Solutions for Practice Test 1, The Official SAT Study Guide, Section 3

Here are solutions for section 3 of the first practice test in The Official SAT Study Guide, second edition, found on pages 396–401. 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:
   • Look at the choices: Choices (A) through (D) are various products of (x ± 1) and (x ± 2). It shouldn't be hard to see that the largest of these four is (A) (x + 1)(x + 2). Now, (E) (x − 4)(x + 4) is equal to 0, because (x − 4) equals 0. So, the answer is (A) (x + 1)(x + 2).
   Solution 2:
   • Guess and check: Evaluate each choice for x = 4. Choice (A) is equal to 30, choice (B) to 15, (C) to 12, (D) to 10, and (E) to 0. Therefore, the answer is (A) (x + 1)(x + 2).
2. Read the question and understand what it is asking: Train C's speed is 3 × 2 = 6 times that of Train B. If Train B was going 7 miles per hour, Train C must have been going 6 × 7 = 42 miles per hour. Therefore, the answer is (E) 42.
3. • Estimate the answer: Looking at the three expressions given, it probably makes sense that the mean would be somewhat less than 5x. However, because the mean of the two smallest terms is 3x, the mean must be somewhat larger than that. So, we'll estimate that the mean is 4x. Since the mean is 8, then our estimate of the value of x is 2.
   • Look at the choices: (B) 2 looks reasonable. If you can see that the answer cannot be (A) 1 or (C) 3, then select (B) 2. Otherwise, see the next step.
   • Guess and check: Assume that x = 2. Then, the three numbers are 2, 10, and 12. The mean of these numbers is ⅓(2 + 10 + 12) = 8, which is correct. Therefore, the answer is (B) 2.
4. Look at the choices: Look at the answer choices, eliminate any where two points on the graph have the same x-coordinate, and select what's left. If two points on the graph have the same x-coordinate, then you can find a vertical line that passes through two or more points. There are several such vertical lines that can be drawn through (B), (C), and (E), so eliminate those answers. For (A), a vertical line drawn through the corners of the "box" will intersect many points, so (A) can also be eliminated. Therefore, the answer is (B).
5. • Read the question (and the diagram) and understand what it is asking: 9 students out of 30 studied butterflies only. The question is asking you to find 9 ÷ 30 and express it as a percentage.
   • Estimate the answer: 10 ÷ 30 is 33.3%, so 9 ÷ 30 must be slightly less than 33.3%.
   • Look at the choices: The only choice that corresponds to our estimate is (C) 30%. Therefore, select (C) 30%.
6. • Estimate the answer: We can assume that the diagram is drawn to scale, as nothing states otherwise. Now, the portion of AB below the x-axis appears to be twice the length of AB above the x-axis, or slightly more. So, we could estimate the value of t to be −6 or −7 or −8.
   • Look at the choices: The only answer anywhere near our estimate is (C) −7. Select (C) −7.
7. Since 4y = 12, y = 3. Now, let's look at 3x² = 12. Since y = 3, we can substitute y for 3 in 3x² = 12, giving yx² = 12, or x²y = 12. Therefore, the answer is (D) 12.
8. • Estimate the answer: We can assume that the diagram is drawn to scale. The radius of the largest circle appears to be somewhat more than twice the radius of C, but less than three times the radius of C. Since the radius of C is 4, then the radius of the largest circle has to be somewhere between 8 and 12. We could make an estimate of 10.
   • Look at the choices: The only answer that is between 8 and 12 is (D) 10. Select (D) 10.
9. • Estimate the answer: x appears to be somewhat closer to 2 than 42, so the answer should be somewhat, but not significantly, less than 22.
   • Look at the choices: (C) 16 and (D) 18 seem possible. We can eliminate (A), (B), and (E).
   • There are 5 tick marks separating 2 and 42, and the distance between 2 and 42 is 40. Therefore, each tick mark represents 8 units. So, the answer is 2 + 8 + 8 = 18. Select (D) 18.
10. • Estimate the answer: Eyeballing the diagram, the value of x appears to be about 1.5 right angles, or 135°. Alternately, you might see x as being about (or slightly less than) twice the 70° angle, so you might arrive at an estimate of 140° or 135° or 130°.
    • Look at the answers: (C) and (D) appear to be the most reasonable answers. The others appear likely to be out of range.
