PSAT/NMSQT Math: Heart of Algebra Questions and Answers 1 — 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?
- The cost for every gigabyte of data used.
- The total number of gigabytes a customer can use per month.
- The flat monthly fee, regardless of data usage. (Correct answer)
- The total cost for a customer who uses zero gigabytes of data and makes no calls.
Correct answer: The flat monthly fee, regardless of data usage.
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?
- 4h + 4s = 8 3h + 5s = 30
- h + s = 8 3h + 5s = 30 (Correct answer)
- h + s = 8 5h + 3s = 30
- 4(h + s) = 8 3h + 5s = 30
Correct answer: h + s = 8 3h + 5s = 30
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?
- 4
- 8
- 15 (Correct answer)
- 23
Correct answer: 15
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?
- 50 + 0.75m < 200
- 50 + 0.75m > 200
- 50m + 0.75 ≤ 200
- 50 + 0.75m ≤ 200 (Correct answer)
Correct answer: 50 + 0.75m ≤ 200
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?
- 2
- 4 (Correct answer)
- 6
- 8
Correct answer: 4
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.
Question 6: A line in the xy-plane passes through the origin and the point (3, 12). Which of the following is an equation of the line?
- y = 3x + 12
- y = 1/4 x
- y = 12x + 3
- y = 4x (Correct answer)
Correct answer: y = 4x
The equation of a line can be written in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Since the line passes through the origin (0,0), the y-intercept (b) is 0. To find the slope (m), use the slope formula: m = (y2 - y1) / (x2 - x1). Using the points (0,0) and (3,12): m = (12 - 0) / (3 - 0) = 12 / 3 = 4. Therefore, the equation of the line is y = 4x + 0, which simplifies to y = 4x.
PSAT/NMSQT Math: Heart of Algebra Questions and Answers 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?