Solutions for 2013 SAT Practice Test, Section 9

Here are solutions for the section 9 of the 2013-14 SAT practice test; you can find the test at the College Board's web site or in the Getting Ready for the SAT booklet. The following solutions illustrate faster, less formal methods that may work better than formal methods 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: Guess and check: Try each answer and see which one solves the equation. The one that does is (A) −27.
   Solution 2: Solve the equation:
   (1/3)y + 9 = 0
   (1/3)y = −9
   y = −27
   Therefore, the answer is (A) −27.
   It's worthwhile noting here that you shouldn't rush through easy questions such as this one, because they're worth as much as hard questions. It's easy to make a careless error on this one and miss easy marks.
2. Solution 1:
   • Estimate the answer: The radius of the smaller circle appears to be ½ of the larger circle. Since the radius of the larger circle is 4, the radius of the smaller appears to be 2.
   • Look at the answer choices: The only choice that makes sense is (B) 2. Select that answer.
   Solution 2: PR is the radius of the larger circle and the diameter of the smaller circle. Since the radius of a circle is twice its diameter, the radius of the smaller circle is ½ that of the larger, and ½(4) = 2. Therefore, the answer is (B) 2.
3. • Estimate the answer: If Roy had 50 acres of corn, he would have 250 acres in total. Therefore, the answer must be a bit less than 250.
   • Look at the answer choices: The only answer anywhere near our estimate is (D) 225. Select that answer.
4. • Estimate the answer: The numbers appear to be almost but not quite doubling each time, so perhaps the answer has to do with doubling and then subtracting a little bit.
   • Look at the answer choices: Answers (C) and (D) appears to match our estimate well.
   • Guess and check: Try each of answers (C) and (D) to see whether that rule works to generate the sequence. The correct answer is (C) Double the preceding number and then subtract 2 from that result.
5. Try a special case: We aren't told what the distance between the curve and either R or S is, so presumably it doesn't matter. So, since RS is 12, we'll assume that the radius of each circle is 6. Now, the perimeter of two semicircles of radius 6 is the same as the perimeter of one circle of radius 6, which is 2π(6) = 12π. Therefore, the answer is (C) 12π. (6) Try a special case: Assume that, say, k = 3. Then h = 4. So, (7 − k)⁄h becomes (7 − 3)⁄4 = 4⁄4 = 1. Look at the choices: The only answer choice that matches our calculated value is (A) 1. Select that answer.
6. Try a special case: Say that h = 4 and k = 3. Then the expression becomes (7 − 3)/4 = 1. The only answer that corresponds is (A) 1. Select that answer.
7. Read the question and understand what it is asking: The question is asking for (area of country B) − (area of country A).
   • If you don't understand how to find the area of the countries in square miles, you can consider another, easier, problem. For example, how many square miles would a country with a population of 6 and 2 people per square mile? The answer is 6 ÷ 2 = 3 (if you don't see that right away, you could draw a figure). So, you would need to divide the total population by the population density.
   • Estimate the answer: The population of country B is slightly less than 8,000,000 ÷ 400 = 20,000. The population of country A is slightly more than 6,000,000 ÷ 600 = 10,000. So, the answer is a bit less than 10,000.
   • Look at the choices: (E) sticks out like a sore thumb, so it's highly unlikely to be the correct answer. Eliminate it. (A) and (B) seem too far below our estimate and can also be eliminated. (D) looks like the most likely choice; if you're confident in your estimate, select (D) 8,000. If you aren't sure whether your estimate is accurate enough to eliminate (C), see the next step:
   • Calculate the answer: Find the value of (7,600,000 ÷ 400) − (6,500,000 ÷ 600) = 8,166.6. Therefore, the answer is (D) 8,000.
8. This is the type of question that you could waste a lot of time on, but with the correct insight, the question is quite easy. The easiest way to solve the question is to note that the equation is x² = x + 6. In words, this equation states that x² is six greater than x. Therefore, x² must be greater than x. So, the answer is (E) x² > x.

