What do the ACT Math Questions Test?

🤓 ACT Math Question Skills

📝 Preparing for Higher Math: The Most Tested ACT Math Skill

Example Questions:

• ➕ Algebra
  • Solve the following equation for *x*:
    3*(2x - 5) + 7 = 4x + 14
    A) x = -7
    B) x = -4
    C) x = 3
    D) x = 5
    E) x = 7
    Answer: C) x = 3
    Explanation:
    To solve the equation, start by distributing the 3 on the left side:
    6x - 15 + 7 = 4x + 14
    Combine like terms:
    6x - 8 = 4x + 14
    Next, isolate the variable x on one side of the equation. To do that, move the 4x term to the left side by subtracting 4x from both sides:
    6x - 4x - 8 = 14
    Simplify:
    2x - 8 = 14
    Now, add 8 to both sides of the equation to get the x term alone:
    2x = 22
    Finally, divide both sides by 2 to solve for x:
    x = 11
    Therefore, the correct answer is C) x = 3.
  • Question 2:
  • Simplify the expression: (4x^2 - 7x + 3) / (2x - 3)
    A) 2x - 1
    B) 2x + 1
    C) 2x - 3
    D) 2x + 3
    E) 2x + 5
    Answer: A) 2x - 1
    Explanation:
    To simplify the expression, perform polynomial long division or use synthetic division:
    2x + 1
    ________________
    2x - 3 | 4x^2 - 7x + 3
    -(4x^2 - 6x)
    ________________
    -x + 3
    -(-x + 3)
    ________________
    0
    ```
    Therefore, the simplified expression is *(4x^2 - 7x + 3) / (2x - 3) = 2x + 1*.
• 📌 Functions
  • Given the function f(x) = 2x^2 - 5x + 3, find the value of f(3).
    A) 12
    B) 6
    C) 0
    D) -3
    E) 3
    Answer: B) 6
    Explanation: To find the value of f(3), substitute 3 for x in the given function:
    f(3) = 2(3)^2 - 5(3) + 3
    Simplify:
    f(3) = 2(9) - 15 + 3
    f(3) = 18 - 15 + 3
    f(3) = 3
    Therefore, the correct answer is B) 6.
  • Question 2: Consider the function *g(x) = 3x^3 - 2x^2 + 5x - 4*. Which of the following statements is true about the function?
    B) The function is odd.
    C) The function is neither even nor odd.
    D) The function has a horizontal asymptote.
    E) The function has a vertical asymptote.
    Answer: C) The function is neither even nor odd.
    Explanation: A function is even if *f(x) = f(-x)* for all *x* in its domain, and it is odd if *f(x) = -f(-x)* for all *x* in its domain.
    Let's check the properties of the given function:
    *g(x) = 3x^3 - 2x^2 + 5x - 4*
    *g(-x) = 3(-x)^3 - 2(-x)^2 + 5(-x) - 4*
    *g(-x) = -3x^3 - 2x^2 - 5x - 4*
    The function *g(x)* is not equal to *g(-x)*, and it is also not equal to the negative of *g(-x)*. Therefore, the function *g(x)* is neither even nor odd. A)
• 📐 Geometry
  • Find the area of a right-angled triangle with legs of lengths 5 units and 12 units.
    A) 17 square units
    B) 30 square units
    C) 24 square units
    D) 60 square units
    E) 144 square units
    Answer: B) 30 square units
    Explanation:
    The area of a right-angled triangle is given by the formula: Area = (base * height) / 2.
    In this case, the two legs of the right-angled triangle are 5 units and 12 units.
    Area = (5 * 12) / 2
    Area = 60 / 2
    Area = 30 square units
    Therefore, the correct answer is B) 30 square units.
  • Question #2
    In triangle ABC, angle A measures 55 degrees, and angle B measures 75 degrees. What is the measure of angle C, in degrees?
    A) 20
    B) 30
    C) 45
    D) 60
    E) 90
    Answer: D) 60
    Explanation:
    The sum of the angles in a triangle is always 180 degrees. So, to find the measure of angle C, subtract the measures of angles A and B from 180:
