Solutions for 2013 SAT Practice Test, Section 6

Here are solutions for the section 6 of the 2013-14 SAT practice test; you can find the test at the College Board's web site or in the Getting Ready for the SAT booklet. 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. Substitute x + k = 12 into p(x + k) = 36. The result is
   p(12) = 36
   Dividing both sides of the equation by 12, we get p = 3. Therefore the answer is (A) 3.
2. Translate the question into algebra. To illustrate how to do this, I've spelled every detail out:
   • "13" becomes 13
   • "is added to" becomes +
   • "one-half" becomes ½
   • "of" becomes ×
   • "a certain number" becomes x
   • "the result is" becomes =
   • "37" becomes 37
   So, we get:
   13 + ½x = 37
   ½x = 24
   x = 48
   Therefore, the answer is (C) 48.
3. The usual route from B to C is 2 + 2 = 4 miles. The detour route is 1 + 2 + 3 + 2 + 2 + 2 = 12 miles. The difference is 12 − 4 = 8 miles. Therefore the answer is (B) 8.
4. Solution 1:
   • Try a special case: Substitute x = 1 into each of the answer choices and see which of the choices result in y = 7.5. Only (B) and (D) do. The other three answers can be eliminated.
   • Try another special case: Substitute x = 2 into each of the answer choices and see which of the choices results in y = 13. (D) gives a result of 15, so the correct answer is (B) y = 5.5x + 2.
   Solution 2: Draw a diagram: Use your graphing or scientific calculator to graph each of the answer choices or generate a table of values for each of the answer choices. You should find that the only equation that works is (B) y = 5.5x + 2.
5. • Read the question carefully and understand what it is asking: Angle AC is a straight angle, so it measures 180°. Therefore, x + 3y = 180. We need to find a value of x so that y is also an integer. If we rearrange the equation, y = (1/3)(180 − x), so 180 − x must be a multiple of 3.
   • Guess and check: Try each answer until we find one for which 180 − x is a multiple of 3. 180 − 30 = 150 = 50 × 3, so the answer is (A) 30.
6. • Draw a diagram: Draw a list of the numbers in order. We know that the smallest number is 2, the largest is 20, and the median (middle number) is 6, so we have:
     2 _ _ 6 _ _ 20
     We also know that the number 3 occurs most often. Now, 3 must occur at least twice (since 2, 6, and 20 occur once), but it can't occur more than twice because there are only two empty spots less than 6. So, we now have:
     2 3 3 6 _ _ 20
   • Try a special case: The list would have the smallest average if the two blanks were 7 and 8. In that case, the average would be (1⁄7)(2 + 3 + 3 + 6 + 7 + 8 + 20) = 7.
   • Try another special case: The list would have the largest average if the two blanks were 18 and 19. In that case, the average would be (1⁄7)(2 + 3 + 3 + 6 + 18 + 19 + 20) = 10.14... . So, I. 7, II. 8.5, and III. 10 are all possible averages. Therefore, the answer is (E) I, II, and III.
7. Draw a diagram: Draw a quick sketch of the coordinate axes. Can you find points that are a distance of 4 units from the origin? It shouldn't be difficult to find (4,0), (0,4), (−4, 0), and (0, −4). So, the answer must be either (D) Four or (E) More than four. Can you find another point? Yes, there are an infinite number of other points in a circle of radius 4. So, the answer is (E) More than four.
8. • Read the problem carefully and understand what it is asking: If the Liu family's stay did not overlap with the Benton family's stay, then the two families must have stayed on 14 different days. Since there are only 14 possible days, either the Liu family stayed for the first 6 nights and the Benton family for the last 8 nights, or the Benton family stayed for the first 8 nights and the Liu family for the last 6. Either way, exactly one of the two families are there every night. So, the problem boils down to finding a night when none of the Jackson, Callan, and Epstein families were at the hotel.
   • Draw a diagram: Draw a diagram with 14 empty spaces:

     _ _ _ _ _ _ _ _ _ _ _ _ _ _ 
     

     Now, see what possibilities there are for the Jackson family (who stayed the most nights):

     Night	1	2	3	4	5	6	7	8	9	10	11	12	13	14
     Possibility 1:	_	_	_	_	J	J	J	J	J	J	J	J	J	J
     Possibility 2:	_	_	_	J	J	J	J	J	J	J	J	J	J	_
     Possibility 3:	_	_	J	J	J	J	J	J	J	J	J	J	_	_
     Possibility 4:	_	J	J	J	J	J	J	J	J	J	J	_	_	_
     Possibility 5:	J	J	J	J	J	J	J	J	J	J	_	_	_	_

