# Math

ACT Mathematics Test contains **60 multiple-choice questions** to be answered in **60 minutes**. The ACT Math Test is always the **second test**. Concepts tested include arithmetic, algebra, coordinate geometry, plane geometry, and trigonometry.

#### Jennifer’s best long jump distance increased by 10% from 1990 to 1991

Jennifer's best long jump distance increased by 10% from 1990 to 1991 and by 20% from 1991 to 1992. By what percent did her best long jump distance increase from 1990 to 1992 ?

- 32%
- 30%
- 20%
- 15%
- 2%

#### If loga x = s and loga y = t, then loga (xy)^2

If log_{a} x = s and log_{a} y = t, then log_{a} (xy)^{2} = ?

- 2(s + t)
- s + t
- 4st
- 2st
- st

#### A simple pendulum consists of a small mass suspended from a string that is fixed

A simple pendulum consists of a small mass suspended from a string that is fixed at its upper end and has negligible mass. The length of time, t seconds, for a complete swing of a simple pendulum can be modeled by the equation t = 2π√(L/32), where L is the length, in feet, of the string. If the time required for a complete swing of Pendulum 1 is triple the time required for a complete swing of Pendulum 2, the length of Pendulum 1's string is how many times the length of Pendulum 2's string?

- 1/3
- 3
- 6
- 9
- 27

#### Vectors AB and CD are shown in the standard (x,y) coordinate plane below

Vectors AB and CD are shown in the standard (x,y) coordinate plane below. One of the following is the unit vector notation of the vector AB + CD. Which one?

- -6i + 3j
- 3i - 1j
- 3i + 9j
- 9i + 1j
- 9i + 11j

#### Which of the following expressions gives the number of permutations of 15 objects taken 5 at a time

Which of the following expressions gives the number of permutations of 15 objects taken 5 at a time?

- 15(5)
- (15 - 5)!
- 15!/5!
- 15!/(15 - 5)!
- 15!/(5!)(15 - 5!)

#### On his first day as a telemarketer, Marshall made 24 calls

On his first day as a telemarketer, Marshall made 24 calls. His goal was to make 5 more calls on each successive day than he had made the day before. If Marshall met, but did not exceed, his goal, how many calls had he made in all after spending exactly 20 days making calls as a telemarketer?

- 670
- 690
- 974
- 1,430
- 1,530

#### Triangle ABC has vertices A(8,2), B(0,6), and C(-3,2)

Triangle ABC has vertices A(8,2), B(0,6), and C(-3,2). Point C can be moved along a certain line, with points A and B remaining stationary, and the area of ABC will not change. What is the slope of that line?

- -1/2
- -3/4
- 0
- 4/3
- 2

#### In the figure below, all line segments are either horizontal or vertical

In the figure below, all line segments are either horizontal or vertical and the dimensions given are in inches. What is the perimeter, in inches, of the figure?

- 10
- 12
- 13
- 14
- 16

#### In the figure below, A, C, F, and D are collinear; B, C, and E are collinear

In the figure below, A, C, F, and D are collinear; B, C, and E are collinear; and the angles at A, E, and F are right angles, as marked. Which of the following statements is NOT justifiable from the given information?

- AB is parallel to EF
- DE is perpendicular to BE
- ∠ACB is congruent to ∠FCE
- ΔBAC is similar to ΔEFC
- CE is congruent to ED

#### Which of the following expressions, if any, are equal for all real numbers x

Which of the following expressions, if any, are equal for all real numbers x ?

- √(-x)
^{2} - ⎪-x⎪
- -⎪x⎪

- I and II only
- I and III only
- II and III only
- I, II, and III
- None of the expressions are equivalent

#### In the standard (x,y) coordinate plane below, an angle is shown whose vertex is the origin

In the standard (x,y) coordinate plane below, an angle is shown whose vertex is the origin. One side of this angle with measure θ passes through (4,-3), and the other side includes the positive x-axis. What is the cosine of θ ?

- -4/3
- -3/4
- -3/5
- 4/5
- 5/4

#### Let A and B be independent events. Denote P(A) as the probability that Event A will occur

Let A and B be independent events. Denote P(A) as the probability that Event A will occur, and denote P(A ∩ B) as the probability that Events A and B will both occur. Which of the following equations must be true?

- P(A) = P(B)
- P(A) = 1 - P(B)
- P(A ∩ B) = P(A) + P(B)
- P(A ∩ B) = P(A) · P(B)
- P(A ∩ B) = P(A) + P(B) - (P(A) · P(B))

#### A restaurant has 10 booths that will seat up to 4 people each

A restaurant has 10 booths that will seat up to 4 people each. If 20 people are seated in booths, and NO booths are empty, what is the greatest possible number of booths that could be filled with 4 people?

- 0
- 1
- 2
- 3
- 5

#### Shima will mix 1 fluid ounce of fertilizer in water

Shima will mix 1 fluid ounce of fertilizer in water for every 40 square feet of soil. At this rate, which of the following expressions gives the number of gallons of fertilizer that Shima will mix in water for 0.5 acres of soil?

