Solutions for Practice Test 9, The Official SAT Study Guide, Section 2

Here are solutions for section 2 of practice test #9 in The Official SAT Study Guide, second edition, found on pages 886–891. 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. The numbers 32 and 33 are in both set X and set Y. So, there are 2 numbers. Select (A) Two.
2. If Linda travelled twice as far in half the time, then Linda travelled 20 miles in 1 hour. So, her average speed was 20 miles per hour. Select (C) 20.
3. If x = k(k − 2), then x + 1 = k(k − 2) + 1 = k² −2k + 1. Select (C) k² −2k + 1.
4. • Estimate the answer: As the graph of y = ax + b increases somewhat steeply to the right, y = 2ax + b will increase quite steeply to the right.
   • Look at the answer choices: See which graphs increase quite steeply to the right. The only such graph is (B). Select that answer.
5. From the Reference Information, the length of the hypotenuse of the right triangle is x√2. So, the perimeter of the triangle is 2x + x√2. We are told that the perimeter is equal to 4 2√2, so by inspection x = 2. Select (A) 2.
6. • Since Sam receives a score of 95 on the test, cross out the "2" in the table to the left of the 95 and replace it with a "3".
   • We are asked for the median score for the 17 students who ended up taking the test. This will be the 9th value. Looking through the table from the top down until we count at least 9 students, we find this occurs at 85. Select (C) 85.
7. • We have two sets of containers; one set of 16 has a total capacity of x gallons, and another set of 8 also has a total capacity of x gallons. If the two sets of containers have the same total capacity but there are more containers in the first set (16) than the second set (8), then the containers in the first set must be smaller and the ones in the second set must be larger.
   • Looking at the second set, if 8 containers have a total capacity of x gallons, then each container has a capacity of x/8 gallons. Select (D) x/8.
8. • Draw a diagram: You'll want to draw a simple rectangle to work with. Probably a good choice would be a rectangle with vertices at (0,1), (1,0), (2,1), and (1,2):
     
   • Try a special case: Looking at the simple case we chose, two of the sides have a slope of −1 and two have a slope of 1. 1 × 1 × (−1) × (−1) = 1. Select (D) 1.
9. If 20 minutes out of 60 was commercials, then 40/60, or 2/3, was not commercials. Enter 2/3.
10. Convert the sentence into an equation:

    If	[ignore]
    the product of ... and	×
    0.3	0.3
    a number	x
    is equal to	=
    1	1

    Putting this together, we get:
    0.3 × x = 1
    x = 1/0.3
    x = 10/3
    Enter 10/3.
11. Substitute the numbers x = 10, y = 3 and z = 5 into x^y − z^y, resulting in:
      10³ − 5³
    = 1000 − 125
    = 875
    Enter 875.
12. Guess and check: Try values for UT until you find one that makes the area of PQST greater than 10 but less than 30. Start with, say, UT = 2. Then, PT is 7 and ST = 2, so the area of PQST is 14, which works. Enter 2.
13. If there were half as many green balloons as red balloons, and ⅓ of the balloons are red, then 1/6 of the balloons are green. Since each balloon is either red, green, or blue, then 1 − ⅓ − 1/6 = ½ of the balloons are blue. Since 18 of the balloons are blue, then there are 18 ÷ ½ = 36 balloons in total. Enter 36.
14. • Draw a diagram: It can help to draw a diagram of the situation.
    • Each line will consist of one point from line l and one point from the other line. As there are 3 points on line l and 4 points on the other line, there are 3 × 4 = 12 possibilities. Enter 12.
15. 2^x + 2^x + 2^x + 2^x = 2⁷
    4(2^x) = 2⁷
    2²(2^x) = 2⁷
    2^x = 2⁵
    x = 5
    Enter 5.
16. If the average of these five numbers is 15, then the sum of these 5 numbers is 5 × 15 = 75. The greatest possible integer would occur when four of the numbers are 1 and one of the numbers is 71. Enter 71.
17. • Read the question and understand what it is asking: Because the 10 steps that Alice takes in opposite directions cancel each other out, the question states, in essence, that 7 of Alice's steps are equal to 10 of Corrine's steps. So, the length of 1 of Alice's steps is equal to 10/7 of Corrine's steps. Enter 10/7.
18. Substitute the function definition in f(2m) = 2f(m):
    f(2m) = 2f(m)
    (2m)² + 18 = 2(m² + 18)
    4m² + 18 = 2m² + 36
    2m² = 18
    m² = 9
    m = 3
    Enter 3.