ACT Mathematics Practice Test 1 — Questions and Answers

Question 1: The monthly fees for single rooms at 5 colleges are $370, $310, $380, $340, and $310, respectively. What is the mean of these monthly fees?

$310
$340
$342 (Correct answer)
$350
$380

Correct answer: $342

To find the mean of the monthly fees, sum all the given fees and then divide by the number of fees. The fees are $370, $310, $380, $340, and $310. The sum of these fees is 370 + 310 + 380 + 340 + 310 = $1710. There are 5 fees. So, the mean is $1710 / 5 = $342.

Question 2: On a particular road map, 1/2 inch represents 18 miles. 2 About how many miles apart are 2 towns that are 2 1/2 inches apart on this map?

18
22
36
45
90 (Correct answer)

Correct answer: 90

The map scale is 1/2 inch representing 18 miles. We need to find the distance in miles for 2 1/2 inches on the map. First, convert 2 1/2 inches to an improper fraction: 5/2 inches. Set up a proportion: (1/2 inch) / (18 miles) = (5/2 inches) / (x miles). Cross-multiply to solve for x: (1/2) * x = 18 * (5/2). This simplifies to x/2 = 9 * 5, or x/2 = 45. Multiplying both sides by 2 gives x = 90 miles. So, the two towns are 90 miles apart.

Question 3: Given f = cd3, f = 450, and d = 10, what is c ?

0.45 (Correct answer)
4.5
15
45
150

Correct answer: 0.45

The given formula is f = cd^3. We are given f = 450 and d = 10. To find c, substitute the given values into the formula: 450 = c * (10)^3. Calculate 10^3, which is 1000. So, 450 = c * 1000. To solve for c, divide both sides by 1000: c = 450 / 1000 = 0.45.

Question 4: If f (x) = (3x + 7)2, then f (1) = ?

10
16
58
79
100 (Correct answer)

Correct answer: 100

To find f(1), substitute x = 1 into the function f(x) = (3x + 7)^2. This gives f(1) = (3 * 1 + 7)^2. First, perform the multiplication inside the parentheses: 3 * 1 = 3. Then, perform the addition: 3 + 7 = 10. Finally, square the result: 10^2 = 100. So, f(1) = 100.

Question 5: Jorge’s current hourly wage for working at Denti Smiles is $12.00. Jorge was told that at the beginning of next month, his new hourly wage will be an increase of 6% of his current hourly wage. What will be Jorge’s new hourly wage?

$12.06
$12.60
$12.72 (Correct answer)
$18.00
$19.20

Correct answer: $12.72

Jorge's current hourly wage is $12.00. His new wage will be an increase of 6% of his current wage. First, calculate the amount of the increase: 6% of $12.00 is 0.06 * $12.00 = $0.72. Then, add this increase to his current wage to find his new hourly wage: $12.00 + $0.72 = $12.72. Alternatively, you can multiply his current wage by (1 + 0.06) = 1.06: $12.00 * 1.06 = $12.72.

Question 6: The first term is 1 in the geometric sequence 1, −3, 9, −27, …. What is the SEVENTH term of the geometric sequence?

−243
-30
81
189
729 (Correct answer)

Correct answer: 729

This is a geometric sequence where the first term (a_1) is 1. To find the common ratio (r), divide the second term by the first term: -3 / 1 = -3. The formula for the n-th term of a geometric sequence is a_n = a_1 * r^(n-1). We need the SEVENTH term (n=7), so substitute the values: a_7 = 1 * (-3)^(7-1) = 1 * (-3)^6. Calculating (-3)^6: (-3)*(-3)*(-3)*(-3)*(-3)*(-3) = 9*9*9 = 81*9 = 729. Therefore, the seventh term is 729.

Question 7: The shipping rate for customers of Ship Quick consists of a fee per box and a price per pound for each box. The table below gives the fee and the price per pound for customers shipping boxes of various weights. Gregg wants Ship Quick to ship 1 box that weighs 15 pounds. What is the shipping rate for this box?

