Solutions for Practice Test 6, The Official SAT Study Guide, Section 8

Here are solutions for section 8 of practice test #6 in The Official SAT Study Guide, second edition, found on pages 728–733. The solutions below demonstrate faster, more informal methods that might work better for you on a fast-paced test such as the SAT. To learn more about these methods, see my e-book Succeeding in SAT Math or the SAT math tips page.

1. If 3(n − 4) = 18, then n − 4 = 6 and so n = 10. Select (D) 10.
2. As there are 4 types of stones and 3 types of metals, there are 4 × 3 = 12 combinations of a stone and a metal. Select (D) 12. If you're in doubt as to how to do this one, say that there are stones A, B, C, and D and metals X, Y, and Z, and try to list all combinations. You'll find that there are 12. Select (D) 12.
3. Convert the sentence into an equation:

   The sum of 3a and	3a +
   the square root of	√
   b	b
   is equal to	=
   the square of the sum of a and b	(a + b)²

   Putting it altogether, we get 3a + √b = (a + b)². Select (B).
4. • Draw a diagram: Draw a diagram similar to the following:
     
   • Looking at the diagram, Kerry cannot walk north, but can walk south or west and still be in range. Select (E) II and III.
5. Multiply the left-hand side of the equation by 2:
   2x/8 = 2x/a
   a = 8
   Select (A) 8.
6. Because s and t lie on a straight line, s + t = 180. Now, u = r = 50, because lines l and m are parallel. So, s + t + u = 230. Select (A) 230.
7. • Draw a diagram: You may find it helpful to sketch a graph of line l based on the information given.
   • Look at the answer choices: If l is perpendicular to the y-axis, it must be of the form y = some constant (this will be quite clear if you draw a diagram!). The only such answer is (C) y = −3. Select that answer.
8. Simplify the equation and then substitute 300 for x and 1,900 for p(x):
   p(x) = 17x − 10x −b
   p(x) = 7x − b
   1900 = 7300 −b
   -200 = −b
   b = 200
   Select (E) 200.
9. • Multiply the first few positive integers (1, 2, 3, etc.) by themselves, resulting in 1, 4, 9, 16, 25, ...
   • Look at the answer choices: The list above contains numbers ending in 1, 4, 5, and 6, so choices (A) through (D) can be eliminated. Select (E) 8.
10. • If the probability of randomly selecting a red marble from the bag is ¼, then the number of marbles in the bag must be a multiple of 4. Eliminate answers (A), (C), and (E).
    • If the probability of randomly selecting a blue marble is 1/6, the number of marbles in the bag must be a multiple of 6. Eliminate answer (D). The only remaining answer is (B) 12; select that answer.
11. • Try a special case: Say that there are four prices, 1, 2, 3, and 4. The sum of those prices is 10. The average price is 2.5 When 10 is divided by 2.5, the answer is 4.
    • Look at the answer choices: 4 is not the sum of the prices (A), half the sum of the prices (B), the average of the prices (C), or half the number of prices (E). Select (D) The number of prices. Note: If you chose a different set of prices, and were not able to eliminate all answers, just try another special case until you could.
12. • Estimate the answer: Since the figure is drawn to scale, we can eyeball the figure. The dotted line appears to be about ¼ of the perimeter of each triangle, so the perimeter of the figure outlined by the solid line is probably around ¾ of the perimeter of the four triangles, or ¾(4)(30) = 90.
    • Look at the answer choices: (C) 80 and (D) 84 are close to our estimate. Eliminate the other three answers.
    • If the area of the square in the figure above is 81, the length of each side of the square is 9. So, the perimeter of the two solid sides of each triangle is 30 − 9 = 21, and the perimeter of the solid figure is 4 × 21 = 84. Select (D) 84.
13. • First, find g(2). Looking at the graph, g(2) = 5, so k = 5.
    • Next, find g(k) = g(5). From the graph, g(5) = 2.5. Select (B) 2.5.
14. Try the extreme values of x and y and see what xy is equal to:

    x	y	xy
    0	−1	0
    0	3	0
    8	−1	−8
    8	3	24

    So, −8 ≤ xy ≤ 24. Select (E) −8 ≤ xy ≤ 24.
15. • Solution 1: n is the exterior angle of both triangles. You may know that the exterior angle of a triangle equals the sum of the opposite interior angles, so each pair of interior angles sums to n. Therefore the sum of all four angles is 2n. Select (B) 2n.
    • The two angles adjacent to the angle marked n° are equal to 180 − n°, because they lie on a straight angle. Because the angles of a triangle sum to 180°, and one of the angles in each triangle is 180 − n°, the other two angles in each triangle must sum to n. The total measure of all four angles is 2n. Select (B) 2n.
16. • Solution 1:
    • Try a special case: Say that t = 3. Then, the second term is 3 + (3 × ⅓) = 4. So the ratio of the second term to the first is 4/3.
    • Estimate the answer: If t = 3, then:
      • (A) evaluates to 4
      • (B) evaluates to 2
      • (C) evaluates to 12/9 = 4/3
      • (D) evaluates to 6/9 = ⅔
      • (E) evaluates to 1
      The only matching answer choice is (C) (t + 9)/3t. Select that answer.
    • Solution 2: If t is the first term of the sequence, then the second term is 3 + ⅓t. The ratio of the second term to the first term is (3 + ⅓t)/t. Multiplying both numerator and denominator by 3 to clear out the fractions, we get (9 + t)/3t. Select (C) (t + 9)/3t.