[Math Lair] Solutions for Practice Test 4, The Official SAT Study Guide, Section 9

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Here are solutions for section 9 of the fourth practice test in The Official SAT Study Guide, second edition, found on pages 609–613. The following solutions 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. If there were originally the same number of girls and boys, and there are twice as many girls after 4 boys get off, then half of the boys must have got off and half must remain. So, there were 4 + 4 = 8 boys originally, so there are 8 girls. Select (C) 8.
  2. Solution 1:
    Solution 2:
  3. Solution 1: Solution 2:
  4. As the Reference Information states, the number of degrees of arc in a circle is 360. So, the three angles must sum to 360°:
    2x + 3x + 4x = 360
    9x = 360
    x = 40
    Select (C) 40.
  5. This looks really scary, but if you remember laws of exponents, it isn't too difficult.
  6. Guess and check: Try each answer and see which meets the given criteria:
  7. Solution 1: Solution 2:
  8. Try a special case: Assume that the telephone cost $100. Then, a 10% increase would raise the price to $110. A further 25% decrease would be $110 × 75% = $82.50. Since the original price was $100, this is 82.5% of the original price. Select (C) 82.5%.
  9. Solution 1: The easiest way to solve the problem, but one that may not be easy to discover, is to consider the quantity 1/x. Since x is an integer greater than 1, 1/x is positive and a small quantity less than 1. So, y = x plus a small quantity. So, y > x. Therefore, yx, so I. is always true. Now, if we multiply both sides of the inequality by x, we get xy > x², so III. must also be true. Now, what about II? Well, we know that y = x (an integer) plus a small number that is not an integer. Adding an integer to a non-integer gives you a non-integer, so II. cannot be true. So, the correct answer is (D) I and III only.
    Solution 2: Try a special case: Say that x = 2. Then, y = 2 + ½ = 2.5. Now, look at I., II., and III.: Try a few more special cases until you are convinced that I. and III. are always true. The answer is (D) I and III only.