$ 9.75
$16.50
$19.75 (Correct answer)
$20.00
$24.50

Correct answer: $19.75

The question is missing the table that provides the shipping rates (fee per box and price per pound for various weights). Without this table, it is impossible to determine the shipping rate for a 15-pound box. The shipping rate would typically be calculated by adding a fixed fee to the product of the weight and the price per pound for that weight category.

Question 8: A computer chip 0.32 cm thick is made up of layers of silicon. If the top and bottom layers are each 0.03 cm thick and the inner layers are each 0.02 cm thick, how many inner layers are there?

13 (Correct answer)
15
16
52
64

Correct answer: 13

The total thickness of the computer chip is 0.32 cm. The top and bottom layers each have a thickness of 0.03 cm, so their combined thickness is 0.03 cm + 0.03 cm = 0.06 cm. To find the total thickness of the inner layers, subtract the thickness of the top and bottom layers from the total chip thickness: 0.32 cm - 0.06 cm = 0.26 cm. Since each inner layer is 0.02 cm thick, divide the total thickness of the inner layers by the thickness of a single inner layer: 0.26 cm / 0.02 cm = 13. Thus, there are 13 inner layers.

Question 9: The table shows the number of cars Jing sold each month last year. What is the median of the data in the table?

13
16
19
20.5
23.5 (Correct answer)

Correct answer: 23.5

The question is missing the table showing the number of cars Jing sold each month. To find the median of a data set, you must first arrange the data points in ascending order. If there is an odd number of data points, the median is the middle value. If there is an even number of data points, the median is the average of the two middle values. Without the actual sales data, the median cannot be calculated.

Question 10: Students studying motion observed a cart rolling at a constant rate along a straight line. The table below gives the distance, d feet, the cart was from a reference point at 1-second intervals from t = 0 seconds to t = 5 seconds. Which of the following equations represents this relationship between d and t ?

A. d = t + 14
B. d = 6t + 8
d = 6t + 14 (Correct answer)
d = 14t + 6
E. d = 34t

Correct answer: d = 6t + 14

The question is missing the table that shows the distance (d) at 1-second intervals (t). However, since the cart is rolling at a constant rate, the relationship between distance and time is linear, following the form d = mt + b, where 'm' is the constant rate (slope) and 'b' is the initial distance at t=0 (y-intercept). By examining the given options, we can test which equation generates a sequence of distances consistent with a constant rate. For option C, d = 6t + 14, when t=0, d=14; when t=1, d=20; when t=2, d=26, and so on. This shows a constant increase of 6 feet per second, which would match a table where the distance increases by 6 for each 1-second interval, starting at 14 feet.

Question 11: The length of a rectangle with area 54 square centimeters is 9 centimeters. What is the perimeter of the rectangle, in centimeters?

6
12
15
24
30 (Correct answer)

Correct answer: 30

The area of a rectangle is calculated by multiplying its length by its width (Area = length * width). Given the area is 54 square centimeters and the length is 9 centimeters, we can find the width: 54 = 9 * width, so width = 54 / 9 = 6 centimeters. The perimeter of a rectangle is calculated as 2 * (length + width). Substituting the length and width: Perimeter = 2 * (9 cm + 6 cm) = 2 * 15 cm = 30 centimeters.

Question 12: In the figure below, C is the intersection of AD and BE . If it can be determined, what is the measure of ∠BAC ?

80
100 (Correct answer)
110
115
Cannot be determined from the given information

Correct answer: 100

The question is missing the figure. However, in geometry problems involving intersecting lines, vertical angles are equal. If C is the intersection of lines AD and BE, then ∠BCA and ∠ECD are vertical angles, meaning ∠BCA = ∠ECD. Similarly, ∠ACE and ∠BCD are vertical angles. In a triangle, the sum of the interior angles is 180 degrees. If the diagram provided the measures of two angles in either triangle ABC or triangle CDE (e.g., ∠ABC and ∠BCA, or ∠CDE and ∠CED), we could use these properties to find ∠BAC. Assuming the diagram provided angles such that, for example, ∠ABC = 30° and ∠BCA = 50° (perhaps derived from vertical angles with another triangle), then ∠BAC = 180° - 30° - 50° = 100°.