(Note: 1 acre = 43,560 square feet; 1 gallon = 128 fluid ounces)

- \( \frac{0.5(40)(128)}{43,560} \)
- \( \frac{40(128)}{0.5(43,560)} \)
- \( \frac{0.5(43,560)}{40(128)} \)
- \( \frac{43,560}{0.5(40)(128)} \)
- \( \frac{0.5(43,560)(40)}{128} \)

#### Let a equal 2b + 3c - 5. What happens to the value of a if

Let a equal 2b + 3c - 5. What happens to the value of a if the value of b decreases by 1 and the value of c increases by 2 ?

- It increases by 4.
- It increases by 2.
- It increases by 1.
- It is unchanged.
- It decreases by 2.

#### What is the degree measure of the smaller of the 2 angles

What is the degree measure of the smaller of the 2 angles formed by the line and the ray shown in the figure below?

- 14°
- 28°
- 29°
- 58°
- Cannot be determined from the given information

#### What is the matrix product \( \begin{bmatrix} a \\ 2a \\ 3a \end{bmatrix} \begin{bmatrix} 1 & 0 & -1 \end{bmatrix} \)

What is the matrix product \( \begin{bmatrix} a \\ 2a \\ 3a \end{bmatrix} \begin{bmatrix} 1 & 0 & -1 \end{bmatrix} \) ?

- \( \begin{bmatrix} a & 0 & -a \\ 2a & 0 & -2a \\ 3a & 0 & -3a \end{bmatrix} \)
- \( \begin{bmatrix} a & 2a & 3a \\ 0 & 0 & 0 \\ -a & -2a & -3a \end{bmatrix} \)
- \( \begin{bmatrix} 2a & 0 & -2a \end{bmatrix} \)
- \( \begin{bmatrix} 6a & 0 & -6a \end{bmatrix} \)
- \( \begin{bmatrix} 0 \end{bmatrix} \)

#### A certain hotel has 80 rooms. Based on many previous years' occupancy rates

A certain hotel has 80 rooms. Based on many previous years' occupancy rates, the owners of the hotel constructed the table below showing the daily occupancy rates and their probabilities of occurring for the coming summer season. Based on the probability distribution in the table, to the nearest whole number, what is the expected number of rooms that will be occupied on any day during the coming summer season?

- 20
- 25
- 58
- 60
- 75

#### A certain perfect square has exactly 4 digits (that is, it is an integer between 1,000 and 9,999)

A certain perfect square has exactly 4 digits (that is, it is an integer between 1,000 and 9,999). The positive square root of the perfect square must have how many digits?

- 1
- 2
- 3
- 4
- Cannot be determined from the given information

#### For every pair of natural numbers n and m

For every pair of natural numbers n and m, to which of the following sets must n + m belong?

- The natural numbers
- The integers
- The rational numbers
- The real numbers
- The complex numbers

- I, II, and III only
- II, III, and IV only
- III, IV, and V only
- II, III, IV, and V only
- I, II, III, IV, and V

#### In the standard (x, y) coordinate plane below, the points (0, 0), (10, 0), (13, 6), and (3, 6)

In the standard (x, y) coordinate plane below, the points (0, 0), (10, 0), (13, 6), and (3, 6) are the vertices of a parallelogram. What is the area, in square coordinate units, of the parallelogram?

- 30
- 60
- 30√3
- 30√5
- 60√5

#### After visiting Florida State University during spring break, Francisco rents a car for 2 days

After visiting Florida State University during spring break, Francisco rents a car for 2 days to travel around Florida. He has $255 to spend on car rental for the 2 days. Sea Horse Car Rental charges $50 per day and $0.25 per mile. Ocean Blue Car Rental charges $60 per day and $0.20 per mile. Which company, if either, allows him to travel more miles for the 2 days, and how many miles more? (Note: Taxes are already included in the rental charges.)

- Sea Horse, 20
- Ocean Blue, 55
- Ocean Blue, 100
- Sea Horse, 135
- Francisco would get the same maximum number of miles from each company.

#### Which equations expresses c in terms of a for all real numbers a, b, and c such that

Which of the following equations expresses c in terms of a for all real numbers a, b, and c such that a^{3} = b and b^{2} = c ?

- c = a
^{6} - c = a
^{5} - c = 2a
^{3} - c = a/2
- c = a

#### The curve y = 0.005x^2 - 2x + 200 for 0 ≤ x ≤ 200 and the line segment from F(0, 200) to G(200, 0)

The curve y = 0.005x^{2} - 2x + 200 for 0 ≤ x ≤ 200 and the line segment from F(0, 200) to G(200, 0) are shown in the standard (x, y) coordinate plane below.

**1. **What is the y-coordinate for the point on the curve with x-coordinate 20 ?

- 160
- 162
- 164
- 166
- 168

#### The sign of a is positive. The sign of b is negative

The sign of a is positive. The sign of b is negative. If it can be determined, what is the sign of the mean of a and b ?

