PSAT/NMSQT Math Practice Test 1 — Questions and Answers

Question 1: A research assistant chose 75 undergraduate students at random from a list of all students enrolled in a large university's psychology program. “How many minutes per day do you generally spend reading?” she questioned each of the 75 kids. The average reading duration in the sample was 89 minutes, with a margin of error of 4.28 minutes for this estimate. Another research assistant plans to duplicate the poll with the goal of reducing the margin of error. Which of the following samples has the smallest margin of error for the projected mean time students in the psychology program read every day?

40 undergraduate students from all degree programs at the college were chosen at random.
40 undergraduate psychology students were chosen at random.
300 undergraduate students from all degree programs at the college were chosen at random.
300 undergraduate psychology students were chosen at random. (Correct answer)

Correct answer: 300 undergraduate psychology students were chosen at random.

To reduce the margin of error in a statistical estimate, two primary factors are crucial: increasing the sample size and ensuring the sample is drawn from the specific population of interest. A larger sample size (300 vs. 40) inherently provides more data, leading to a more precise estimate and thus a smaller margin of error. Additionally, sampling only undergraduate psychology students ensures the sample accurately represents the target population, minimizing variability and improving the estimate's relevance.

Question 2: In a class, the ratio of boys to girls is 4:7. How many extra boys should be enrolled in a class with 44 students to achieve a 1:1 ratio?

5
10
12 (Correct answer)
14

Correct answer: 12

The initial ratio of boys to girls is 4:7, meaning there are 11 parts in total. In a class of 44 students, each part represents 44 / 11 = 4 students. This means there are 4 * 4 = 16 boys and 7 * 4 = 28 girls. To achieve a 1:1 ratio, the number of boys must equal the number of girls, which is 28. Therefore, 28 - 16 = 12 extra boys need to be enrolled.

Question 3: Jackson is staying at a hotel where a room costs $99.95 a night plus tax. The room rate is subject to an 8% tax, as well as a one-time untaxed fee of $5.00 imposed by the hotel. Which of the following is Jackson's total price for staying x nights in dollars?

1.08(99.95 + 5)x
1.08(99.95x + 5)
(99.5+0.08x)+5
1.08(99.95x) + 5 (Correct answer)

Correct answer: 1.08(99.95x) + 5

The room cost per night is $99.95. For 'x' nights, the total room cost before tax is 99.95x. An 8% tax is applied to this room cost, so we multiply 99.95x by 1.08 (representing the original cost plus the 8% tax). The $5.00 hotel fee is a one-time, untaxed charge, so it is added directly to the taxed room cost. Thus, the total price is 1.08(99.95x) + 5.

Question 4: A yellow box has a capacity that is 20% greater than a green box's capacity. How many of the same books can the green box hold if the yellow box can contain 30 of the same size books?

25 (Correct answer)
8
24
12

Correct answer: 25

Let the capacity of the green box be G. The yellow box's capacity (Y) is 20% greater than the green box's, so Y = G + 0.20G = 1.20G. We know the yellow box can hold 30 books, so Y = 30. Substituting this into the equation, we get 30 = 1.20G. To find G, divide 30 by 1.20: G = 30 / 1.20 = 25. Therefore, the green box can hold 25 books.

Question 5: The cost C, in dollars, of producing n goods, according to a corporate management, is C equals 7 n plus 350. Each item costs $12 at the company. When the total money from selling a number of products exceeds the total cost of producing that quantity of items, the company generates a profit. Which of the following inequalities contains all potential values of n for which the manager expects the company to profit?

n > 70 (Correct answer)
n < 70
n < 80
n > 80

Correct answer: n > 70

The cost of producing 'n' goods is given by C = 7n + 350. The revenue from selling 'n' goods is R = 12n, since each item sells for $12. A profit is generated when the total revenue exceeds the total cost, so R > C. This translates to the inequality 12n > 7n + 350. Subtracting 7n from both sides gives 5n > 350, and dividing by 5 yields n > 70. Thus, the company profits when more than 70 items are sold.

Question 6: - 70 is the sum of six negative integers. What is the greatest possible value of one of the other five numbers if the smallest integers is -15?