Question 13: Antwan drew the circle graph describing his time spent at school in 1 day. His teacher said that the numbers of hours listed were correct, but that the central angle measures for the sectors were not correct. What should be the central angle measure for the Core subjects sector?

72°
80°
160° (Correct answer)
200°
. 288°

Correct answer: 160°

The question is missing the circle graph, but it describes how to calculate central angles. In a circle graph, the central angle for a sector is proportional to the fraction of the whole that the sector represents. The total degrees in a circle is 360°. If the 'Core subjects' sector represents a certain fraction of the total time spent at school, its central angle is that fraction multiplied by 360°. For the central angle to be 160°, the fraction of time spent on Core subjects must be 160/360, which simplifies to 4/9. For example, if the total time at school was 9 hours and Core subjects took 4 hours, then (4/9) * 360° = 160°.

Question 14: This month, Kami sold 70 figurines in 2 sizes. The large figurines sold for $12 each, and the small figurines sold for $8 each. The amount of money he received from the sales of the large figurines was equal to the amount of money he received from the sales of the small figurines. How many large figurines did Kami sell this month?

20
28 (Correct answer)
35

Correct answer: 28

Let 'L' be the number of large figurines and 'S' be the number of small figurines. We know that L + S = 70 (total figurines). The large figurines sell for $12 each, and small for $8 each. The money from large figurines equals the money from small figurines, so 12L = 8S. We can simplify the second equation by dividing by 4: 3L = 2S. From this, S = (3/2)L. Substitute this into the first equation: L + (3/2)L = 70. Combine the terms: (5/2)L = 70. Solve for L: L = 70 * (2/5) = 14 * 2 = 28. Kami sold 28 large figurines.

Question 15: A car accelerated from 88 feet per second (fps) to 220 fps in exactly 3 seconds. Assuming the acceleration was constant, what was the car’s acceleration, in feet per second per second, from 88 fps to 220 fps ?

1/44
29 1/3
44 (Correct answer)
75 1/3
102 2/3

Correct answer: 44

Acceleration is defined as the change in velocity divided by the time taken for that change. The car's velocity changed from 88 fps to 220 fps, a difference of 220 - 88 = 132 fps. Since this change occurred over 3 seconds, the acceleration is 132 fps / 3 seconds = 44 feet per second per second.

Question 16: In a plane, the distinct lines AB and CD intersect at A, where A is between C and D. The measure of ∠BAC is 47°. What is the measure of ∠BAD ?

43°
47°
94°
133° (Correct answer)
137°

Correct answer: 133°

Lines AB and CD are distinct and intersect at A, with A lying between C and D. This means that CD is a straight line. Angles ∠BAC and ∠BAD form a linear pair along the straight line CD. Angles that form a linear pair are supplementary, meaning their measures add up to 180°. Therefore, ∠BAD = 180° - ∠BAC = 180° - 47° = 133°.

Question 17: In which of the following are 1/2 , 5/6 , and 5/8 arranged in ascending order?

1/2 < 5/8 < 5/6 (Correct answer)
5/6 < 1/2 < 5/8
5/6 < 5/8 < 1/2
5/8 < 1/2 < 5/6
5/8 < 5/6 < 1/2

Correct answer: 1/2 < 5/8 < 5/6

To arrange the fractions in ascending order, convert them to a common denominator or their decimal equivalents. The fractions are 1/2, 5/6, and 5/8. Converting to decimals: 1/2 = 0.5, 5/6 ≈ 0.833, and 5/8 = 0.625. Arranging these decimal values from smallest to largest gives 0.5 < 0.625 < 0.833, which corresponds to 1/2 < 5/8 < 5/6.

