Solutions for Practice Test 10, The Official SAT Study Guide, Section 8

Here are solutions for section 8 of practice test #10 in The Official SAT Study Guide, second edition, found on pages 976–981. 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. • Solution 1: If there are 8 dinners and 3 desserts, then there are 8 × 3 = 24 possibilities of dinners and deserts. Select (A) 24.
   • Solution 2: Call the dinners D1, D2, ..., D8 and the desserts S1, S2, and S3. List all of the combinations out: D1S1, D1S2, D1S3, D2S1, D2S2, D2S3, D3S1, D3S2, D3S3, D4S1, D4S2, D4S3, D5S1, ... At this point, we've listed out 13 possibilities, so the answer can't be (B) through (E). Select (A) 24.
2. Convert the sentence into an equation:

   The sum of 3x and 5	3x + 5
   is equal to	=
   The product of x and ⅓	x · ⅓

   Putting it all together, we get 3x + 5 = x · ⅓ or 3x + 5 = ⅓x. Select (E) 3x + 5 = ⅓x.
3. If 15 of 90 trash cans are blue, then the probability that one selected at random is blue is 15/90, or 1/6. Select (C) 1/6.
4. By inspection, x could be 1 and y could be 2. Or, x could be 2 and y could be 4. Or, x could be 3 and y could be 6. We could continue this indefinitely, so we could definitely find more than four such integer pairs that satisfy the equation. Select (E) More than four.
5. • Estimate the answer: Looking at the graph, it can be seen that the bookstore sold more books in July-August than in June-July, more books in October-November than in August-September, and more books in September-October than in August-September.
   • Look at the answer choices: We can eliminate (B) July and August, (E) October and November, and (D) September and October. This leaves (A) June and July and (C) August and September.
   • Compare the figures for June and July with those from August and September. Looking at the graph, the sales in August are just slightly more than those in June, but the sales in September are quite a bit less than those in July. So, fewer books must have been sold in August-September than in June-July. Select (C) August and September.
6. • Draw a diagram: Plot point D on the number line.
   • If AC = 24 and AB = BC, then AB = 12 and BC = 12.
   • If AB = 12 and AD = DB, then AD = 6 and DB = 6.
   • If DB = 6 and BC = 12, then DC = 6 + 12 = 18. Select (D) 18.
7. Factor 10⁻ⁿ out of both terms, resulting in:
     10⁻ⁿ(6 + 1)
   =7/10ⁿ
   Select (B) 7/10ⁿ.
8. • Draw a diagram: You may find it helpful to draw a picture of a circle divided into 4 and 5 pieces.
   • As the number of degrees of arc in a circle is 360 (consult the Reference Information if unsure), the number of degrees in ¼ of a circle is 360 ÷ 4 = 90. The number of degrees of arc in 1/5 of a circle is 360/5 = 72. The difference is 90 − 72 = 18. Select (B) 18.
9. Looking at the graph, f(x) is negative between x = 0 and x = 6. Select (B) 0 < x < 6.
10. • Estimate the answer: The bottom layer of marble would contain 4 × 4 × 1 = 16 cubic feet of marble. It looks like the pedestal contains somewhere around twice that amount of marble. So, perhaps the pedestal contains somewhere around 32 cubic feet of marble.
    • Look at the answer choices: Based on our estimate, answers (A) 14, (B) 16, and (E) 80 are unreasonable, so eliminate them. We still need to decide between (C) 30 and (D) 36.
    • The top layer of marble contains 1 cubic foot of marble. The next layer contains 2 × 2 × 1 = 4 cubic feet of marble. The third layer contains 3 × 3 × 1 = 9 cubic feet of marble. The bottom layer contains 16 cubic feet of marble. In total, the figure contains 1 + 4 + 9 + 16 = 30 cubic feet. Select (C) 30.
11. • Solution 1: We can solve the equation for x as follows:
      4(2^x) = 2^y
      2²(2^x) = 2^y
      2^{x + 2} = 2^y
      x + 2 = y
