Solutions for 2013 SAT Practice Test, Section 2

Here are solutions for section 2 of the 2013-14 SAT practice test; you can find the test on 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. Solution 1: To write 70,000 in scientific notation, you would shift the decimal point 4 places to the left, so the exponent would be (D) 4.
   Solution 2: Use your scientific calculator to convert 70,000 into scientific calculator. The exponent value should read "04". Therefore, the answer is (D) 4.
   Solution 3: I wouldn't recommend this solution if you can remember how to do either Solution 1 or Solution 2, but if you have to:
   • Guess and check: Evaluate each answer choice (it's probably best to start at (C), the middle value) to see for which value of n 7.0 × 10ⁿ = 70,000. The answer is (D) 4.
2. Solution 1: If Sam drove m miles, then if Kara drove twice as many miles, she drove 2m miles. If Darin drove 20 miles fewer than Kara, then he drove 2m − 20 miles. Therefore, the answer is (B) 2m − 20.
   Solution 2:
   • Try a special case: Say that Sam drove 10 miles (i.e. m = 10). Then Kara drove 20 miles and Darin drove 0 miles.
   • Look at the choices: The only answer choice whose value is 0 when m = 10 is (B) 2m − 20. Therefore, the answer must be (B).
3. Solution 1:
   • Look at the choices: Based on the given choices, the answer must be one or more of (1,4), (3,1), and/or (2,2). If you're observant, you might notice right away that (2,2) can't be a solution, because it would make the left-hand side even, but 11 is odd.
   • Guess and check: Test (1,4) and (3,1) to see which satisfy 3x + 2y = 11. Both values do. Therefore, the answer is (D) (1,4) and (3,1).
   
   Solution 2: Use your graphing or scientific calculator to graph and/or generate a table of values for 3x + 2y = 11. By inspecting the output, you should be able to determine that (1,4) and (3,1) are both solutions. Therefore, the answer is (D) (1,4) and (3,1).
4. • Estimate the answer: The profit has to be less than 500x; if x = 10, the profit has to be somewhat less than $5,000.
   • Look at the choices: Our estimate allows us to eliminate (A), but there are still four plausible answers.
   • Perform the calculation:
     500(10) − 20(10)²
     = 5000 − 2000
     = 3000
     Therefore, the answer is (C) $3,000.
5. • Estimate the answer: The middle number is 12, so 12 seems a good estimate.
   • Look at the answers: (B) 12 is one of the answers. You may be able to see immediately that this is the only reasonable answer. If so, select (B). Otherwise, see the next step.
   • Try a special case: If n = 1, then the three numbers are 11, 12, and 13. The arithmetic mean of these three numbers is 12. The only answer that matches is (B) 12. Therefore, that must be the correct answer.
6. • Estimate the answer: There is no indicator that the diagram is not drawn to scale, so we can assume it is. The angle that we're looking for appears to be slightly larger than 90°, so a reasonable range for an estimate might be between 90° and 120°.
   • Look at the choices: (A) and (B) fall within the range of our estimate. (C) seems a bit high, but not outside the realm of possibility. We could probably eliminate (D) and (E), though.
   • Work backwards:
     • We can calculate angle AMC if we knew the measures of angles MAC and MCA.
     • Since AM and CM are the angle bisectors of angle BAC and BCA, MAC = ½BAC, and MCA = ½BCA.
     • Since the triangle is isosceles, and we are given the third angle, we can determine the other two angles. Each is ½(180 − 40) = 70°. So, MAC and MCA are each 35°, and AMC = 180 − 35 − 35 = 110°. Therefore, the answer is (A) 110°.
7. • Estimate the answer: If there are fewer pineapples than there are pears or peaches, then pineapples must make up less than ⅓ of the salad.
   • Look at the choices: (C), (D), and (E) are greater than ⅓. (B) is just slightly less than ⅓. (A) seems the most reasonable.
   • Estimate the answer (again): Peaches make up ½ of the salad, and there are fewer pineapples than pears, so pineapples must make up less than ¼ of the salad. Therefore, the answer cannot be (B), so it must be (A) 1/5.
8. • Draw a diagram: The diagram is already given, but it may help to draw lines from S and T to the centre of the circle.
   • Estimate the answer: ST appears to be exactly ¼ of the circle. Since "The number of degrees of arc of a circle is 360" (see Reference Information), ST appears to be 90°.
   • Look at the choices: The only choice anywhere near 90° is (C) 90°. Therefore, the answer must be (C).
9. Solution 1:
   • Look at the choices: No answer stands out, but looking at the choices allows us to organize the given information better.
   • Rewrite the given information to be more similar to the choices. The information given in the problem could be summarized as: "If x is in P, then x is in Q."
   • Look at the choices (again): Look for a choice that appears to be inconsistent with the statement above. "(C) 6 is in P, but not in Q" appears to be inconsistent with "If x is in P, then x is in Q." Therefore, the answer is (C).
   