     In each case, the Jacksons must be there between night 5 and night 10.
   • Look at the choices: We can eliminate choices (B) through (E), since the Jacksons must be there on nights 5 through 10. Therefore, the answer is (A) The 3rd.
9. Solution 1: Draw a diagram; Draw a circle, divide it into thirds, divide each piece into four, and count the number of pieces. There are 12 pieces. So, the answer is 12.
   Solution 2: If a cake is cut into thirds and each piece into fourths, the cake has been cut into (1/3)(1/4) = (1/12) = twelfths. Therefore, there are 12 pieces and the answer is 12.
10. Solve for h when y = 3 and x = 4:
    3 = h⁄4
    h = 12
    So, when x = 6, y = h⁄x = 12⁄6 = 2. Therefore, the answer is 2.
11. Try a special case: Say that x = 58. Since x and y are supplementary, y = 180 − x = 180 − 58 = 122. Therefore, an answer is 122. Of course, you could have chosen another value for x within the given range.
12. Solution 1: If the price goes up $2 per year, it will take 45 years for the price to increase by $90. This will occur in the year 1990 + 45 = 2035. Therefore the answer is 2035
    Solution 2: Create a chart: You might create a chart something along the lines of the following:

    Year	Price
    1990	$10
    1991	$12
    1992	$14
    1993	$16
    1994	$18
    1995	$20
    ...	...
    2000	$30
    ...	...
    2010	$50
    ...	...
    2020	$70
    ...	...
    2030	$90
    ...	...
    2035	$100

    So the answer is 2035.
13. Solution 1: The point with the largest y-value appears to be at x = 5. This being a parabola, there is no reason to believe that the graph is not symmetrical. So, the answer is 5.
    Solution 2:
    • Try a special case: Since the graph intersects the x-axis at 3 and 7, assume that the equation of the graph is y = (x − 3)(x − 7) = 0, or y = x² − 10x + 21.
    • Draw a diagram: Using your graphing calculator, graph the above equation or, using your scientific calculator, generate a table of values for the above equation. By inspection, you should be able to determine that the largest value occurs at 5. Therefore, the answer is 5.
14. If the number has a remainder of 9 when divided by 10, its ones digit must be 9. You can then check all two-digit numbers ending in 9 for their remainder when divided by 9. You can save time if you remember that the digital root of a number—the sum of the number's digits, subtracting 9 until you get a 1-digit result—is equal to the remainder when that number is divided by 9. The possibility for the tens digit that works is 8. If the tens digit is 8, the digital root is 17 − 9 = 8, which is the remainder we want. So, the number is 89.
15. • Draw a diagram: A diagram is already given, but it might be easier to break the diagram up into two rectangles:
      
    • Solve part of the problem: We need to find the length of the vertical line forming the bottom rectangle. Now, the area of the upper rectangle is 1 ×2 = 2 = 8/4, so the area of the lower rectangle must be ¼. Therefore, the length of the bottom vertical line is ¼ ÷ 1 = ¼.
    • We now know the length of each site. The perimeter of the figure is 1 + 2 + 1 + ¼ + 1 + ¼ + 1 = 6½. We need to convert this to a decimal (6.5) or a mixed fraction (13/2). It's probably easiest to enter 6.5 on the answer sheet.
16. Solution 1: List every possibility out: There are 9 possibilities:
    4 × 10 = 40
    4 × 11 = 44
    4 × 12 = 48
    5 × 10 = 50
    5 × 11 = 55
    5 × 12 = 60
    6 × 10 = 60
    6 × 11 = 66
    6 × 12 = 72
    5 of the 9 possibilities are divisible by 5. So, the answer is 5⁄9.
    Solution 2: The probability that the result is divisible by 5 is the probability that at least one number selected is divisible by 5.
    Let P(J) represent the probability that j is divisible by 5.
    Let P(K) represent the probability that k is divisible by 5.
    Now, you may remember that P(A ∪ B) = P(A) + P(B) − P(A ∩ B). In this case:
    P(J ∪ K) = P(J) + P(K) − P(J ∩ K) = 1⁄3 + 1⁄3 − 1⁄9 = 5⁄9
    Therefore, the answer is 5⁄9.
17. • Read the problem carefully and determine what it is asking: The question is asking for the total compensation (not the sales or commission) if both Tom and Alison had the same sales and the same commission.
    • Set up equations for the problem.
      Let s represent the sales.
      Tom's compensation is 300 + 0.2s. Alison's compensation is 200 + 0.25s. They are equal, so:
      300 + 0.2s = 200 + 0.25s
      100 = 0.05s
      2000 = s
      Now, we have to answer the question. Tom's compensation is 300 + 0.2(2000) = 700. Therefore, the answer is 700.
    • Check the answer (if you have time): Alison's compensation is 200 + 0.25(2000) = 700. So, the answer makes sense.
18. • Convert the equation to slope-intercept form:
      12y = −tx − 3
      y = (−t⁄12)x − 3
    • Since we know that the slope is −10, we can solve for t:
      −10 = −t⁄12
      t = 120
      You should be careful with the negative signs here; however, the student-produced responses can't have a negative answer, so if you get a negative result you know you did something wrong. Anyway, the answer is 120.