[Math Lair] Solutions for Practice Test 10, The Official SAT Study Guide, Section 8

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Here are solutions for section 8 of practice test #10 in The Official SAT Study Guide, second edition, found on pages 976–981. The solutions below 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. Convert the sentence into an equation:
    The sum of 3x and 53x + 5
    is equal to=
    The product of x and ⅓x · ⅓
    Putting it all together, we get 3x + 5 = x · ⅓ or 3x + 5 = ⅓x. Select (E) 3x + 5 = ⅓x.
  2. If 15 of 90 trash cans are blue, then the probability that one selected at random is blue is 15/90, or 1/6. Select (C) 1/6.
  3. By inspection, x could be 1 and y could be 2. Or, x could be 2 and y could be 4. Or, x could be 3 and y could be 6. We could continue this indefinitely, so we could definitely find more than four such integer pairs that satisfy the equation. Select (E) More than four.
  4. Factor 10n out of both terms, resulting in:
      10n(6 + 1)
    =7/10n
    Select (B) 7/10n.
  5. Looking at the graph, f(x) is negative between x = 0 and x = 6. Select (B) 0 < x < 6.
  6. From the Reference Information, the sum of the measures, in degrees, of the angles of a triangle is 180. So, if the degree measures of the angles of a triangle are in the ratio 2:3:4, the smallest angle must be (2/9)(180) = 40°, and the largest must be (4/9)(180) = 80°. The difference is 80 − 40 = 40°. Select (C) 40°.
  7. Simplify the expression:
    (n/(n − 1))(1/n)(n/(n + 1)) = 5/k
    n/((n − 1)(n + 1)) = 5/k
    Now, the next step might be a little tricky if you haven't had a lot of practice evaluating integer expressions. We are told that n and k are both integers. Now, n does not have any factors in common with either n − 1 or n + 1, so the fraction is in lowest terms. Because 5 is prime, the fraction on the right is also in lowest terms. So, the numerators and denominators are equal to each other, giving:
    n = 5 (Equation 1)
    (n − 1)(n + 1) = k (Equation 2)
    Substituting equation 1 into equation 2:
    (5 − 1)(5 + 1) = k
    k = 24
    Select (C) 24.