Solutions for Practice Test 4, The Official SAT Study Guide, Section 9

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. • A line with a negative slope goes down as it goes to the right. So, you can eliminate (A) and (B).
   • Looking at the remaining three answers, (C) has a negative y-intercept and (E) has a y-intercept of 0. So, these can be eliminated. Select (D).
3. Solution 1:
   • Using your calculator, divide $1.89, the cost of the box, by 6: $1.89 ÷ 6 = $0.315.
   • Look at the answer choices: The choice closest to $0.315 is (B) $0.30. Select that answer.
   
   Solution 2:
   • Estimate the answer: Six individual donuts cost $0.40 × 6 = $2.40. $1.89 is slightly lower than that, so the answer must be somewhat less than $0.40.
   • Look at the answer choices: Based on our estimate, only (B) and (C) are reasonable. Eliminate the rest.
   • Guess and check: Multiply each answer choice by 6 and see which is closest to $1.89. (C) $0.40 × 6 = $2.40, which is 51¢ too high. (B) $0.30 × 6 = $1.80. This is 9¢ too low. The closest answer is (B) $0.30.
4. Solution 1:
   • Estimate the answer: Presumably the least amount of money required to purchase exactly 21 donuts is less than the amount required to purchase 24. The amount required to purchase 24 is $3.59 × 2 = $7.18. So, the answer is probably somewhat less than that.
   • Look at the answer choices: Only answers (A) and (B) are less than $7.18. Eliminate the other three answers.
   • In order to purchase exactly 21 donuts, you need to purchase a box of 12 (at $3.59), a box of 6 (at $1.89), and three individual donuts (at $0.40 each):
     $3.59 + $1.89 + 3($0.40) = $6.68
     Select (B) $6.68.
   Solution 2:
   • Estimate the answer: In order to purchase exactly 21 donuts, you need to purchase a box of 12 (at $3.59), a box of 6 (at $1.89), and three individual donuts (at $0.40 each). We could estimate the answer by assuming that the box of 12 is $3.60, the box of 6 is about $2, and the three individual donuts are $1. $3.60 + $2 + $1 = $6.60. Therefore, the answer must be around $6.60.
   • Look at the answer choices: The only answer choice around $6.60 is (B) $6.68. Select that answer.
5. • Estimate the answer: Looking at the graph, h(4) ≅ 2 and h(6) ≅ 4. h(5) is about halfway between the two, so estimate 3.
   • Look at the answer choices: (C) 3 is one of the choices. Select that choice.
6. 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.
7. This looks really scary, but if you remember laws of exponents, it isn't too difficult.
   • Taking the first equation:
     x^−½ = ⅓
     Squaring both sides and then simplifying, we get:
     x⁻¹ = 1/9
     1/x = 1/9
     x = 9
   • Now, looking at the second equation, y^z = 16. Now, 4² = 2⁴ = 16¹ = 16. So, either y = 2 and z = 4, or y = 4 and z = 2, or y = 16 and z = 1. However, we are given that z > y, so it must be the case that z = 4 and y = 2. So, the value of x + z = 9 + 4 = 13. Select (D) 13.
8. • Draw a diagram: The diagram is already given, but draw a vertical line through the middle of the circle. This line divides the circle into two symmetrical halves. So, the x-coordinates must be an equal distance from 4.
   • Look at the answer choices: The only answer whose two x-coordinates are an equal distance from 4 are (C) 2 and 6. Select that answer.
9. Guess and check: Try each answer and see which meets the given criteria:
   • For (A) 2, 2(2) + 7 = 11. This leaves a remainder of 1 when divided by 5, so this won't work.
   • For (B) 3, 2(3) + 7 = 13. This does leave a remainder of 3 when divided by 5. Select (B) 3.
10. • Draw a diagram: Draw a diagram of the people in Stacy's classroom, ordered by height (you can represent people as dots to save time). Ensure that Stacy (labelled "S" below) is twelfth from the start and twelfth from the end:
      
    • Count the number of circles in the diagram. There are 23. Select (B) 23.
11. Solution 1:
    • Try a special case: Say that x = 0. Then, g(0) = c. We are told that c is negative, so the value of the function at x = 0 must be negative.
    • Look at the answer choices: Since the value of the function at x = 0 must be negative, we can eliminate choices (B) and (E), which show a positive value at x = 0, as well as (C) and (D), which show a zero value at x = 0. Therefore, the answer must be (A).
    Solution 2:
    • Draw a diagram: If you have a graphing calculator, pick values for a, b, and c such that a and c are negative, and view the resulting graph. You may want to try a few different values. You will notice that,
12. • Draw a diagram: The diagram is given, but draw the length we are asked to find, PR, on the diagram. Label the points where PR intersects AB and CD; we could call them X and Y, respectively. Your diagram should look something like the diagram on the right.
    • Estimate the answer: The figure is not drawn to scale, but it appears to match the information given in the question fairly well, so we could make an estimate based on the diagram. On the diagram, PR appears to be twice as long as BC, whose length is 4. So, PR appears to be somewhere around 8.
    • Look at the answer choices: (B) 8 is there, but there are other nearby choices (6 and 10), and the diagram isn't necessarily drawn to scale, so it's a good idea to investigate further. If you aren't able to make further progress, (B) 8 is a good guess, though.
    • Try a special case: Assume that the length XQ is equal to 1. Then, PX is equal to 1, QY is equal to 3, and YR is equal to 3. Adding these all up, PR is equal to 8. Select (B) 8.
13. 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%.
14. • Convert the sentence into an equation:

      When	[ignore]
      the number w	w
      is multiplied by	×
      4	4
      the result is the same as	=
      when	[ignore]
      4	4
      is added to	+
      w	w

      The resulting equation is:
      w × 4 = 4 + w
      or
      4w = 4 + w
    • We are asked to find the value of 3w, so solve the equation for 3w:
      3w = 4
      Select (E) 4.
15. • Read the question carefully and determine what it is asking: Read the question carefully, being careful to not overlook words such as consecutive and even. The question boils down to the following: You are given a right triangle, the sides of which are x, x + 2 and x + 4, and you are asked to find an equation relating the sides.
    • Draw a diagram: Draw a diagram of the triangle. It should look somewhat like the following:
      
    • Consult the Reference information for the formula for the Pythagorean Theorem. It is:
      a² + b² = c²
      Where a and b are the two legs of the triangle, and c the hypotenuse. If we replace a with x, b with x + 2, and c with x + 4, we get:
      x² + (x + 2)² = (x + 4)²
      Select (C) x² + (x + 2)² = (x + 4)².
16. 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, y ≠ x, 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.:
    • I. is true, because 2.5 ≠ 2.
    • II. is false, because 2.5 is not an integer.
    • III. is true, because 2(2.5) ≥ 2².
    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.