[Math Lair] Solutions for Practice Test 1, The Official SAT Study Guide, Section 7

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Here are solutions for section 7 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. Solution 1: Solution 2:
  2. Solution 1: Solution 2: I don't recommend this solution unless you have a programmable calculator and you've programmed Heron's formula on it (too time-consuming otherwise), but if so, then you can use guess and check. For each answer choice, use Heron's formula to evaluate the area of the resulting triangle. You will find that, for (E), the area is 0, so that indicates something is amiss. Select (E) 10.
  3. Solution 1: Solution 2:
  4. The easiest way to solve this equation is by inspection. √2 · 9 = √18, so p = 9. Enter 9.
  5. When 1.783 is rounded to the nearest whole number, the result is 2.
    When 1.783 is rounded to the nearest tenth, the result is 1.8.
    The required answer is 2 − 1.8 = .2.
  6. If the probability is 2⁄5 that she will pick a brown towel, then 2⁄5 of the towels in the closet are brown. We know there are 6 brown towels. To get the total number of towels in the closet, divide the number of brown towels by the fraction of brown towels in the cupboard:
    6 ÷ (2⁄5) = 15
    Enter 15 as the answer.
  7. One way of looking at the problem is as follows: For every five days, the number of extra days on which it rained increases by 1. After 6 × 5 = 30 days, the number of extra days on which it rained will be 6 greater. Enter 6 as the answer.
  8. The difference between the third term and the sixth term is 77 − 17 = 60. Since:
    sixth term = third term + 3 common differences
    or (sixth term − third term) = 3 common differences
    the common difference is 60 ÷ 3 = 20. So, the eighth term is 77 + 20 + 20 = 117.
  9. |x − 3| = ½
    x − 3 = ½ or x − 3 = −½
    x = 3.5 or x = 2.5
    We are asked for the least value of x, which is 2.5.
  10. Guess and check: Z looks like it has to be pretty small compared with the other numbers. Say that Z = 0. Then W = 5, Y = 4, and X = 9. This makes 5940, which fits. Enter 5940.
  11. Draw a diagram: A diagram is already given, but it isn't drawn to scale. Draw a diagram that is roughly to scale:
    First diagram for question 17
    If you add a few more lines to the diagram, you can see that it can be divided into several equilateral triangles of side length 10:
    Second diagram for question 17
    The border of the shape is equal to 9 sides of the equilateral triangles. Since each side is of length 10, the length of the border is 9 × 10 = 90.
  12. First, find where g(x) = 0:
    k(x + 3)(x − 3) = 0
    x = −3 or x = 3
    If g(a − 1.2) = 0, then a − 1.2 = −3 or 3. If a > 0, only the positive possibility will work out:
    a − 1.2 = 3
    a = 4.2

    Enter 4.2 as the answer.