Solutions for 2014 SAT Practice Test, Section 2

Here are solutions for section 2 of the 2014–15 SAT practice test; you can find the test on the College Board's web site or in the Getting Ready for the SAT booklet. Note that this test is the same as the 2012–13 practice test. 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. • Convert the sentence into an equation: This results in:
     10 + x = 5 + 10
   • Solve the equation for 2x:
     x = 5
     2x = 10
     Select (C) 10.
2. • Convert the sentence into an equation:

     The result when	[ignore]
     a number	x [or another variable]
     is divided by	/
     2	2
     is equal to	=
     the result when	[ignore]
     that same number	x [or whatever letter you used above]
     is divided by	/
     4	4

     Putting this all together, we get:
     x / 2 = x / 4
   • Solve the resulting equation. First, subtract x / 4 from both sides:
     x / 4 = 0
     
     now multiply both sides by 4:
     x = 0
     
     Select (C) 0.
3. • Draw a diagram: Draw a dotted line through the middle of each of the five letters.
   • The only letter where the half to the left of the dotted line does not appear to be a reflection of the right is E. Select (E) E.
4. Keep in mind that vertical angles are equal. Since m and the combination of p and x are vertical, they must be equal. So:
   m = p + x
   40 = 25 + x
   x = 15
   Select (A) 15.
5. Solution 1:
   • Estimate the answer: Looking at the table, y appears to increase by 3 for each unit that x increases. So, the answer will probably be a linear equation containing 3x in it.
   • Look at the answer choices: The only choice with 3x in it is (B) y = 3x + 3. Select that answer.
   Solution 2:
   • Try a special case: According to the table, when x = 0, y = 3.
   • Look at the answer choices: See for which answers y = 3 when x = 0. This is the case for only (A) and (B). The other three answers can be eliminated.
   • Try another special case: According to the table, when x = 1, y = 6.
   • Look at the answer choices: When x = 1, according to (A) y = 4. Eliminate that answer. The answer must be (B) y = 3x + 3.
6. • Estimate the answer: There are a few ways of getting a rough estimate here:
     • If Food is 25% of David's expenses and the Car is 20%, then if David spent $250 on food, he'd spend $200 on the car, and if he spent $500 on food, he'd spend $400 on the car. So, if he spends $400 on food, he'd spend between $200 and $400, but closer to $400 than $200.
     • Looking at the graph, the food sector is just somewhat larger than the car sector, so if David spends $450 on food, he'd spend not too much less, probably between $50 and $100 less, on the car. So, his car expenses are probably between $350 and $400.
   • Look at the answer choices: The only one that matches our estimates is (C) $360. Select that answer.
7. Try a special case: Say that n = 1. Then, k = 3 (since 8¹ = 2³). So, n/k = ⅓. Select (B) ⅓.
8. • Estimate the answer: It may be difficult to come up with a good way of estimating the answer here; however, the answer must be a fair bit less than the difference between buying the refrigerator at 20% off and buying it at 10% off with no further discounts. The difference in this case would be 10% of $600, or $60, so the answer must be a fair bit smaller than that.
   • Look at the answer choices: (D) and (E) are well out of the range of our estimate and can be eliminated. You might feel confident in eliminating (C) as well. That leaves two answers, (A) $6 and (B) $12.
   • If a $600 refrigerator is discounted by 20%, it now costs $600 × (1 − 20%) = $480.
     If a $600 refrigerator is discounted by 10% twice, it now costs $600 × (1 − 10%) × (1 − 10%) = $486.
     The difference is $6. Select (A) $6.
9. Solution 1:
   • Try a special case: Say that x = 1. Then, f(x) = 3(1) + 4 = 7. Then, 2(f(x) + 4 = 2(7) + 4 = 18.
   • Guess and check: Try each answer choice and see which evaluates to 18 when x = 1. (E) is the only one that does so. Select (E) 6x + 12.
   Solution 2: If f(x) = 3x + 4, then:
     2f(x) + 4
   = 2(3x + 4) + 4
   = 6x + 12
   Select (E) 6x + 12.
10. It's possible to solve this problem in a less formal way than the following three solutions illustrate, but I've chosen to present these solutions in order to illustrate the thought processes that might go into solving this problem:
    Solution 1:
    • Draw a diagram: Draw a diagram similar to the one on the right, showing a triangle with sides 7, 10, and an unknown length (labelled ? in the diagram). Also draw the height of the triangle in (labelled h in the diagram).
    • A right triangle is formed with the base, the height, and the side with length 7. According to the Pythagorean theorem (provided in the Reference Information), h² + (part of the base)² = 7². Now, anything squared must be ≥ 0; therefore, (part of the base)² ≥ 0. So,
      h² ≤ 7²
      h ≤ 7
      So, if the maximum value for h is 7, then the maximum area of the triangle must be ½(7)(10) = 35 (since, from the Reference Information, the area of a triangle = ½bh, where b is the base and h the height of the triangle). Select (C) 35.
    Solution 2:
    • Draw a diagram: Draw a diagram similar to the one on the left, depicting a line of length 10 as the base of the triangle. The two known sides must meet at some angle. Draw the two sides meeting at a variety of angles.
    • From the Reference Information, the area of a triangle = ½bh, where b is the base and h the height of the triangle. You can draw the heights on your diagram if you wish. Since the base is always 10, the greatest area comes with the greatest height. The greatest height occurs when the triangle is a right triangle and so the height is 7. The maximum area of the triangle must be ½(7)(10) = 35 (since, from the Reference Information, the area of a triangle = ½bh, where b is the base and h the height of the triangle). Select (C) 35.
    Solution 3:
    • Try a special case: Say that the triangle is a right triangle with legs of 7 and 10. Then the area would be ½(7)(10) = 35.
    • Look at the answer choices: 35 is one of the answer choices (C), so that's a good sign.
    • Make an assumption: We know 35 is a possible answer. Could 70 be a possible answer? Let's assume that it is for a moment and...
    • ...Draw a diagram: Draw the base and height of a triangle with area 70, using one of the given sides as the base. We could either draw a triangle with base 10 and height 14 (½(10)(14) = 70) or with base 7 and height 20. Let's say you've drawn a base of 7 and height of 20, as shown on the right. Now, attach the side of length 10 to the base so it meets the top of the height. No matter what you try, you can't, can you? So, there can't be any such triangle with area 70. So, the answer must be (C) 35.
11. • Estimate the answer: Perez lost, so he must have received less than 60,000 votes. He received 3/8 of 120,000 votes. This is more than 3/10 of 100,000 votes, or 30,000 votes. So, he must have received more than 30,000 votes and less than 60,000 votes.
    • Look at the answer choices: The only answer greater than 30,000 and less than 60,000 is (C) 45,000. Select that answer.
12. • First, keep in mind that the probability of an event occurring is equal to the number of favourable outcomes divided by the number of total outcomes.
    • The total number of outcomes is equal to "the number of positive integers less than or equal to 10." There are 10 of these (1, 2, 3, 4, 5, 6, 7, 8, 9, 10).
    • To find the number of favourable outcomes, solve:
      5n + 3 ≤ 14
      5n ≤ 11
      n ≤ 2.2
      There are two favourable outcomes (1 and 2). The probability is 2/10 = 1/5. Select (C) 1/5.
13. • Convert the sentence into an equation:

