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Free Compass Math Practice Questions
on Setting up Equations with Solutions and Explanations - Sample 10

Solutions with detailed explanations to compass math test practice questions in sample 10.

  1. If the length L of a rectangle is 3 meters more than twice its width and its perimeter is 300 meters, which of the following equations could be used to find L?

    A) 3L + 3 = 300
    B) 3L = 300
    C) 3L - 3 = 300
    D) 2L + 3 = 300
    E) 4L = 300

    Solution

    Let L and W be the length and width of the rectangle. "length L of a rectangle is 3 meters more than twice its width " is translated mathematically as follows

    L = 2 W + 3

    Use perimeter to write

    300 = 2 W + 2 L

    We need to rewrite the above equation in terms of L only. Solve the equation L = 2 W + 3 for W to obtain

    W = (1/2)(L - 3)

    We now substitute W by (1/2)(L - 3) in the perimeter formula. Hence

    300 = 2 ((1/2)(L - 3)) + 2 L = 3L - 3

    Simplify

    300 = 3L - 3

    Which correspond to answer in C) above.

  2. The average of two numbers is 50. Their difference is 40. Write an equation that may be used to find x the smallest of the two numbers.

    A) x - 20 = 50
    B) 2x + 20 = 50
    C) x - 20 = 100
    D) x + 20 = 40
    E) x + 20 = 50

    Solution

    If x is the smallest number and the difference of the two numbers is 40, then the second number is 40 + x. The average of the two numbers is 50. Hence

    (x + x + 40) / 2 = 50

    Multiply both sides of the equation by 2 and group like terms

    2x + 40 = 100

    Divide all terms of the above equation by 2

    x + 20 = 50

    Which corresponds to the answer in E).

  3. Pump A can fill a tank in 2 hours and pump B can fill the same tank in 3 hours. If t is the time, in hours, that both pump take to fill the tank, which of these equations could be used to find t?

    A) 2t + 3t = 1
    B) t/2 + t/3 = 1
    C) t / (2 + 3) = 1
    D) (2 + 3) / t = 1
    E) t + 2 + 3 = 1

    Solution

    If pump A can fill the tank in 2 hours, its rate is 1/2 tank/hour and similarly the rate of pump B is 1/3 tank/hour. In t hours, pump A fills (1/2) t tank and pump B fills (1/3) t tank. If t is the time to fill the whole tank (1), then

    (1/2)t + (1/3) t = 1

    The above equation may also be written as

    t/2 + t/3 = 1

    and corresponds to the answer in B)


  4. John drove for two hours at the speed of 50 miles per hour (mph) and another x hours at the speed of 55 mph. If the average speed of the entire journey is 53 mph, which of the following could be used to find x?

    A) (55 + x) / 2 = 53
    B) (50 + x) / 2 = 53
    C) (55 + 53) / 2 = x
    D) 100 + 55 x = 53 (2 + x)
    E) 53 x = 2

    Solution

    The average of the entire journey is given by

    total distance / total time

    The total distance for this journey is

    total distsnce = 50*2 + 55 x = 100 + 55x

    The total time for this journey is

    total time = 2 + x

    The average speed is 53. hence

    53 = (100 + 55x) / (2 + x)

    The above equation may be written as

    100 + 55x = 53(2 + x)

    which corresponds to the answer in D)

  5. The sum of 3 consecutive even numbers is 126. Which of these equations could be used to find x, the largest of these 3 numbers?

    A) 3x - 6 = 126
    B) 3x + 6 = 126
    C) 3x + 3 = 126
    D) 3x - 3 = 126
    E) 3x - 9 = 126

    Solution

    The difference between any two consecutive even numbers is 2. Hence if x is the largest, then the other two numbers are

    x - 2 and x - 4

    The sum of these numbers is 126. Hence

    x - 4 + x - 2 + x = 126

    which may be written as

    3x - 6 = 126

    and corresponds to answer A) above.

  6. The radius of a circle is 3 centimeters (cm) more than twice the side of a square. The circumference of the circle is 4 times the perimeter of the square. Write an equation that can be used to find the radius r of the circle.

    A) 2 pi r = 16 (r - 3)
    B) 2 pi r = 8 (r + 3)
    C) 2 pi r = 16 (r + 3)
    D) 2 pi r = 8 (r - 3)
    E) 6 pi = 4r

    Solution

    If r is the radius of the circle and x is the side of the square, then

    r = 2x + 3

    If C is the circumference of the circle and P is the perimeter of the square, then

    C = 4 P

    Now using the formula for circumference (C = 2 pi r) of circle and perimeter (P = 4x) of square, we can write

    2 pi r = 4 (4x)

    We now solve the equation r = 2x + 3 for x

    x = (1/2)(r - 3)

    and substitute x by (1/2)(r - 3) in 2 pi r = 4 (4x), simplify to obtain

    2 pi r = 4 (4((1/2)(r - 3)))

    2 pi r = 8(r - 3)

    which corresponds to D) above

  7. The height of a trapezoid is h (mm). b is the length of one of the two bases of the trapezoid and is 2 mm longer than 3 times the length of the second base. The area of the trapezoid is 300 mm2. Write an equation to find b.

    A) (h / 2) (4b - 2) = 300
    B) (h / 2) (4b + 2) = 300
    C) h (4b - 2) = 300
    D) h (4b - 2) = 600
    E) (h / 3) (4b - 2) = 600

    Solution

    The area A of a trapezoid of height h and bases b and B is given by

    A = (h/2)*(b + B)

    "b is the length of one of the two bases of the trapezoid and is 2 mm longer than 3 times the length of the second base" is translated mathematically as

    b = 3B + 2 which can also be written as B = (b - 2) / 3

    Substitute in the formula of the area

    300 = (h/2) [ b + (b-2)/3 ]

    Which may rewritten as

    600 = (h / 3)(4b - 2) , which corresponds to E) above

  8. 25% of one third of the sum of twice x and 3 is equal to half of the difference of x and its tenth. Write an equation to find x.

    A) 0.25(2x + 3) / 3 = 0.5 (x - 0.1)
    B) 0.25(2x + 3) / 3 = 0.5 (x - 0.1 x)
    C) 0.25(2x) / 3 + 3 = 0.5 (x - 10)
    D) 0.25(2x + 3) / 3 + 3 = 0.5 (x - 10/x)
    E) 0.25(2x) / 3 + 3 = 0.5 (x - 10 x)

  9. The price x of a car was first decreased by 15% and decreased a second time by 10%. Write an equation to find x if the car was bought at $12,000.

    A) (x - 0.15 x) - 0.1 (x - 0.15 x) = 12,000
    B) x - 0.15x - 0.1 x = 12,000
    C) x - 0.15x - 0.1 x = 12,000
    D) (x - 0.15 x) - 0.1 x = 12,000
    E) x = 12,000 - 15% - 10%

  10. A company produces a product for which the variable cost is $6.2 per unit and the fixed costs are $22,000. The company sells the product for $11.45 per unit. Write an equation to find x, the numbers of units sold if the total profit made by the company was $45,000.

    A) 45,000 = 11.45 x + (6.2 x + 22,000)
    B) 45,000 - 22,000 = 11.45 x - 6.2 x
    C) 45,000 = 11.45 x - (6.2 x + 22,000)
    D) 45,000 = 11.45 x - 6.2 x + 22,000
    E) 45,000 + 11.45 x = 6.2 x + 22,000

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