   Solution 2:
   • Draw a diagram: You could draw a Venn diagram for this question. If you've drawn it correctly, it should look like:
     
   • Guess and check: Try finding a place for each number in the Venn diagram:
     
     You'll notice that there is no place to put 6 such that it is in P, but not in Q. Therefore, the statement that must be incorrect is (C) 6 is in P, but not in Q.
   
   Solution 3:
   • Transform the problem into something more concrete. For example, you could substitute "is an apple" for "is in P" and "is a fruit" for "is in Q". Every apple has to be a fruit, right?
   • Guess and check: Read through each solution, while mentally making the above substitution. You will find that (C) "Object 6 is an apple, but not a fruit." doesn't make any sense. Therefore, the answer must be: (C) 6 is in P, but not in Q.
10. • Read the question carefully and determine what it is asking: It appears that you are being asked to find 6 × 6 × 6 ÷ (3 × 2 × 1), unless there's a trick to the question.
    • Take advantage of question order: This is question 10 in a 20 question section, so it's probably not a hard question and we can trust our first instinct.
    • Perform the required calculation: You can perform the calculation in your head if you notice that 6 × 6 × 6 ÷ (3 × 2 × 1) = 6 × 6 = 36. Otherwise, use your calculator. Select (D) 36.
11. Solution 1:
    • Read the question carefully and determine what it is asking: You are asked to find a value of x that will make the equation true. The equation must be true for all a, except where a = 0. An insight that can be useful for solving this problem is to note that, since the left hand side of the equation doesn't contain any a's, and the right hand side does, the a's have to cancel each other out somehow for the right value of x.
    • Guess and check: If you grasped the insight in the previous step, you might want to start with (E) 5. Otherwise, you might start with another value. For x = 5, the equation becomes 5/5 = (5 + a)/(5 + a), which is true for all a. Therefore, the answer must be (E) 5.
    Solution 2:
    • Try a special case: Pick a value for a. Probably a value that is not equal to a possible value for x is best. For example, we could assume that a = 3. The equation becomes 5/x = 8/(x + 3)
    • Guess and check: Try each of the answer choices until you find one that makes the above equation true. The one that works is (E) 5.
    Solution 3:
    • Try a special case: Pick a value for a. Probably a value that is not equal to a possible value for x is best. For example, we could assume that a = 3. The equation becomes 5/x = 8/(x + 3)
    • Cross-multiply and solve the equation:
      5(x + 3) = 8x
      5x + 15 = 8x
      15 = 3x x = 5
      Therefore, the answer is (E) 5.
    • • Read the question (and examine the diagram) carefully and determine what it is asking. The question boils down to the following: If four triangles have an area of 10, what is the area of 25 triangles?
      • Estimate the answer: 25 is more than 6 × 4, so the answer should be slightly more than 60.
      • Look at the possible answers: The only answer in the range of our estimate is (E) 62.5. Select that answer.
    • Typically, when you see this type of question, there will be some sort of shortcut; don't try to solve for x, y, and z individually (it won't work, anyway). Here, subtract the second equation from the first equation:
      y + z = 5
      Therefore, the answer is (E) 5.
    • • Read the question carefully and determine what it is asking: The problem involves dividing x dollars evenly amongst a group of people. If there are 3 people, each pays x/3 dollars. If there are 4 people, each pays x/4 dollars. The question asks for the difference in these two numbers.
      • Estimate the answer: ⅓ − ¼ is less than ¼, so x/3 − x/4 should be less than x/4.
      • Look at the choices: The only choice less than x/4 is (A) x/12. Therefore, the answer must be (A).
    • Solution 1:
      • Estimate the answer: When x = 2, y = 7. Therefore, the boundary should contain a 7 in it.
      • Look at the answers: Answers (C) and (D) don't contain a 7, so we can eliminate those.
      • Try a special case: In order to eliminate one or more answers, try some values of y and see which answer choices hold. If we let y = 4, then x = ½, which is less than 2. This eliminates answers (B) and (E), since neither range includes 4. Therefore, the answer must be (A) y < 7.
      Solution 2: Solve the equation in terms of x:
      x = (y − 3) / 2
      . Then, substitute this value into the inequality:
      (y − 3) / 2 < 2
      y < 7
      Therefore, the correct answer is (A) y < 7.
    • The graph of g appears to be the exact same as the graph of f, only translated up one unit. So, the answer is (B) g(x) = f(x) + 1.
    • This is probably one of the most difficult questions that you'll run into on the SAT. If you weren't able to get it, you shouldn't feel bad about it. When I first read through the practice test, my thought here was that someone should be able to score 800 without answering this question correctly. If you look at the scoring section, you'll notice that my hunch was right. Anyway:
      