    Measure of angle C = 180 - 55 - 75
    Measure of angle C = 50
    Therefore, the measure of angle C is 60 degrees.
• 🔢 Number and Quantity
  • Which of the following numbers is both a multiple of 5 and a perfect square?
    A) 15
    B) 25
    C) 36
    D) 48
    E) 55
    Answer: C) 36
    Explanation:
    To be a multiple of 5, a number must end in either 0 or 5. Among the given choices, the number 36 ends in 6, so it is not a multiple of 5.
    Now, let's check which number is a perfect square. A perfect square is an integer that can be expressed as the square of an integer. Among the given choices, 36 is a perfect square because it can be expressed as 6^2.
    Therefore, the correct answer is C) 36.
  • Question 2:
  • Which of the following numbers is a prime number?
    A) 21
    B) 33
    C) 47
    D) 56
    E) 63
    Answer: C) 47
    Explanation:
    A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Let's check the given numbers:
    - 21: Divisible by 3 and 7.
    - 33: Divisible by 3 and 11.
    - 47: Only divisible by 1 and 47, making it a prime number.
    - 56: Divisible by 2, 4, 7, and 8.
    - 63: Divisible by 3, 7, and 9.
    Therefore, the prime number among the given choices is 47.
• 🎲 Statistics and Probability
  • A box contains 6 red balls, 4 blue balls, and 5 green balls. If one ball is randomly selected from the box, what is the probability of choosing a red ball?
    A) 6/15
    B) 4/15
    C) 2/5
    D) 3/7
    E) 6/15
    Answer: A) 6/15
    Explanation:
    The total number of balls in the box is 6 (red) + 4 (blue) + 5 (green) = 15 balls.
    To find the probability of choosing a red ball, divide the number of red balls by the total number of balls:
    Probability of choosing a red ball = Number of red balls / Total number of balls
    Probability of choosing a red ball = 6 / 15
    Simplify the fraction:
    Probability of choosing a red ball = 2 / 5
    Therefore, the correct answer is A) 6/15.
  • Question #2
    In a bag, there are 5 red marbles, 4 blue marbles, and 6 green marbles. Two marbles are drawn at random from the bag without replacement. What is the probability of drawing one red marble and one blue marble, in any order?
    A) 5/33
    B) 2/15
    C) 1/3
    D) 20/63
    E) 2/7
    Answer: D) 20/63
    Explanation:
    To find the probability of drawing one red marble and one blue marble, in any order, we can consider two scenarios:
    Scenario 1: Red then Blue
    Scenario 2: Blue then Red
    Let's calculate the probability for each scenario:
    Scenario 1: Red then Blue
    Probability of drawing a red marble first: *(5 red marbles) / (15 total marbles) = 5/15*
    Probability of drawing a blue marble second (after removing one red marble): *(4 blue marbles) / (14 remaining marbles) = 4/14*
    Scenario 2: Blue then Red
    Probability of drawing a blue marble first: (4 blue marbles) / (15 total marbles) = 4/15
    Probability of drawing a red marble second (after removing one blue marble): (5 red marbles) / (14 remaining marbles) = 5/14
    Now, add the probabilities from both scenarios to get the total probability:
    Total probability = Probability of Scenario 1 + Probability of Scenario 2
    Total probability = (5/15) * (4/14) + (4/15) * (5/14)
    Total probability = 20/210 + 20/210
    Total probability = 40/210
    Total probability = 20/105
    Total probability = 4/21
    Therefore, the probability of drawing one red marble and one blue marble, in any order, is 20/63.

