Solutions for Practice Test 1, The Official SAT Study Guide, Section 8

Here are solutions for section 8 of the first practice test in The Official SAT Study Guide, second edition, found on pages 413–418. 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. • Estimate the answer: Looking at the graph, it should be fairly obvious that the answer is at least 10, but well under 25.
   • Look at the answer choices: Based on our estimate, we can eliminate (A), (B), and (E), leaving (C) 11 and (D) 13.
   • There were 5 honorable mentions for photography, 4 for painting, 2 for pottery, 1 for metalwork and 1 for silkscreen. 5 + 4 + 2 + 1 + 1 = 13. Therefore, the answer is (D) 13.
2. • Draw a diagram: A diagram is already drawn, but you can fill in what you know on it. Since AD = 6, and AD is the radius of the circle, BA, CA, and AE are also 6. Since the two triangles are congruent, DE = 4. Fill in these numbers on the diagram.
   • Look at the answer choices: Comparing the answer choices with the numbers we filled in on the diagram, answers (A) through (D) are all incorrect, and (E) ED = 4 does match the diagram. Select (E) ED = 4.
3. • Read the question and determine what it is asking: These types of questions can look intimidating, but really it's just asking you to plug a = 5, b = 2, c = 6 into a^b − ac + c.
   • Perform the calculation: 5² − 5(6) + 6 = 25 − 30 + 6 = 1. Select (A) 1.
4. • Draw a diagram: The diagram could look something like the following:
     
   • Estimate the answer: The coordinate axes divide the big square into four little squares. Each little square has area 2 × 2 = 4, so the answer should be 4 × 4 = 16.
   • Look at the answer choices: The only choice near our estimate is (C) 16. Select that answer.
5. Read the question and understand what it is asking: Read the question through a first time so you know what to look for. Then read the question through a second time, looking for the relevant information in each sentence:
   • Sentence 1: One of Owen, Chadd, Steph, and Daria is the oldest.
   • Sentence 2: Chadd is not the oldest.
   • Sentence 3: Daria might be the oldest.
   • Sentence 4: Steph is not the oldest.
   • Sentence 5: Owen is not the oldest.
   Since Chadd, Steph, and Owen are not the oldest, Daria must be the oldest. Select (B) Daria.
6. Solution 1:
   • Estimate the answer: We can assume the diagram to be drawn to scale. y appears to be somewhat more than 90, and x appears to be somewhat less than 90. So, x + y is about 180, and 2(x + y) is around 360.
   • Look at the answer choices: (E) 360 is the only answer in range of our estimate ((D) 270 is too small, as it means that x must be less than 45, which isn't what the diagram shows). Select (E) 360.
   Solution 2:
   • Draw a diagram: A diagram is given, but extend all of the lines in the diagram.
   • Since QR || PS, we know, from what we know about a line intersecting a parallel line that the angle to the left of the one marked y° must equal x°. So, x and y are supplemental. So, x and y = 180 and 2(x + y) = 360. Select (E) 360.
7. • Convert the sentences into equations. The first sentence can be written as:
     ⅓(x + y + z) = 12
     Since z is the largest of the three numbers, the second sentence can be written as:
     (x + y) − z = 4
   • Look at the answer choices: If we manipulate the above equations slightly, they are the same equations as given in (A). Select (A) x + y + z = 36, x + y − z = 4.
8. • Estimate the answer: Since 81 = 3⁴, the answer has to be less than 4, maybe around 2.
   • Look at the answer choices: We can eliminate (C), (D), and (E), leaving (A) 3/2 and (B) 2.
   • Based on the laws of exponents,
     3^2x·3^2y = 3^{2x + 2y}
     This equals 81, which is 3⁴. So,
     3^{2x + 2y} = 3⁴
     and so
     2x + 2y = 4
     x + y = 2
     Therefore, the answer is (B) 2.
9. • Estimate the answer: f is tallest somewhere around halfway between 0 and 8, say around 4.
   • Look at the answer choices: We can eliminate (A) 2, (D) 6, and (E) 8.
   • To be on the safe side, it's a good idea to count the dashes on the graph to see whether the answer is (B) 4 or (C) 5. Counting, the function is tallest at x = 4. Select (B) 4.
10. Solution 1: Multiply both sides of the equation by 9:
    9k = 3x
    Therefore, the answer is (B) 9k.
    Solution 2:
    • Estimate the answer: To make our estimate a bit easier, assume that k and x are positive. Now, if k = x/3, then x must always be greater than k (as an example, if k = 1, then x = 3). Therefore, 3x must be greater than 3k.