0
-3
-4
-5 (Correct answer)

Correct answer: -5

To maximize one of the six negative integers, the other five integers must be as small (most negative) as possible, while still being distinct and negative. Given the smallest integer is -15, the next four smallest distinct negative integers would be -14, -13, -12, and -11. The sum of these five integers is -15 + (-14) + (-13) + (-12) + (-11) = -65. Since the total sum of all six integers is -70, the greatest possible value of the remaining integer is -70 - (-65) = -5.

Question 7: Male primates on a primate reserve are on average 15 years old, whereas female primates are on average 19 years old. Which of the following statements concerning the combined group of male and female primates at the primate reserve's mean age, m, must be correct?

15 < m < 19 (Correct answer)
m = 20
m < 20
m > 20

Correct answer: 15 < m < 19

The mean age of male primates is 15 years, and the mean age of female primates is 19 years. When combining these two groups, the overall mean age (m) of the combined group must always fall between the individual means of the two groups. It cannot be less than the lowest mean (15) or greater than the highest mean (19), assuming both groups have members. Therefore, the combined mean 'm' must be strictly greater than 15 and strictly less than 19.

Question 8: What is the ratio of the following function's minimum to maximum value? −2≤𝑥≤3?𝑓(𝑥)=−3𝑥+1

- 8/7 (Correct answer)
8/7
7/6
-7/6

Correct answer: - 8/7

The function f(x) = -3x + 1 is a linear function with a negative slope, meaning it is decreasing over its domain. For the interval -2 ≤ x ≤ 3, the maximum value will occur at the smallest x-value, and the minimum value will occur at the largest x-value. The maximum value is f(-2) = -3(-2) + 1 = 7. The minimum value is f(3) = -3(3) + 1 = -8. The ratio of the minimum to maximum value is -8/7.

Question 9: For the population of 16-year-olds in the United States, a researcher began to find if there is an association between exercise and sleep. She gathered data from a random sample of 2000 16-year-olds in the United States and discovered strong evidence of a link between exercise and sleep. Which of the following conclusions is backed up by evidence?

For 16-year-olds all throughout the world, there is a link between exercise and sleep.
Using the study's definitions of exercise and sleep, an increase in sleep is induced by an increase in activity for 16-year-olds around the world.
There is a positive link between exercise and sleep among 16-year-olds in the United States. (Correct answer)
For 16-year-olds in the United States, an increase in sleep is induced by an increase in activity, according to the study's definitions.

Correct answer: There is a positive link between exercise and sleep among 16-year-olds in the United States.

The researcher gathered data from a random sample of 16-year-olds *in the United States*. Random sampling allows the findings to be generalized to the population from which the sample was drawn. The study found a 'strong evidence of a link' (association), but not causation. Therefore, the conclusion can only state a positive link (association) between exercise and sleep among 16-year-olds in the United States, not worldwide or implying causation.

Question 10: When Jame's car travels at an average speed of 50 miles per hour, the gas mileage is 21 miles per gallon. At the start of a trip, the car's petrol tank contains 17 gallons of gas. Which of the following functions f represents the number of gallons of gas remaining in Jame's tank t hours after the trip begins if his car moves at an average speed of 50 miles per hour?

f(t) = 17 - 21t/50
f(t) = 17 - 50t/21 (Correct answer)
f(t) = 17 - 50t/21
f(t) = 17 - 21/50t

Correct answer: f(t) = 17 - 50t/21

James's car travels at an average speed of 50 miles per hour. In 't' hours, the total distance traveled is 50t miles. The car's gas mileage is 21 miles per gallon, meaning it consumes 1 gallon for every 21 miles. Therefore, the number of gallons consumed after 't' hours is (50t) / 21. Starting with 17 gallons, the remaining gas f(t) is the initial amount minus the consumed amount: f(t) = 17 - (50t / 21).