    • Calculate the answer: Since "The number of degrees of arc in a circle is 360" (see Reference Information at the top of the section):
      360 = 90 + 30 + 110 + x
      x = 130
      Therefore, the answer is (C).
11. Try a special case: Say that k = 6. When 6 is divided by 7, the remainder is 6. When 6 + 2 is divided by 7, the remainder is 1. Therefore, the answer is (B) 1.
12. • Read the question and determine what it is asking: You are asked to find a graph where the value of y is positive at x = 0, and that increases as x increases.
    • Look at the choices: (A) and (B) decrease as x increases. Of the remaining three choices, only (D) is positive where x = 0. Therefore, the answer is (D).
13. • Estimate the answer: It may be difficult to estimate the absolute value of the answer, but you may be able to see that each odd-numbered term in the sequence will be positive and each even-numbered term in the sequence negative. So, the sixth term will be less than zero.
    • Look at the choices: (A), (B), and (C) are positive and can be eliminated. There are still two possibilities, (D) and (E).
    • Calculate the answer: 1(−2)(−2)(−2)(−2)(−2) = −32. Therefore, the answer is (E) −32.
14. Solution 1:
    • Estimate the answer: We are given (2x − 5)(2x + 5) = 5. Since (2x − 5) and (2x + 5) multiply to a positive number, either they are both positive or they are both negative. So, either 2x > 5 (if both positive) or 2x < − 5 (if both negative). Squaring either inequality, we get 4x² > 25. So, the answer will probably be somewhat larger than 25.
    • Look at the choices: (A) and (B) are obviously wrong, because both 4 and x² are positive. (C) and (D) are both less than 25. Therefore, the answer is (E) 30.
    Solution 2: Expand the equation and solve for 4x²
    (2x − 5)(2x + 5) = 5
    4x² − 25 = 5
    4x² = 30
    Therefore, the answer is (E) 30.
15. • Estimate the answer: We can assume the diagram is drawn to scale. The slope of the given line appears negative, but somewhat greater than −1. Therefore, the slope is between 0 and −1.
    • Look at the choices: The only answer in range of our estimate is (B) −½. Select that answer.
16. If 3a + 4b = b, then we can subtract b from both sides, getting
    3a + 3b = 0
    multiplying both sides by 2, we get:
    6a + 6b = 0
    Therefore, the answer is (A) 0.
17. • Estimate the answer: The area of the shaded rectangular region appears to be ½ the area of the triangle. The area of the triangle is ½ × 10√2 × 10√2. You can calculate the result, or estimate it as being slightly larger then ½(14)(14). Either way, you should get the area of the triangle as being 100 or so. So, the shaded area should be about 50.
    • Look at the answer choices: The only choice in range of our estimate is (C) 50. Select that answer.
18. • Estimate the answer: f(x) appears to increase as x increases, so probably a > 1.
    • Look at the answer choices: We can eliminate (A) and (B), but there are still three other possibilities.
    • Try a special case: f(0) = ka⁰ = k. So, based on the information in the table, k = ½
    • Try another special case: f(1) = ka¹ = ½a. So, based on the information in the table, ½a = 2, so a = 4. Therefore, the answer is (D) 4.
19. • Estimate the answer: h appears to be less than e, since h appears to be the leg of a right triangle and e the hypotenuse. Therefore, the answer should be somewhat less than m.
    • Look at the choices: Both (A) and (B) are somewhat less than m. The other three choices can be eliminated.
    • Draw a diagram: The diagram is already given, but draw a line from the centre of the square to the top left and bottom right corners.
    • Work backwards: We could find the value of h in terms of m if we knew the length of the line we drew; by the Pythagorean theorem (see Reference Information), h² + (half the length of the line we drew)² = e² (= m²). Now, the length of the line we drew is the hypotenuse of a 45°-45°-90° triangle, so from the Reference Information its length is m√2. So:
      h² + (½m√2) = e² = m²
      h² + ½m² = m²
      h² = ½m²
      h = m⁄√2
      Therefore, the answer is (A) m⁄√2.
20. • Try a special case: Assume that k = 1. Then the commission on the two cars is .01(2)(14,000) = 280 dollars.
    • Look at the choices: The only choice that is equal to 280 when k is equal to 1 is (A) 280k. Therefore the answer is (A) 280k.