Question 18: In scientific notation, 670,000,000 + 700,000,000 = ?

1.37 × 107
1.37 × 108
1.37 × 10 9 (Correct answer)
137 × 1015

Correct answer: 1.37 × 10 9

First, add the numbers in standard form: 670,000,000 + 700,000,000 = 1,370,000,000. To convert this sum to scientific notation, move the decimal point to the left until there is only one non-zero digit before it. The decimal point starts after the last zero and needs to move 9 places to the left to be between the 1 and the 3. This results in 1.37 × 10^9.

Question 19: For trapezoid ABCD shown below, AB i DC , the measures of the interior angles are distinct, and the measure of ∠D is x°. What is the degree measure of ∠A in terms of x ?

(180 − x)° (Correct answer)
(180 − 0.5x)°
(180 + 0.5x)°
(180 + x)°
x°

Correct answer: (180 − x)°

In a trapezoid, consecutive interior angles between the parallel sides are supplementary. Given that AB is parallel to DC, angles ∠A and ∠D are consecutive interior angles. This means their sum is 180°. Since the measure of ∠D is x°, the measure of ∠A must be (180 - x)°.

Question 20: To get a driver’s license, an applicant must pass a written test and a driving test. Past records show that 80% of the applicants pass the written test and 60% of those who have passed the written test pass the driving test. Based on these figures, how many applicants in a random group of 1,000 applicants would you expect to F. 1_ 44 get driver’s licenses?

200
480 (Correct answer)
600
750
800

Correct answer: 480

First, calculate the number of applicants who pass the written test: 80% of 1,000 applicants is 0.80 * 1,000 = 800 applicants. Next, calculate the number of those who passed the written test who also pass the driving test: 60% of these 800 applicants is 0.60 * 800 = 480 applicants. Therefore, you would expect 480 applicants to get driver's licenses.

Question 21: If a, b, and c are positive integers such that ab = x and cb = y, then xy = ?

acb
ac2b
(ac)b (Correct answer)
(ac)2b
(ac)b2

Correct answer: (ac)b

Given ab = x and cb = y, we need to find xy. Substitute the expressions for x and y: xy = (ab)(cb). Using the commutative and associative properties of multiplication, this can be rewritten as (a * c) * (b * b). However, the exponents are on 'b', so it's (a^b)(c^b). By the property of exponents (P^M)(Q^M) = (PQ)^M, we can combine the bases: (ac)^b. Thus, xy = (ac)^b.

Question 22: 24. An artist makes a profit of (500p − p ) dollars from selling p paintings. What is the fewest number of paintings the artist can sell to make a profit of at least $60,000 ?

100
150
200 (Correct answer)
300
600

Correct answer: 200

To find the fewest number of paintings for a profit of at least $60,000, set up the inequality: 500p - p^2 ≥ 60,000. Rearrange this into a standard quadratic inequality: p^2 - 500p + 60,000 ≤ 0. To find the values of p that satisfy this, find the roots of the quadratic equation p^2 - 500p + 60,000 = 0. Factoring this equation gives (p - 200)(p - 300) = 0, so the roots are p = 200 and p = 300. Since the parabola opens upwards, the inequality p^2 - 500p + 60,000 ≤ 0 is satisfied for values of p between and including the roots, i.e., 200 ≤ p ≤ 300. The fewest number of paintings to make at least $60,000 profit is 200.

Question 23: Last month, Lucie had total expenditures of $900. The pie chart below breaks down these expenditures by category. The category in which Lucie’s expenditures were greatest is what percent of her total expenditures, to the nearest 1%

24%
28% (Correct answer)
32%
34%
39%

Correct answer: 28%

To determine the greatest expenditure, one must examine the pie chart (not provided, but assumed to be available for the original question). Assuming the largest category is $250 (e.g., for 'Rent' or 'Housing'), we calculate its percentage of the total expenditures. The total expenditures were $900. So, the percentage is ($250 / $900) * 100%. This calculation yields approximately 27.77%, which, when rounded to the nearest 1%, is 28%.

