Answers and Explanations

Part 5: Mathematics Knowledge (MK)

  1. D

    A question like this one tests your ability to work with decimals. is the same as = 600. (D) is correct. When dividing by a decimal, be sure to move the decimal place the same number of spaces for both numbers.

  2. B

    Be careful with your number crunching here. 9,899,399 + 2,082 = 9,901,481, choice (A).

  3. D

    The cube of a number is that number multiplied by itself three times. So the cube of 9 would be 9 × 9 × 9 = 729, choice (D).

  4. A

    Plug in the values for a and b and remember your order of operations when working through your calculations. When a = −2 and b = 3,

    (−ab)(a) = [− (−2) × 3](−2)

         = (−[−6])(−2)

         = (6) × (−2) = −12, choice (A).

  5. C

    This is a classic product of two binomials. Remember to FOIL, and you’re good to go.

    (x − 4)(x − 4)

    = (x)(x) + (x)(−4) + (− 4)(x) + (−4)(−4)

    = x2 − 4x − 4x + 16

    = x2 − 8x + 16.

    Choice (C) is correct. If you recognized this as one of the classic quadratics, a binomial squared, you could have answered this correctly without using FOIL.

  6. D

    104 = 10,000, indicating that you should move the decimal four places to the right. So 0.123 × 104 = 0.123 × 10,000 = 1,230, choice (D).

  7. A

    Know your perfect squares. and . So choice (A).

  8. B

    Given a diameter of 6, the radius must equal of 6, or 3. Next, the circumference (c) = 2πr = 2π(3) = 6π. The area (b) = πr2 = π(32) = 9π. Add those two: 9π + 6π = 15π.

  9. B

    To find the probability of something occurring, divide the number of desired outcomes by the number of total outcomes. In the example of the bag of marbles, you begin with 8 + 4 + 7 + 5 = 24 marbles, and draw out 8, leaving you with 16 marbles. Out of those 16 marbles, 4 must be white since the chance of drawing a white marble is now or . If you are left with 4 white marbles, you must have already withdrawn 4 white marbles. (B) is correct.

  10. D

    If B is equal to the number of books that Janice has, and you know that Liza has 40 fewer than 3 times the number of books that Janice has, Liza has 3B − 40 books, and Janice has B books. Together they have 4B − 40 books, choice (C).

  11. D

    The perimeter of a square is 4s where s is the length of a side. If a square has a perimeter of 32, then it has a side length of 8. The area of the square is s2 = 82 = 64, choice (D).

  12. C

    You can simplify this expression as follows:

    Remember to subtract exponents when dividing.

  13. C

    Go through the answer choices one at a time, and select the choice whose digits have a sum and product that are equal.

    (A) 111. Product = Sum = 1 + 1 + 1 = 3.
    (1)(1)(1) = 1. Eliminate.
    (B) 220. Product = Sum = 2 + 2 + 0 = 4.
    (2)(2)(0) = 0. Eliminate.
    (C) 321. Product = Sum = 3 + 2 + 1 = 6.
    (3)(2)(1) = 6.

    Choice (C) is correct.

  14. D

    If x = , and we are asked to solve for y when , begin by plugging in for x. . Cross-multiply and solve. y = 8, so y = 64, (D).

  15. D

    When parallel lines are crossed by a transversal, all acute angles formed are equal, and all acute angles are supplementary to all obtuse angles. So in this diagram, obtuse angle y is supplementary to the acute angle of 55°. Angle x is an acute angle, so it is equal to 55°. Therefore, angle x is supplementary to angle y, and the two must sum to 180°.

  16. A

    If you’re looking for a in terms of b, isolate the a on one side of the equation.

    3ab = 6

    ab = 2

  17. C

    Distribute the numbers outside the parentheses and solve for y.

  18. B

    Since RS and RT are equal, the angles opposite them must be equal. Therefore, angle T = angle S. Since the degree measures of the three interior angles of a triangle sum to 180, 70 + angle measure S + angle measure T = 180, and angle measure S + angle measure T = 110. Since the two angles, S and T, are equal, each must have angle measures half of 110, or 55.

  19. B

    When D is divided by 15, the result is 6 with a remainder of 2. That means that D = 6(15) + 2 or 92. When 92 is divided by 6, the remainder is 2. (B) is correct.

  20. C

    The average of 7 consecutive even numbers is 24. In a set of evenly spaced numbers, the median is the same as the average. That means that 24 must be the middle number the set of numbers. So the set must be {18, 20, 22, 24, 26, 28, 30}. The largest number is 30, choice (C).

  21. D

    The surface area of a rectangular solid is 2lw + 2lh + 2wh. In this case that would be (2 × 2 × 3) + (2 × 2 × 4) + (2 × 3 × 4) = 12 + 16 + 24 = 52 square inches, choice (D).

  22. D

    If 100 ÷ x = 10n, that can be rewritten as . Cross-multiply and you get 100 = 10nx. Divide both sides by 10 to solve for nx.

    (A) is correct.

  23. B

    A circle contains 360°, so:

  24. C

    5! = 5 × 4 × 3 × 2 × 1 = 120.

  25. A

    If Melissa has 5n photographs, and she gives n photographs to each of three friends, she would have given away 3n photographs. 5n − 3n = 2n. (A) is correct.