✏️ Essential Skills: The Second Most Tested Skill

Example Questions:

• ⚖️ Proportions
  • If 4 similar notebooks cost $12, how much would 7 similar notebooks cost?
  • A) $5 B) $14 C) $21 D) $28 E) $49
  • Answer: C) $21
  • Explanation:
  • To find the cost of 7 similar notebooks, use proportions:
  • Let x be the cost of 7 notebooks.
  • 4 notebooks cost $12, so one notebook costs $12 / 4 = $3.
  • Now, set up the proportion:
  • (Cost of 7 notebooks) / (Cost of 1 notebook) = (7 notebooks) / (1 notebook)
  • x / 3 = 7 / 1
  • x = 3 * 7
  • x = 21
  • Therefore, the cost of 7 similar notebooks is $21.
  • In a recipe, the ratio of milk to flour is 3:2. If 5 cups of flour are used, how many cups of milk should be used?
  • A) 2 B) 5 C) 7.5 D) 8 E) 10
  • Answer: C) 7.5
  • Explanation:
  • Let x be the number of cups of milk needed.
  • The given ratio of milk to flour is 3:2, which means:
  • (Cups of milk) / (Cups of flour) = 3 / 2
  • x / 5 = 3 / 2
  • To find the value of x, cross-multiply and solve for x:
  • 2x = 3 * 5
  • 2x = 15
  • x = 15 / 2
  • x = 7.5
  • Therefore, 7.5 cups of milk should be used in the recipe.
• ％ Percentages
  • A shirt is originally priced at $40. During a sale, the price is reduced by 20%. How much is the shirt during the sale?
  • A) $8 B) $16 C) $24 D) $32 E) $48
  • Answer: D) $32
  • Explanation:
  • To find the sale price of the shirt, multiply the original price by the percentage reduction:
  • Sale price = Original price - (Percentage reduction * Original price)
  • Sale price = $40 - (0.20 * $40)
  • Sale price = $40 - $8
  • Sale price = $32
  • Therefore, the shirt is priced at $32 during the sale.
  • At the beginning of the year, a company had 80 employees. Over the course of the year, the company hired 20 new employees and had to let go of 12 employees. What was the percentage increase in the number of employees during the year?
  • A) 8% B) 15% C) 20% D) 50% E) 66.67%
  • Answer: C) 20%
  • Explanation:
  • To find the percentage increase in the number of employees, use the formula:
  • Percentage increase = (Increase in quantity / Original quantity) * 100
  • The increase in the number of employees = 20 (new hires) - 12 (employees let go) = 8 employees.
  • Percentage increase = (8 / 80) * 100
  • Percentage increase = 0.1 * 100
  • Percentage increase = 10%
  • Therefore, the percentage increase in the number of employees during the year is 10%.
• 📦 Volume
  • A rectangular box has dimensions of 4 inches by 6 inches by 3 inches. What is the volume of the box?
  • A) 18 cubic inches B) 36 cubic inches C) 72 cubic inches D) 80 cubic inches E) 144 cubic inches
  • Answer: B) 36 cubic inches
  • Explanation:
  • The volume of a rectangular box is calculated by multiplying its length, width, and height:
  • Volume = Length * Width * Height
  • Volume = 4 inches * 6 inches * 3 inches
  • Volume = 24 cubic inches
  • Therefore, the volume of the box is 24 cubic inches.
  • A cylindrical tank has a height of 10 feet and a diameter of 8 feet. What is the volume of the tank? (Use π ≈ 3.14)
  • A) 80π cubic feet B) 160π cubic feet C) 200π cubic feet D) 400π cubic feet E) 800π cubic feet
  • Answer: B) 160π cubic feet
  • Explanation:
  • The volume of a cylinder is given by the formula:
  • Volume = π * r^2 * h
  • where r is the radius and h is the height.
  • Given the diameter is 8 feet, the radius (r) is half of the diameter, so r = 8/2 = 4 feet.
  • Now, calculate the volume:
  • Volume = π * (4 feet)^2 * 10 feet
  • Volume = π * 16 square feet * 10 feet
  • Volume = 160π cubic feet
  • Therefore, the volume of the tank is 160π cubic feet.
• 🗺️ Surface Area
• This is the category many people struggle with as it requires previous knowledge of many of the surface area formulas! I recommend looking over these!
  • A cube has a side length of 6 inches. What is the total surface area of the cube?
  • A) 12 square inches B) 24 square inches C) 36 square inches D) 72 square inches E) 216 square inches
  • Answer: D) 72 square inches
  • Explanation:
  • The total surface area of a cube is calculated by multiplying the area of one face by the number of faces (6 for a cube).
  • Surface Area = 6 * (Side length)^2
  • Surface Area = 6 * (6 inches)^2
  • Surface Area = 6 * 36 square inches
  • Surface Area = 216 square inches
  • Therefore, the total surface area of the cube is 216 square inches.
  • A right circular cone has a base radius of 5 feet and a slant height of 13 feet. What is the total surface area of the cone? (Use π ≈ 3.14)
  • A) 110π square feet B) 150π square feet C) 195π square feet D) 210π square feet E) 260π square feet
  • Answer: C) 195π square feet
  • Explanation:
  • The total surface area of a right circular cone is the sum of its lateral surface area and the area of its base.
  • The lateral surface area of a cone is given by:
  • Lateral Surface Area = π * r * l
  • where r is the base radius and l is the slant height.
  • Given r = 5 feet and l = 13 feet, calculate the lateral surface area:
  • Lateral Surface Area = π * 5 feet * 13 feet
  • Lateral Surface Area = 65π square feet
  • The area of the base of the cone is given by:
  • Base Area = π * r^2
  • Base Area = π * (5 feet)^2
  • Base Area = 25π square feet
  • Now, calculate the total surface area:
  • Total Surface Area = Lateral Surface Area + Base Area
  • Total Surface Area = 65π square feet + 25π square feet
  • Total Surface Area = 90π square feet
  • Therefore, the total surface area of the cone is 90π square feet.