      t²	t²
      is	=
      how much	x
      greater than	+
      t	t

      The result is:
      t² = x + t
    • Solve the equation:
      t² − t = x
      t(t − 1) = x
      Select (D) t(t − 1).
14. • Draw a diagram: Draw a diagram of the situation indicated in the problem, such as the one shown on the right (where x represents the quantity you want to find).
    • The quantity we are looking for is the hypotenuse of a right triangle with legs 5 and 2. To find it, use the Pythagorean theorem, which is given in the Reference Information:
      5² + 2² = x²
      x = √29
      Select (C) √29.
15. • You are given that p² − n² = 12. Factoring, we get:
      (p + n)(p − n) = 12
    • Now, if two integers multiply to 12, they must be one of: 12 and 1, 6 and 2, or 4 and 3. Now, p − n will always be smaller than p + n, so the possible values of p − n are 1, 2, and 3.
    • Look at the answer choices: We know that 4 cannot be an answer choice, so eliminate (D) and (E).
    • If p − n = 1, can p + n = 12? You can solve the equations or play around with some numbers or use the properties of even and odd numbers to convince yourself that it can't. So 1 cannot be an answer choice either.
    • Look at the answer choices: We can now eliminate (A) and (C). Select (B) II only.
16. Using the diagram, count the number of blocks (intersections) you need to travel to reach W from F. The answer is (D) 3½. Select that answer.
17. Solution 1:
    • Estimate the answer: Drawing a straight line between F and Z, each route must have a mirror image reflected in that line. So, the number of routes must be even. There also seems to be more than two routes.
    • Look at the answer choices: Based on our estimate, we can eliminate (B) Five, (D) Three, and (E) Two.
    • Find the number of possible routes if you start by going left, then multiply by 2 (due to symmetry). There are three. So, the answer is 3 × 2 = 6. Select (A) Six.
    Solution 2:
    • Estimate the answer and look at the answer choices as above.
    • Break the problem up: Find out how many ways there are to travel to each intersection in between F and Z. There is one way to travel to the two intersections adjacent to F. For each successive intersection, the number of ways to get there is equal to the sum of the number of ways to get to each preceding intersection. Write the numbers on the diagram, as shown at right. There are 6 ways to get to Z. Select (A) Six.
18. • Draw a diagram: Draw all points on the diagram that are a m-distance of 3 from the firehouse.
    • The points are in the shape of a square, rotated 45 degrees. Select (B) Square.
19. Solution 1: This is actually a very easy question if you remember the laws of exponents and don't get intimidated by all of the letters. Factor the expression by (2x)^y:
    (2x)^3y − (2x)^y = (2x)^y((2x)^2y − 1)
    Select (C) (2x)^y((2x)^2y − 1).
    Solution 2: This is a bit more time-consuming, but if you have some time left it's still a good way of approaching the problem:
    • Try a special case: Say that x = 2 and y = 1, for example. Then, (2x)^3y − (2x)^y evaluates to 60 (I'm omitting the calculations here; you'd probably want to use your calculator anyway).
    • Look at the answer choices: The only answer choice of the five that evaluates to 60 is (C) (2x)^y((2x)^2y − 1). Select that answer.
20. If the product of two integers ends with 9, then either both integers end in 3 or one integer ends in 9 and the other ends in 1. However, since j, k, and n are consecutive, both j and n can't both end in 3. So, j ends in 9 and n ends in 1, and so k ends in 0. Select (A) 0.