Question 11: The pressure due to the atmosphere and surrounding water is 18.7 pounds per square inch when a scientist dives into salt water to a depth of 9 feet below the surface. The pressure grows linearly as the scientist descends. The pressure is 20.9 pounds per square inch at a depth of 14 feet. Which of the following linear models best reflects the pressure p in pounds per square inch at a depth of d feet below the surface if the pressure grows at a consistent rate as the scientist's depth below the surface increases?

p = 0.44d + 14.74 (Correct answer)
p = 2.2d - 9.9
p = 0.44d + 0.77
p = 2.2d - 1.1

Correct answer: p = 0.44d + 14.74

This problem describes a linear relationship between pressure (p) and depth (d). We are given two points: (d1, p1) = (9, 18.7) and (d2, p2) = (14, 20.9). First, calculate the slope (m) using the formula m = (p2 - p1) / (d2 - d1) = (20.9 - 18.7) / (14 - 9) = 2.2 / 5 = 0.44. Next, use the point-slope form (p - p1 = m(d - d1)) or substitute a point into p = md + b to find the y-intercept (b). Using (9, 18.7): 18.7 = 0.44(9) + b => 18.7 = 3.96 + b => b = 14.74. Thus, the linear model is p = 0.44d + 14.74.

Question 12: A car will pay $6.50 and a truck will pay $10 to cross a bridge. 187 automobiles and trucks crossed the bridge in two hours, resulting in a total toll collection of $1,338. Which of the following equations produces the number of cars (x) and trucks (y) that passed the bridge during the two-hour period?

x +y = 187 6.5x + 10y = 1,338 (Correct answer)
x +y = 1,338 6.5x + 10y = 187
x +y = 187 6.5x + 10y = 1,338 x 2
x +y = 187 65.5x + 10y = 1,338/2

Correct answer: x +y = 187 6.5x + 10y = 1,338

Let 'x' represent the number of cars and 'y' represent the number of trucks. The first piece of information states that 187 automobiles and trucks crossed the bridge, which translates to the equation x + y = 187. The second piece of information concerns the total toll collection: cars pay $6.50 each (6.5x) and trucks pay $10 each (10y), totaling $1,338. This gives the equation 6.5x + 10y = 1,338. These two equations form the system that models the situation.

Question 13: A typical image obtained by a camera of the surface of Mars is 11.2 gigabytes in size. For a maximum of 11 hours per day, a tracking station on Earth can receive data from the spacecraft at a rate of 3 megabits per second. What is the maximum number of typical photos that the monitoring station could get from the camera each day if 1 gigabit equals 1,024 megabits?

10 (Correct answer)
5
60
150

Correct answer: 10

Explanation: The tracking station can receive 118,800 megabits each day ( 3 megabits/1 second x 60 seconds/1 minute x 60 minutes/1 hour x 11 hours), which is about 116 gigabits each day ( 118.800/1,024). If each image is 11.2 gigabits, then the number of images that can be received each day is 115/11.2 = 10.4. Since the question asks for the maximum number of typical images, rounding the answer down to 10 is appropriate because the tracking station will not receive a completed 11th image in one day.

Question 14: The length of a sailboat is 19 meters. In inches, how long is it? 1 meter = 1.094 yards 2.54 centimeters = 1 inch 1 kilogram = 2.205 pounds 1 liter = 1.06 quarts

748 (Correct answer)
300
30
750

Correct answer: 748

Explanation: There are two ways to solve this problem: either convert meters to centimeters and then use the conversion factor to convert centimeters to inches, or else use the table to convert meters to yards, and then convert to inches. In the first instance, recall that there are 100 centimeters in a meter (centi means “hundredth”). Therefore 19 m = 1900 cm = (1900/2.34) = 748 inches. In the second instance, recall that there are 36 inches in a yard, therefore: 19 m = 19x1.094 = 20.786 yd = 20.786x36 = 748 inches. Proportions are commonly used for conversions. After converting meters to centimeters set up proportions to solve for an unknown variable, x. 900 cm/x in = 2.54 cm/1 in Cross multiply. 900 = 2.54 Divide each side by 2.54 to solve for x. x = 748

Question 15: There are sixteen empty seats in Mrs. Clarkson's classroom. When every student is present, all of the chairs are filled. How many pupils make up her entire class if 2/5 of them are absent?

40 (Correct answer)
15
30
20

Correct answer: 40

Explanation: Since 16 chairs are empty, and this represents 2/5 of the total enrollment, then the full class must consist of Class = 5/2 x 16 = 40 students Use proportions 2/5 = 16/x Cross multiply. 2x = 80 Divide each side by 2 to solve for x. x = 40 students

Question 16: Anne purchased $24.15 worth of vegetables. She purchased two pounds of onions, three pounds of carrots, and 1 ½ pounds of mushrooms. What is the price per pound of mushrooms if onions cost $3.69 per pound and carrots cost $4.29 per pound?