📍 Modeling: Also Tested on the ACT

Example Questions:

• A car rental company charges a flat fee of $30 per day for renting a car, plus an additional $0.25 per mile driven. If a customer rents a car and drives it for 3 days, accumulating 150 miles, how much will the customer be charged in total?
• A) $45 B) $60 C) $75 D) $90 E) $105
• Answer: C) $75
• Explanation:
• To find the total charge, calculate the daily rental cost and the mileage cost, and then sum them up.
• Daily rental cost = $30 per day * 3 days = $90 Mileage cost = $0.25 per mile * 150 miles = $37.50
• Total charge = Daily rental cost + Mileage cost = $90 + $37.50 = $127.50
• Therefore, the customer will be charged a total of $127.50.
• A company is selling tickets to a concert. The cost per ticket is $50 during the early bird period and $60 during the regular sale period. The company estimates that during the early bird period, they will sell 300 tickets, and during the regular sale period, they will sell 500 tickets. The company also expects that for every $5 increase in ticket price, the number of tickets sold will decrease by 30. Assuming all tickets are sold, how much total revenue will the company generate from ticket sales?
• A) $57,000 B) $59,000 C) $61,500 D) $63,000 E) $65,000
• Answer: C) $61,500
• Explanation:
• Early Bird Period:Tickets sold = 300Ticket price = $50Revenue during early bird period = Tickets sold * Ticket price = 300 * $50 = $15,000Regular Sale Period:Tickets sold = 500Ticket price = $60Revenue during regular sale period = Tickets sold * Ticket price = 500 * $60 = $30,000
• Now, let's consider the decrease in ticket sales due to the price increase during the regular sale period. For every $5 increase in ticket price, the number of tickets sold will decrease by 30.
• The price increased by $60 - $50 = $10, resulting in a decrease in ticket sales by (10 / 5) * 30 = 60 tickets.
• Adjusted tickets sold during the regular sale period = 500 - 60 = 440 tickets
• Revenue during the adjusted regular sale period = Adjusted tickets sold * Ticket price = 440 * $60 = $26,400
• Total revenue = Revenue during early bird period + Revenue during adjusted regular sale period Total revenue = $15,000 + $26,400 = $41,400
• Therefore, the company will generate a total revenue of $41,400 from ticket sales.

🫡 Conclusion