      x = y − 2
      Select (A) y − 2.
    • Solution 2:
      • Try a special case: Say that x = 1. Then 2^y = 4(2^x) = 4(2¹) = 8. Since 2³ = 8, y = 3. So, x is 2 less than y.
      • Look at the answer choices: The only answer that matches our results for this special case is (A) y − 2. Select that answer.
12. From the Reference Information, the sum of the measures, in degrees, of the angles of a triangle is 180. So, if the degree measures of the angles of a triangle are in the ratio 2:3:4, the smallest angle must be (2/9)(180) = 40°, and the largest must be (4/9)(180) = 80°. The difference is 80 − 40 = 40°. Select (C) 40°.
13. • Try a special case: Say that the phone call lasts for 3 minutes. The cost will be 0.5 + 0.3 + 0.3 = 1.1.
    • Look at the answer choices: Substitute n = 3 into each answer choice and see which evaluate to 1.1:
      • (A) evaluates to 2.4
      • (B) evaluates to 1.4
      • (C) evaluates to 1.7
      • (D) evaluates to 1.1
      • (E) evaluates to 2.1
      The only answer choice that evaluates to 1.1 when n = 3 is (D). Select (D) f(n) = 0.50 + 0.30(n − 1).
14. • Solution 1:
      • Draw a diagram: Because l and m are parallel lines, and because lines intersect them, there will be several angles with measure y° and z°. Mark those angles appropriately.
      • The triangle at the top of the figure is now marked with angles z°, x°, and y°. Because the angles of a triangle equal 180°, z + x + y = 180, or z = 180 − x − y. Select (E) 180 − x − y.
    • Solution 2:
      • Estimate the answer: In the diagram, x looks like it's about 90°, y looks like it's around 45°, and z looks like it's around 45°.
      • Look at the answer choices: Based on our estimate, (A) x + y would evaluate to around 135°, (C) 180 − x would evaluate to around 90°, and (D) 180 − x + y would evaluate to around 135°. These are nowhere near 45°, so eliminate those.
      • Try a special case: On the diagram, draw a second pair of lines. Label the angles z°, x°, and y° that correspond to the original angles. This time, however, make z° very small and x° quite large. As it turns out, y° is also quite small.
      • Look at the answer choices: For this new case, (B) x − y doesn't make any sense, since it's still big and z is small. (E) 180 − x − y does make sense, though. Select (E) 180 − x − y.
15. Simplify the expression:
    (n/(n − 1))(1/n)(n/(n + 1)) = 5/k
    n/((n − 1)(n + 1)) = 5/k
    Now, the next step might be a little tricky if you haven't had a lot of practice evaluating integer expressions. We are told that n and k are both integers. Now, n does not have any factors in common with either n − 1 or n + 1, so the fraction is in lowest terms. Because 5 is prime, the fraction on the right is also in lowest terms. So, the numerators and denominators are equal to each other, giving:
    n = 5 (Equation 1)
    (n − 1)(n + 1) = k (Equation 2)
    Substituting equation 1 into equation 2:
    (5 − 1)(5 + 1) = k
    k = 24
    Select (C) 24.
16. • Try a special case: Say that 5 coworkers contributed to a catered lunch that cost $3 in total, but 2 coworkers failed to contribute. Each co-worker would have originally contributed $3/5 = 60¢, but the three now have to contribute $1 each, resulting in an additional amount of 40¢, or $0.40.
    • Look at the answer choices: Look at each answer and see which evaluates to 0.4 when m = 5, y = 3, and p = 2:
      • (A) is obviously wrong because it doesn't include p, so no need to even check.
      • (B) evaluates to 3/(5 − 2) = 1.
      • (C) evaluates to (2)(5)/(5 − 2) = 10/3.
      • (D) evaluates to 3(5 − 2)/5 = 9/5.
      • (E) evaluates to 2(3)/5(5 − 2) = 2/5 = 0.4
      (E) is the only answer evaluating to the expected result. Select (E) (py)/(m(m − p)).