$ 2.60 (Correct answer)
$ 2.50
$ 3.20
$ 3.15

Correct answer: $ 2.60

Explanation: Begin by determining the total cost of the onions and carrots, since these prices are given. This will equal (2 x $3.69) + (3 x $4.29) = $20.25. Next, this sum is subtracted from the total cost of the vegetables to determine the cost of the mushrooms: $24.15 - $20.25 = $3.90. Finally, the cost of the mushrooms is divided by the quantity (lbs) to determine the cost per pound: Cost per lb = $3.90/1.5 = $2.60

Question 17: Which of the following is a solution to the 4x − 12 < 4?

8
2
4 (Correct answer)
5

Correct answer: 4

Explanation: Begin as you would a regular equation: 4x − 12 < 4 Add 12 to each side. 4x < 16 Divide by 4. x < 4 Note: The inequality does not change because the division was by positive 4. Since x must be less than 4, and not equal to it (< not ≤ ), the answer 4 is incorrect: the solution does not include 4. Only the answer 3 satisfies the condition that it must be < 4.

Question 18: If a = -6 and b = 7, what is the value of 4 a( 3 b + 5) + 2 b?

-610 (Correct answer)
610
640
-640

Correct answer: -610

Explanation: First, compute the value enclosed by the parentheses, 3b + 5 = 3 x 7 + 5 = 26. Next, compute >br> 4a = -24. Note that a is negative, so this product is also negative. The product, 4a(3b + 5), will therefore be negative, as well, and equals -624. Finally, add the value of 2b, or 2 x 7 =14, to -624, to get the final answer: 624 + 14 = -610. Substitute the given values for the variables into the expression: 4x - 6(3 x 7 + 5) + 2 x 7 Using the order of operations, compute the expression in the parentheses first. Remember that you must first multiply 3 by 7, and then add 5, in order to follow order of operations. = 4x -6(21 + 5) + 2 x 7 Next add the values in the parentheses. = 4x - 6(26) + 2 x 7 Simplify by multiplying the numbers outside of the parentheses. = -24(26) + 14 Multiply -24 by 26. = -624 + 14 Add. = -610

Question 19: What is the value of x if 12/X = 30/6?

2.4 (Correct answer)
3.1
3.5
2.8

Correct answer: 2.4

Explanation: A proportion such as this can be solved by taking the cross product of the numerators and denominators from either side. 12/x = 30/6 Take the cross product by cross multiplication. 30x = 6 x 12 Multiply 6 by 12. 30x = 72 Divide each side by 30. x = 2.4

Question 20: Pepperoni pizza and Hawaiian pizza are two styles of pizza sold at a certain pizza shop. Although the cost of making each pizza style varies, they are all sold at the same price. The store currently has 32 Hawaiian pizzas that cost $4.25 each to make and 18 pepperoni pizzas that cost $3.75 each to make. To increase profits, the business owner wants to add additional pepperoni pizzas. That day's production budget is $316. Which equation can be used to calculate how many additional pepperoni pizzas, p, are required?

($4.25 + $3.75)(32 + 18 + p) = $316
$4.25(32) + $3.75(18) + p = $316
$4.25(32) + $3.75(18p) = $316
$4.25(32) + $3.75(18 + p) = $316 (Correct answer)

Correct answer: $4.25(32) + $3.75(18 + p) = $316

Explanation: There are 32 Hawaiian pizzas and the cost for producing each is $4.25, so the total cost of producing 32 Hawaiian pizzas is $4.25(32). There are 18 pepperoni pizzas and p more will be added, so the total number of pepperoni pizzas is (18 + p). If each of these pepperoni pizzas costs $3.75 to produce, the total cost of producing pepperoni pizzas is $3.75(18 + p). The total cost of producing all pizzas is equal to the sum of producing the Hawaiian pizzas and the pepperoni pizzas. So the equation $4.25(32) + $3.75(32 + p) = 316 is the correct answer.

