Solutions for Practice Test 2, The Official SAT Study Guide, Section 5

Here are solutions for section 5 of the second practice test in The Official SAT Study Guide, second edition, found on pages 463–468. 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. If 3x = 0, then (dividing both sides of the equation by 3), x = 0. Substituting x = 0 into 1 + x + x² results in 1 + 0 + 0 = 1. Select (B) 1.
2. • Draw a diagram: If you don't see the answer immediately, it can help to draw two circles, one three times as big as the other or so, and label the diameters and radii of both. You should probably see that the radius of A is about three times the radius of B, so the answer would be (E) 3:1. If not:
   • Let d represent the diameter of circle B, and 3r the diameter of circle A. The radius of circle A is (3/2)d, and the radius of circle B is ½d. The ratio between the two is 3:1, so the answer is (E) 3:1.
3. Try a special case: Say that N = {2, 3, 4} (you could pick any set whose average is 3, of course). Now, M = {4, 6, 8}. The average of the numbers in set M = ⅓(4 + 6 + 8) = 6. Select (D) 6.
4. Try a special case: Let P = 1, R = 2, T = 3. So, PRT is 123. Now,
   123 × 10⁻² = 1.23
   The answer that corresponds to 1.23 is (C) P.RT. Select that answer.

   Solution 2: When you multiply a number by 10⁻², you shift the decimal point two places to the left. Now, for a whole number such as PRT, the decimal point would occur to the right of the last digit, here T. Shifting it two decimal places to the left gives P.RT. Select (C) P.RT.

5. We can subtract k from both sides of the inequality, leaving
   n < 0
   This corresponds to answer (E) n < 0. Select that answer.
6. • Estimate the answer: 7/16 is about ½, so x must be somewhere around twice as large as y. Since y is 3.5 feet, x must be about 7 feet.
   • Look at the answer choices: The only answer choice anywhere near 7 is (A) 8. Select that answer.
7. Solution 1: You may know that, if a graph of a function is given by y = f(x), then the graph of y = ⅓f(x) will be the same graph, only one-third as narrow (i.e. three times as wide). So, the graph will be wider. Select (B) It will be wider.

   Solution 2:

   • Try a special case: Say that a = 1.
   • Draw a diagram: If you have a graphing calculator, plot the graphs of y = x² + 2 and y = ⅓x² + 2. You'll notice that the graph of y = ⅓x² + 2 is wider than the other one. Select (B) It will be wider. If you don't have a graphing calculator, it's possible to do this by calculating a table of a few values by hand, although it may be time-consuming.
8. Solution 1:
   • Read the question carefully and determine what it is asking: Meredith wants to wear an outfit consisting of one red item, one white item, and one blue item. We need to determine how many ways there are to do this.
   • We could imagine that we have three labels, "red," "white," and "blue," and we have to attach one label to each of a hat, a sweater, and a pair of jeans. In other words, we would need to find how many ways there are to permute the three labels "red," "white," and "blue." The number of ways to do this is 3! = 6. Select (B) 6.
   Solution 2:
   • Read the question carefully and determine what it is asking: Meridith wants to wear an outfit consisting of one red item, one white item, and one blue item. We need to determine how many ways there are to do this.
   • Estimate the answer: If there were no restrictions on colour, there would be 3 × 3 × 3 = 27 choices. However, with the colour restrictions, the number of possibilities might be much less than that, maybe only 1/3 or 1/4 or 1/6. So, perhaps the answer is somewhere between 4 and 9.
   • Look at the answer choices: We can definitely eliminate (E) 27. (A) 3 and (D) 12 seem unlikely as well. We are still left with a few possibilities, including (B) 6 and (C) 9.
   • It looks like there aren't so many possibilities that we couldn't list them out. So, list all of the possibilities out:

     Hat	Sweater	Jeans
     red	white	blue
     red	blue	white
     white	red	blue
     white	blue	red
     blue	red	white
     blue	white	red

     There are 6 possibilities. Select (B) 6.
9. Convert the sentence into an equation as follows:

   When	[ignore]
   twice	2 ×
   a certain number	x
   is increased by	+
   5	5
   the result is	=
   14	14

   So, we get:
   2x + 5 = 14
   2x = 9
   x = 4.5
   Enter 4.5 as the answer.
10. • Estimate the answer: y is definitely greater than 90°, perhaps around 120° or 135 °
    • Draw a diagram: A diagram is given, but if you recall the supplemental and congruent angles created when a pair of parallel lines intersects another line, you can fill in several angles on the diagram:
      
