. The second type costs $4 per visit (not including the first 5 visits), or 4(v − 5), plus a flat $185 enrollment fee, so write 4(v − 5) + 185 on the other side of the equal sign. That’s it! Now, solve for v:

The answer is (C).

Try on Your Own

Directions: Take as much time as you need on these questions. Work carefully and methodically. There will be an opportunity for timed practice at the end of the chapter.

6. An automotive insurer computes the charge for an insurance policy by starting with a fixed fee for administrative expenses and then adding a percentage of the value of the car, v. If the total price for an automobile policy is given by the function T = 0.02v + 25, then the value 0.02 best represents which of the following?

A. The fee for the administrative expenses

B.The value of the car

C.The total price of the policy

D. The percentage of the value of the car

7. Ibrahim has a contract for a cell phone plan that includes the following rates: the plan has a fixed cost of $50 per month, a data plan that provides 2 gigabytes (GB) of data for free and $8 for each GB of data after that, and a text message plan that costs $0.10 per text message sent. Which of the following equations represents the amount of money, in dollars, that Ibrahim will spend as long as he uses at least 2 GB of data? (Assume d = dollars, g = number of GB of data used, and t = number of text messages sent.)

A. d = 50 + 8g + 0.1t

B.d = 50 + (8g − 2) + 0.1t

C.d = 50 + 8(g − 2) + 0.1t

D. d = 5,000 + 800g + 10t

HINT: For Q8, take the time to determine which number goes with which variable before attempting to build the entire equation.

8. An online video game service has two different packages available. The first is an Unlimited Rental Package that costs $250 for 3 months and allows customers to rent as many games as they want. The second is the Standard Rental Package that initially has a flat fee of $130 but costs $4 per rental in the 3-month period. How many rentals on the Standard Rental Package would it take for both packages to equal the same price over 3 months?

A. 2

B.30

C.96

D. 120

9. A produce stand normally sells watermelons for $0.60 per pound. On Mondays, it sells watermelons at a 20 percent discount. The stand also sells sweet potatoes for $0.79 each. Which of the following represents the total cost, c, if a customer buys 4 sweet potatoes and a watermelon weighing p pounds on a Monday?

A. c = 0.2p + 0.79

B.c = 0.48p + 3.16

C.c = 0.6p + 0.79

D. c = 0.6p + 3.16

HINT: For Q10, if a arrows hit the inner circle, how many hit the outer circle in terms of a?

10. Juan is on his school’s archery team. In a match, he gets 12 arrows to shoot at a target. He gets 8 points if the arrow hits the inner circle of the target and 4 points if it hits the outer circle. Which of the following equations represents Juan’s total score if a of his arrows hit the inner circle and the rest hit the outer circle?

A. p = 8a

B.p = 8a + 4

C.p = 48 + 4a

D. p = 96 − 4a