Question 21: Dr. Scot, the leader of the scientific team examining the bacteria, is putting together a report on the potential damage the pathogen poses to the public if it is not contained. He wants to create a generic equation for the bacteria's behavior after n minutes of replicating. Which of the following equations best explains the bacteria's development pattern if b is the number of bacteria present after n minutes?

b=n×2n (Correct answer)
b=n2n
b=2×n2
b=(n-1)2×2

Correct answer: b=n×2n

Explanation: The correct answer is b=n×2n

Question 22: A toy factory's profit in dollars is computed by subtracting sales revenue from production expenses and multiplying the difference by the number of toys sold. The entire profit earned from two types of toys, A and B, is represented by the expression (9.50 – 2.45)x + (12.75 – 4.50)y, where x is the number of toy A sold and y is the number of toy B sold. In the statement, what does 8.25 stand for?

The profit of selling toy A is $8.25.
Toy A is sold at $8.25.
The profit of selling toy B is $8.25 (Correct answer)
Toy B is sold at $8.25

Correct answer: The profit of selling toy B is $8.25

Explanation: The given expression represents the total profit earned, in dollars, of two types of toys, A and B. Because y represents the number of toy B sold, 8.25y = (9.50 – 2.45)y represents the profit earned by selling toy B. Therefore, it follows that the coefficient, 8.25, must represent the profit of selling toy B.

Question 23: 6x+3y=9 is the equation for line m. At one point, line n crosses line m, making a right angle at the intersection. Which of the following equations can be used to represent line n's equation?

2x-4y=8 (Correct answer)
x+2y=8
2x-y=8
4x-2y=8

Correct answer: 2x-4y=8

Explanation: Since line m and line n intersect forming a right angle, they are perpendicular lines. The slopes of two perpendicular lines are negative reciprocals to each other. So to find the equation of line n, we need first to find the slope of line m. Step 1: Solve for the slope of line m. 6x+3y=9→3y=-6x+9 Divide both sides of the equation by 3: → y=-2x+3 Therefore, the slope of line m is -2. This implies that the slope of line n is –(–12)=12. Step 2: Identify which of the choices has a slope of 12. Option C: 2x-y=8 →y=2x-8 →slope is 2. Hence, option A is incorrect. Option A: 2x-4y=8 →4y=2x-8 →y=12x-2→slope is 12. Hence, option B can be the equation of line n.

Question 24: A car loan with a fixed annual interest rate is available from one bank. Sam applied for the loan and ended up using the entire money to purchase a $43,000 automobile. He had to pay the bank $51,600 over the course of five years. What would Sam's annual interest be if she borrowed $50,000 instead?

$ 2,000 (Correct answer)
$ 1.500
$ 9.000
$3,500

Correct answer: $ 2,000

Explanation: The interest formula is I = Prt, where P = borrowed amount, r = rate, t = term (time). For five years, Sam has to pay $51,600. It means that the total interest for five years is $51,600 – $43,000 = $8,600. So the yearly interest is $8,600 5 = $1,720. Using the formula above, we have $1,720=$43,000r1 Thus, the interest rate is r=$1,720$43,000=0.04=4%. When $50,000 is loaned, the yearly interest would be I=$50,0000.041=$2,000.

Question 25: Gerald has received notification from his grandfather that he will be receiving a section of their family farmland. He has been told that he can choose his piece as long as it is within a 900-meter fence wire. How should Greald fence his land in such a way that he has the most land area possible?

Enclose a rectangular plot of land with dimensions of 200 m x 250 m.
Enclose a 300-meter-long by 150-meter-long rectangular plot of land.
Enclose a square of land with a length of 225 meters on each side. (Correct answer)
Enclose a square of land with a 450-meter-long side.

Correct answer: Enclose a square of land with a length of 225 meters on each side.

Explanation: The largest possible area can be found by getting a square shape of land with the side length of 900 meters 4 = 225 meters. Option C satisfies the condition of using the 900 meters perimeter (fencing wire) and the area that will be generated is 225 x 225 = 50,625 square meters.

PSAT/NMSQT Math Practice Test 1

A research assistant chose 75 undergraduate students at random from a list of all students enrolled in a large university's
psychology program. “How many minutes per day do you generally spend reading?” she questioned each of the 75 kids.
The average reading duration in the sample was 89 minutes, with a margin of error of 4.28 minutes for this estimate.
Another research assistant plans to duplicate the poll with the goal of reducing the margin of error.
Which of the following
samples has the smallest margin of error for the projected mean time students in the psychology program read every day?