    • It can now be seen that x° and y° sum to a straight angle, or 180°. So:
      x + y = 180
      Since we are given that y = 3x:
      x + 3x = 180
      4x = 180
      x = 45
      Don't forget, we're asked to find the value of y, not x. y = 3x = 3(45) = 135.
11. • Draw a diagram: It may help you to visualize the problem if you draw the CD cases inside the illustration of the box.
    • If each CD case is ¼ inches wide, then four will fit in a one-inch box, so 4 × 8 = 32 will fit in an eight-inch box (you could also have obtained the answer by dividing 8 by ¼).
12. Since we are asked to find x⁄y, try to manipulate the equation so that this expression appears:
    (3x + y)⁄y = 6⁄5
    3x⁄y + y⁄y = 6⁄5
    3(x⁄y) + 1 = 6/5
    3(x⁄y) = 1/5
    x⁄y = 1/15
    Enter 1/15 as your answer.
13. Solution 1:
    • Estimate the answer: Looking at the table, two stores increased their profits by over $1,000 and one store increased their profits by over $2,000. A reasonable estimate might be something like $1,500.
    • You might be able to see the following in your head; if so, you don't have to write it out. The average increase in profit would be:
      ⅓(increase in profit for A + increase in profit for B + increase in profit for C)
      = ⅓(year 2 profit for A − year 1 profit for A + year 2 profit for B − year 1 profit for B + year 2 profit for C − year 1 profit for C)
      All of these numbers are in the grid, so you can simply plug them in and evaluate.
      = ⅓(6,250 − 5,000 + 7,350 − 6,000 + 12,700 − 10,000)
      = 1750
      Enter 1750 as the answer.
    Solution 2:
    • Estimate the answer: Looking at the table, two stores increased their profits by over $1,000 and one store increased their profits by over $2,000. A reasonable estimate might be something like $1,500.
    • You might be able to see the following in your head; if so, you don't have to write it out. The average increase in profit would be:
      ⅓(increase in profit for A + increase in profit for B + increase in profit for C)
      = ⅓(year 2 profit for A − year 1 profit for A + year 2 profit for B − year 1 profit for B + year 2 profit for C − year 1 profit for C)
      So far, this is the same as solution 1. You can save yourself a little bit of time if you notice that this expression is equal to:
      = ⅓(year 2 profit for A + year 2 profit for B + year 2 profit for C − year 1 profit for A − year 1 profit for B − year 1 profit for C)
      = ⅓(total profit for year 2 − total profit for year 1)
      = ⅓(26,250 − 21,000)
      = 1750
      Enter 1750 as the answer.
14. Solution 1:
    • First, find the points for where f(a) = a:
      |3a − 17| = a
      3a − 17 = a or 3a − 17 = −a
      2a = 17 or 4a = 17
      a = 8.5 or a = 4.25
    • Next, check whether a number between 4.25 and 8.5 would work. For example, try 5:
      f(5)
      = |3(5) − 17|
      = |−2|
      =2 < 5
      So, enter 5 as the answer. Of course, any other number between 4.25 and 8.5 would also work.
    Solution 2: Guess and check. For example:
    • If a = 0, f(a) = 17, so that doesn't work.
    • If a = 4, f(a) = 5, so that doesn't work, but we're closer.
    • If a = 8, f(a) = 7, so that does work. Enter 8 as the answer.
    Solution 3: If you have a graphing calculator, graph y = |3x − 17| and y = x. The portions of the graph where y = x is above the other graph represents the possible answers. Graphing the two, it appears that the correct range is between about 4 and 8 or so. Enter a number about halfway through the range, such as 6.
15. • Read the question carefully and understand what it is asking. There is a lot of information in this question that isn't needed (e.g. how many pieces of candy are in the jar). What is asked for is to find how many out of 13 pieces must be red in order for him to have more red pieces in total.
    • Estimate the answer: Ari has slightly fewer red pieces than green to begin with, so more than half of the 13 pieces he takes must be red. In other words, at least 7, probably 8 or 9 or so, have to be red.
    • Once Ari takes 13 pieces, he will have 3 + 4 + 13 = 20 pieces in total. If Ari has more red candies than green candies, the least number of red candies that he can have is 11. Ari already has 3 red candies, so he must take 11 − 3 = 8 more red candies. Enter 8 as the answer.
16. • Read the question carefully and understand what it is asking: Don't worry that you've never heard of "tri-factorable" before, it's a made-up term. The question asks you to find how many numbers less than 1,000 are the product of three consecutive integers.
    • To understand the question better, it can help to list numbers that are the product of three consecutive integers. One would be 1 × 2 × 3 = 6. Another is 2 × 3 × 4 = 24. You could continue like this, but you might realize that 9 × 10 × 11 = 990 is the largest such number less than 1,000 (10 × 11 × 12 is over 1,000). So, if the smallest factor is between 1 and 9, the number will be less than 1,000. So, there are 9 such numbers and 9 is the answer.
17. Solution 1:
    • Draw a diagram: It might help you to understand the question better if you draw a rough graph plotting the cost for carriers A and B as a function of t.
    • Guess and check: For a call that lasts t > 20 minutes, the cost for carrier A is equivalent to 5¢ per minute for the first 20 minutes and 7¢ per minute for t − 20 minutes. The cost for carrier B is equivalent to 6¢ per minute.

      Now, if the call lasted for 40 minutes, then A would give you 5¢ a minute for 20 minutes and 7¢ a minute for 20 minutes. Is this equal to 6¢ a minute for 40 minutes? Yes. 5(20) + 7(20) = 100 + 140 = 240 = 6(40). Enter 40 as the answer.

    Solution 2:
    • Draw a diagram: It might help you to draw a diagram as above.
    • We can represent the cost of a phone call with carrier A lasting over 20 minutes as:
      1 + .07(t − 20)
      We can represent the cost of a phone call with carrier B lasting over 20 minutes as:
      .06t
      If the cost of a phone call lasting t minutes is the same on both carriers, the two expressions must be equal to one another:
      1 + .07(t − 20) = .06t
      1 + .07t − 1.4 = .06t
      .01t = 0.4
      t = 40
      Enter 40 as the answer.
18. If each square has a side length of k inches, then, by counting the number of sides along the dark line, the perimeter is 16k. Since there are 10 squares, the area is 10k². If the perimeter in inches equals the area in square inches, then:
    16k = 10k²
    k² − 1.6k = 0
    k(k − 1.6) = 0
    k = 0 or k = 1.6
    Since the diagram was drawn to a non-zero size, presumably k = 0 is not the solution that the SAT writers want. So, enter 1.6 as the answer.