# PSAT/NMSQT Math Practice Test

#### 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?

Explanation:

The margin of error will most likely be reduced by increasing the sample size while randomly selecting participants from
the initial population of interest.

#### 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?

Explanation:

The ratio of boy to girls is 4:7. Therefore, there are 4 boys out of 11 students. To find the answer, first divide the total
number of students by 11, then multiply the result by 4.

44 ÷ 11 = 4 ⇒ 4 × 4 = 16

There are 16 boys and 28(44–16) girls. So, 12 more boys should be enrolled to make the ratio 1:1

#### 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?

Explanation:

The total charge that Jackson will pay is the room rate, the 8% tax on the room rate, and a fixed fee. If Jackson stayed x nights,
then the total charge is (99.95 x + 0.08 x 99.95 x ) + 5, which can be rewritten

as 1.08 (99.95 x ) + 5

#### 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?

Explanation:

The capacity of a yellow box is 20% bigger than the capacity of a green box and it can hold 30 books. Therefore, we want to
find a number that 20% bigger than that number is 30. Let 𝑥 be that number. Then:

1.20× 𝑥 = 30

Divide both sides of the equation by 1.2. Then:

𝑥=301.20=25

#### 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?

Explanation:

One method to find the correct answer is to create an inequality. The income from sales of n items is 12n. For the company to
profit, 12n must be greater than the cost of producing n items; therefore, the inequality 12n greater than 7 n plus 350 can be
used to model the context. Solving this inequality yields n > 70.

#### - 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?

Explanation:

The smallest number is −15. To find the largest possible value of one of the other five integers, we need to choose the smallest
possible integers for four of them. Let 𝑥 be the largest number. Then:

−70=(−15)+(−14)+(−13)+(−12)+(−11)+𝑥→−70=−65+𝑥→𝑥=−70+65=−5

#### 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?

Explanation:

The student must reason that because the mean of the males is lower than that of the females, the combined mean cannot
be greater than or equal to that of the females, while also reasoning that because the mean of the females is greater than that
of the males, the combined mean cannot be less than or equal to the mean of the males. Therefore the combined mean must
be between the two separate means.

#### What is the ratio of the following function's minimum to maximum value?

−2≤𝑥≤3?𝑓(𝑥)=−3𝑥+1

Explanation:

Since 𝑓(𝑥) is linear function with a negative slop, then when 𝑥=−2,𝑓(𝑥) is maximum and when 𝑥=3,𝑓(𝑥) is minimum.

Then the ratio of the minimum value to the maximum value of the function is:

𝑓(3)/𝑓(−2) = −3(3)+1/−3(−2)+1 = −8/7

#### 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?

Explanation:

A relationship in the data can only be generalized to the population that the sample was drawn from.

#### 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?

Explanation:

Since James car is traveling at an average speed of 50 miles per hour and the car's gas
mileage is 21 miles per gallon, the number of gallons of gas used each hour can be found by
50 miles/1 hour x 1 gallon/2 miles. + 50/21. The car uses50/21 gallons of gas per hour, so it uses
50/21 t gallons of gas in t hours. The car’s gas tank has 17 gallons of gas at the beginning of the trip.
Therefore, the function that models the number of gallons of gas remaining in the tank t hours
after the trip begins is f(t) = 17 — 50t/21.

#### 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?

Explanation:

To determine the linear model, one can first determine the rate at which the pressure due to
the atmosphere and surrounding water is increasing as the depth of the diver increases.
Calculating this gives 20.9 - 18.7/14 - 9 = 2.2/5, or 0.44. Then one needs to determine the
pressure due to the atmosphere or, in other words, the pressure when the diver is at a depth
of 0.Solving the equation 18.7 = 0.44(9) + b gives b = 14.74.Therefore, the model that can be
used to relate the pressure and the depth is p = 0.44d + 14.74.

#### 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?

Explanation:

If x is the number of cars that crossed the bridge in two hours and y is the number of trucks that crossed the
bridge in two hours, then x + y represents the total number of cars and trucks that crossed the bridge in two
hours, and 6.5 x + 10 y represents the total amount collected in two hours. x + y space equals space 187, and
6.5 x s+ 10 y = 1, 338.

#### 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?

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.

#### 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

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

#### 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?

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

#### 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?

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

#### Which of the following is a solution to the 4x − 12 < 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.

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

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

#### What is the value of x if 12/X = 30/6?

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

#### 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?

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.

#### 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?

Explanation:

The correct answer is b=n×2n

#### 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?

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.

#### 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?

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.

#### 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?

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.

#### 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?

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.

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