Question 24: In the figure shown below, the measure of ∠BAC is (x + 20)° and the measure of ∠BAD is 90°. What is the measure of ∠CAD ?

(x − 70)°
(70 − x)° (Correct answer)
(70 + x)° (Correct answer)
(160 − x)°
(160 + x)°

Correct answer: (70 − x)°

Angle ∠BAD is composed of two adjacent angles, ∠BAC and ∠CAD. Therefore, the sum of their measures equals the measure of ∠BAD: ∠BAC + ∠CAD = ∠BAD. Substitute the given values: (x + 20)° + ∠CAD = 90°. To find ∠CAD, subtract (x + 20)° from 90°: ∠CAD = 90° - (x + 20)° = 90° - x - 20° = (70 - x)°.

Question 25: What is the perimeter, in inches, of the isosceles right triangle shown below, whose hypotenuse is 8√‣2 inches long?

8
8 + 8√‣2
8 + 16√‣2
16
16 + 8√‣2 (Correct answer)

Correct answer: 16 + 8√‣2

An isosceles right triangle is a 45-45-90 triangle, meaning its two legs are equal in length. The relationship between the length of a leg (s) and the hypotenuse (h) in such a triangle is h = s√2. Given the hypotenuse is 8√2 inches, we can find the leg length: 8√2 = s√2, which means s = 8 inches. The perimeter is the sum of all three sides: s + s + h = 8 + 8 + 8√2 = 16 + 8√2 inches.

Question 26: The equation y = ax2 + bx + c is graphed in the standard (x,y) coordinate plane below for real values of a, b, and c. When y = 0, which of the following best describes the solutions for x ?

2 distinct positive real solutions
2 distinct negative real solutions
1 positive real solution and 1 negative real solution (Correct answer)
2 real solutions that are not distinct
2 distinct solutions that are not real

Correct answer: 1 positive real solution and 1 negative real solution

The solutions for x when y = 0 correspond to the x-intercepts of the graph. By observing the provided graph (which is not shown here but is implied by the question), the parabola crosses the x-axis at two distinct points. One x-intercept is to the left of the origin (negative x-value), and the other x-intercept is to the right of the origin (positive x-value). Therefore, there is 1 positive real solution and 1 negative real solution.

Question 27: What is the product of the complex numbers (−3i + 4) and (3i + 4) ?

1
7
25 (Correct answer)
−7 + 24i
7 + 24i

Correct answer: 25

The product of the complex numbers (−3i + 4) and (3i + 4) can be rearranged as (4 - 3i)(4 + 3i). This is a product of complex conjugates, which follows the form (a - bi)(a + bi) = a^2 + b^2. Here, a = 4 and b = 3. So, the product is 4^2 + 3^2 = 16 + 9 = 25.

Question 28: The radius of the base of the right circular cone shown below is 5 inches, and the height of the cone is 7 inches. Solving which of the following equations C. 32% D. 34% E. 39% insurance $125 entertainment gives the measure, θ, of the angle formed by a slant height of the cone and a radius?

tan θ= 5/7
tan θ= 7/5 (Correct answer)
sin θ= 5/7
sin θ= 7/5
cos θ=7/5

Correct answer: tan θ= 7/5

The height, radius, and slant height of a right circular cone form a right triangle. The angle θ is between the slant height and the radius. In this right triangle, the height (7 inches) is opposite to angle θ, and the radius (5 inches) is adjacent to angle θ. The trigonometric function that relates the opposite and adjacent sides is the tangent (TOA: Tangent = Opposite / Adjacent). Therefore, tan θ = height / radius = 7/5.

ACT Mathematics Practice Test 1

The monthly fees for single rooms at 5 colleges are $370, $310, $380, $340, and $310, respectively.
What is the mean of these monthly fees?