   However, if you miss the above insight, you could solve for x, find that x = −2 or x = 3, eliminate choices (A), (B), and (C), and then substitute −2 and 3 in (D) and (E) to see which is true.

9. Substitute the values of f(10) and f(5) in the given equation. This produces an equation in a. We can then solve for a:
   f(10) + f(5) = 55
   5(10) − 2a + 5(5) − 2a = 55
   −4a = −20 a = 5
   Therefore, the answer is (C) 5.
10. • Draw a diagram: Draw a number line and try to plot numbers halfway between even integers and odd integers on it. You'll find that "even-odd" numbers are numbers such as 1.5, 2.5, 3.5, etc.; in other words, numbers whose fractional part is .5 or ½.
    • For any number whose fractional part is .5, twice that number is an integer. So, I. is correct and II. is incorrect. Now, looking at the diagram, it doesn't appear that an "even-odd" number can be halfway between two even integers, so III. must be incorrect. Therefore, the correct answer is (B) II only.
11. • Try a special case: Let m = 1.
    • Look at the choices: Substitute m = 1 into the answer choices. Answers (A), (B), (C), and (E) evaluate to 16. Answer (D) evaluates to 8. Therefore, the answer is (D) 4^m(2^m).
12. Solution 1: Estimate the answer: Since the graph is drawn to scale, you can look at the bars to see for which pair the ratio would be the largest. For 1994, the ratio appears to be much larger than for any other year. So, the answer is (E) 1994.
    Solution 2: Read the numbers off of the graph and compare them to see which is greatest. We really only need to where the enrollment at School B is greater than that at school A:
    • For 1990 the enrollment ratio is 1,200:800, or 3:2
    • For 1992 the enrollment ratio is 2,000:1,200, or 5:3
    • For 1994 the enrollment ratio is 1,600:800, or 2:1
    The largest ratio is 2:1. So, the answer is (E) 1994.
13. • Draw a diagram: The diagram is already drawn, but we can fill in two more angles:
      • The angle opposite to the one labelled p° must equal p°, since vertical angles are equal.
      • The angle at the top right must equal k °, since l and m are parallel.
      • Since l and m are parallel, s must equal n.
      
      There are several straight angles here, and so several possibilities for sums of angles that must equal 180°.
    • Look at the choices: We want to find an answer choice that contains three angles that make up a straight angle. The correct answer is (B) k + p + s.
14. • Estimate the answer: You might estimate the answer to be around 3 × 6 = 18.
    • Look at the choices: (C) 20 is close, but (B) 12 isn't too far off. Time permitting, it might be a good idea to perform the calculation:
    • Calculate the result: There are 4 even integers x where 4 ≤ x ≤ 10 (namely 4, 6, 8, 10), and 5 integers y where 4 < y < 10 (namely 5, 6, 7, 8, 9). Therefore, the number of different ordered pairs is 4 × 5 = 20. Select (C) 20.
15. • Read the question and determine what it is asking: We are asked to find roughly what n(0) − n(2) equals, or roughly what 500 − 500(0.81)² equals.
    • Estimate the answer: If we keep multiplying the result by 0.81, the number will decrease. We could estimate the decrease for the first decade by 500 − 500(0.8) = 500 − 400 = 100. So, the decrease for the first two decades might be a bit less than 200.
    • Look at the answers: The only answer within the range of our estimate is (C) It decreased by about 180. Select that answer.
16. • Draw a diagram: Draw a diagram illustrating what the question is describing. A rough sketch is fine.
    • Estimate the answer: If you know the formula for the volume of a sphere (V = (4/3)πr³, it's not too difficult to make a rough estimate. (4/3)π is about 4, so the volume of the sphere is about 4r³. However, our diagram shows that the volume of the cube must be greater, since there's a lot of space in the corners that the sphere doesn't cover. So, the answer must be greater than 4r³
    • Look at the answers: The only answer greater than our estimate is (E) 8r³. Select that answer.