    • Look at the answers: The only answer that is always > 3k is (B) 9k (if necessary, you may want to play around with a few values for (C) to convince yourself that (C) can't be the answer). Therefore, the answer is (B) 9k.
    Solution 3:
    • Try a special case: Say that x = 3. Then k = 1. Now, 3x = 9, which is equal to 9 × 1 = 9k. Therefore, the answer is (B) 9k.
11. • Draw a diagram: If you find it difficult to visualize a cube, it may be helpful to draw a diagram.
    • A cube has six faces. If two are black, then 6 − 2 = 4 are white. If the total area of the four white faces is 64 square inches, the area of each white face is ¼(64) = 16 square inches. Since each face is a square, the side length must be √16 = 4 inches. Now, the area of the cube is 4 × 4 × 4 = 64. Select (A) 64.
12. Solution 1:
    • Draw a diagram: The diagram is already given. You may find it helpful to multiply everything on the number line by 4. So, 1 becomes 4, −1 becomes −4, and now x becomes 1, y becomes 3, w becomes −2, and v becomes −3.
    • Look at the answer choices:
      • (A) is equal to 0.
      • (B) is equal to −2.
      • (C) is equal to −1.
      • (D) is equal to −1.
      • (E) is equal to 2.
      (B) has the least value. Select (B) v + x.
    Solution 2:
    • Look at the answer choices:
      • (A) is equal to 0.
      • (B) is somewhat negative.
      • (C) is somewhat negative, but not as small as (B).
      • (D) is somewhat negative, but not as small as (B).
      • (E) is positive.
      (B) has the least value. Select (B) v + x.
13. • Read the question and understand what it is asking: You are asked to find the median of a set of 7 integers. The median is the middle value. Since there are an odd number of integers, there is only one middle value. Therefore, the median must be an integer. This insight allows us to eliminate II. right off the bat.
    • Try a special case: Try a small number, like 1, and see what the median is:
      The median of 1,3,4,6,7,10,12 is 6.
      Try a large number, like 20, and see what the median is:
      The median of 3,4,6,7,10,12,20 is 7. So, both I. and III. are possible. Therefore, the answer is (D) I and III only.
14. Solution 1: We can select any one of 5 colors for color 1. Having done that, we can select any one of 4 colors for color 2. The answer is 4 × 5 = 20. Select (B) 20.
    Solution 2: While this solution takes more time, you could list out all of the possibilities. Say that the colors are Red (R), Orange (O), Yellow (Y), Green (G), and Blue (B). The possibilities are:
    RO (Red=color 1, Orange=color 2), RY, RG, RB, OR, OY, OG, OB, YR, YO, YG, YB, GR, GO, GY, GB, BR, BO, BY, BG.
    There are 20 possibilities. Select (B) 20.
15. • Estimate the answer: If one side of a rectangle gets 30% longer, the area increases by 30%. If one side gets 30% shorter, the area decreases by 30%. The change has to be between +30% and −30%. Probably the area isn't going to change too much.
    • Look at the answer choices: "(C) It is unchanged." Looks promising. There are also two answers with small decreases. We can definitely eliminate (A) and (B).
    • Take advantage of question order: This is the second-last question in the section, so it's a hard question. Therefore, the obvious answer is probably wrong. Eliminate (C) It is unchanged.
    • Try a special case:
      • Say that the length of the rectangle is 10 and the width is 10. Then the area is 100.
      • If the length of the rectangle is increased by 30%, it becomes 13.
      • If the width is decreased by 30%, it becomes 7.
      • The new area of the rectangle is 13 × 7 = 91. This represents a (100 − 91)⁄100 × 100% = 9% decrease.
      Therefore, the answer is (E) It is decreased by 9%.
16. • Try a special case: To make calculations easier, say that k = 0.
    • Determine how many bees in the hive were on day number 10:
      n(10) = 10²⁄2 − 20(10) + 0
      n(10) = 50 − 200
      n(10) = −150
      Therefore, there are −150 bees in the hive on day 10 (this seems nonsensical, but this is because we decided to let k = 0 to make the calculations easier, so we'll accept it. If you wanted to go back and let k = 150 or 200 or something, you could do that too).
    • Guess and check: Try each answer out:
      • On day 20, there are ½(20)² −20(20) = −200 bees in the hive. Nope.
      • On day 30, there are ½(30)² −20(30) = −150 bees in the hive. This is the answer we're looking for.
      Therefore, the answer is (B) 30.