- Positive
- Negative
- Both positive and negative
- Neither positive nor negative
- Cannot be determined from the given information

#### The figure below shows 2 tangent circles such that the 10-centimeter diameter

The figure below shows 2 tangent circles such that the 10-centimeter diameter of the smaller circle is equal to the radius of the larger circle. What is the area, in square centimeters, of the shaded region?

- 10
- 75
- 5π
- 10π
- 75π

#### The median of a set of data containing 9 items was found

The median of a set of data containing 9 items was found. Four data items were added to the set. Two of these items were greater than the original median, and the other 2 items were less than the original median. Which of the following statements must be true about the median of the new data set?

- It is the average of the 2 new lower values.
- It is the same as the original median.
- It is the average of the 2 new higher values.
- It is greater than the original median.
- It is less than the original median.

#### If 5 times a number n is subtracted from 15, the result is negative

If 5 times a number n is subtracted from 15, the result is negative. Which of the following gives the possible value(s) for n ?

- 0 only
- 3 only
- 10 only
- All n > 3
- All n < 3

#### The temperature, t, in degrees Fahrenheit, in a certain town on a certain spring day

The temperature, t, in degrees Fahrenheit, in a certain town on a certain spring day satisfies the inequality ⎪t - 24⎪ ≤ 30. Which of the following temperatures, in degrees Fahrenheit, is NOT in this range?

- -10
- -6
- -5
- 0
- 54

#### The scatterplot in the standard (x, y) coordinate plane below contains data points

The scatterplot in the standard (x, y) coordinate plane below contains data points showing a strong linear correlation between the variables x and y. Mia drew the line shown to model the data. One of the following equations represents Mia’s line. Which one?

- y = -3x + 8
- y = -3x + 10
- y = -2x + 10
- y = 2x + 10
- y = 2x + 8

#### At a school picnic, 1 junior and 1 senior will be selected to lead the activities

At a school picnic, 1 junior and 1 senior will be selected to lead the activities. If there are 125 juniors and 100 seniors at the picnic, how many different 2-person combinations of 1 junior and 1 senior are possible?

- 25
- 100
- 125
- 225
- 12,500

#### If f(x) = x^2 + x + 5 and g(x) = √x, then what is the value of g(4)/f(1)

If f(x) = x^{2} + x + 5 and g(x) = √x, then what is the value of g(4)/f(1)?

- 2/7
- 25/7
- 2/25
- 2
- 4

#### The braking distance, y feet, for Damon's car to come to a complete stop is modeled by

The braking distance, y feet, for Damon's car to come to a complete stop is modeled by y = 3(x^{2} + 10x)/40, where x is the speed of the car in miles per hour. According to this model, which of the following is the maximum speed, in miles per hour, Damon can be driving so that the braking distance is less than or equal to 150 feet?

- 10
- 30
- 40
- 50
- 60

#### A park has the shape and dimensions in blocks given below. A water fountain is located halfway between

A park has the shape and dimensions in blocks given below. A water fountain is located halfway between point B and point D. Which of the following is the location of the water fountain from point A ? (Note: The park’s borders run east-west or north-south.)

- 3½ blocks east and 6 blocks north
- 5 blocks east and 4½ blocks north
- 5 blocks east and 6 blocks north
- 8½ blocks east and 4½ blocks north
- 9 blocks east and 7½ blocks north

#### In the right triangle shown below, which of the following statements is true about ∠A

In the right triangle shown below, which of the following statements is true about ∠A ?

- cos A = 12/13
- sin A = 12/13
- tan A = 12/13
- cos A = 13/12
- sin A = 13/12

#### When graphed in the standard (x, y) coordinate plane

When graphed in the standard (x, y) coordinate plane, which of the following equations does not represent a line?

- x = 4
- 3y = 6
- x - y = 1
- y = 3x/4 - 2
- x
^{2}+ y = 5

#### The depth of a pond is 180 cm and is being reduced by 1 cm per week

The depth of a pond is 180 cm and is being reduced by 1 cm per week. The depth of a second pond is 160 cm and is being reduced by 1/2 cm per week. If the depths of both ponds continue to be reduced at these constant rates, in about how many weeks will the ponds have the same depth?

- 10
- 20
- 40
- 80
- 140

#### A machine part is diagrammed in the figure below with the dimensions given in inches

A machine part is diagrammed in the figure below with the dimensions given in inches. If the centers of the circles lie on the same line parallel to the bottom of the part, what is the distance, in inches, between the centers of the 2 holes in the machine part?

- 5 3/16
- 5 1/16
- 5
- 4 13/16
- 4 3/16

#### What is the y-intercept of the line in the standard (x, y) coordinate plane

What is the y-intercept of the line in the standard (x, y) coordinate plane that goes through the points (-3, 6) and (3,2) ?

- 0
- 2
- 4
- 6
- 8

#### Ken baked, frosted, and decorated a rectangular cake for the last Math Club meeting

Ken baked, frosted, and decorated a rectangular cake for the last Math Club meeting. The cake was 3 inches high, 12 inches wide, and 16 inches long. He centered the cake on a piece of cardboard whose rectangular top surface had been covered with aluminum foil, as shown in the figure below.

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