PSAT/NMSQT Math: Heart of Algebra Questions and Answers

Question 1

A cell phone company offers a data plan that costs $25 per month plus $15 for every gigabyte of data used. This relationship is modeled by the function C(g) = 15g + 25, where C is the total monthly cost and g is the number of gigabytes used. What does the number 25 in the function represent?

Accepted Answer

The flat monthly fee, regardless of data usage.

Answer

In the linear function C(g) = 15g + 25, which is in the slope-intercept form y = mx + b, the term 'b' represents the y-intercept, or the starting value when the variable 'g' is zero. In this context, g=0 means no data was used. Therefore, the 25 represents the initial, fixed cost that is charged every month even with zero data usage.

Question 2

At a local fair, a combo meal that includes a hot dog and a soda costs $8. A family buys 4 combo meals. A different family buys 3 hot dogs and 5 sodas for a total of $30. If h is the cost of a hot dog and s is the cost of a soda, which of the following systems of equations can be used to determine the price of a hot dog and a soda?

Accepted Answer

h + s = 8
3h + 5s = 30

Answer

The first statement says a combo meal with one hot dog (h) and one soda (s) costs $8. This translates to the equation h + s = 8. The second statement says 3 hot dogs (3h) and 5 sodas (5s) cost a total of $30, which translates to the equation 3h + 5s = 30. The information about the family buying 4 combo meals is extra information not needed to set up the system. Therefore, the correct system is h + s = 8 and 3h + 5s = 30.

Question 3

If 5x - 6 = 14, what is the value of 2x + 7?

Accepted Answer

15

Answer

This is a two-step problem. First, solve the initial equation for x. 
1. 5x - 6 = 14 
2. Add 6 to both sides: 5x = 20 
3. Divide by 5: x = 4. 
Now, substitute the value of x into the second expression. 
1. 2x + 7 
2. 2(4) + 7 
3. 8 + 7 = 15.

Question 4

A car rental company charges a flat fee of $50 plus an additional $0.75 for every mile driven. Which of the following inequalities represents the maximum number of miles, m, a person can drive if they have a budget of no more than $200?

Accepted Answer

50 + 0.75m ≤ 200

Answer

The total cost is the sum of the flat fee ($50) and the variable cost ($0.75 per mile, or 0.75m). The phrase 'no more than $200' means the total cost can be less than or equal to $200. Therefore, the inequality that represents this situation is 50 + 0.75m ≤ 200.

Question 5

If 2a + b = 10 and a - b = 2, what is the value of a?

Accepted Answer

4

Answer

This system of equations can be solved efficiently using the elimination method. By adding the two equations together, the 'b' variable is eliminated:
(2a + b) + (a - b) = 10 + 2
3a = 12
Divide by 3:
a = 4.
Alternatively, one could use substitution. Solve the second equation for a (a = 2 + b), and substitute this into the first equation.

PSAT/NMSQT Practice Test

PSAT/NMSQT Practice Test

PSAT/NMSQT Math: Heart of Algebra Questions and Answers