      • You shouldn't have to memorize formulas for the SAT, but if you know that the sum of interior angles of a polygon with n sides is 180(n − 2)°, then this will really help. If not, you might try reasoning by analogy, since you know that a triangle is 180°, a quadrilateral can be made up of two triangles, so it's 2 × 180° = 360°, a pentagon can be made from three triangles, so it's 3 × 180° = 540°, and so on, you might be able to remember this relationship.
      • Since each angle in the mystery polygon is equal, then each angle is equal to 180(n − 2)⁄n°.
      • We can look at what we are shown as a polygon with 4 sides, so its angles must sum to 360°. Since x + y = 80, then the other two angles must sum to 280°. Since each are equal, each is equal to 140°.
      • Solve the resulting equation:
        180(n − 2)⁄n = 140
        (n − 2)⁄n = 7⁄9
        n = 9
        Therefore the answer is (B) Nine.
      
      If you weren't able to figure this out, here's how you could make a best guess:
      • Estimate the answer: You've probably seen a regular hexagon and a regular octagon before. The angles that we can see definitely look wider than those of a hexagon and they don't look quite like those of an octagon. To me, they look wider than those of an octagon. Therefore, the shape probably has at least nine sides.
      • Look at the choices: The only reasonable choices are (A) Ten and (B) Nine. Select one of them.
    • • Estimate the answer: s and v appear to be the furthest away on the number line, so the answer probably contains both variables.
      • Look at the choices: Answers (B) and (D) contain both s and v. We'll eliminate the remainder.
      • Guess and check: v appears to be 2.5 and s appears to be −2.5, so substitute those values into (B) and (D), ensuring that you're very careful with negative signs and remember order of operations (the contents of the absolute value brackets should be treated as parentheses and evaluated first). For (B) we get:
        |−2.5 + 2.5|
        = |0|
        = 0
        For (D) we get:
        |−2.5 − 2.5|
        = |−5|
        = 5
        Therefore the answer is (D) |s − v|.
    • • Read the question carefully and determine what it is asking: According to the graph, the depth of the water decreased 2 feet between 3:00 and 4:00. According to the text, the depth of the water decreased 10% between 3:00 and 4:00. You are asked for the depth of the water at 4:00.
      • Perform the calculation: If 2 feet is 10% of the depth of water at 3:00, the depth of water must be 2/10% = 20 feet. However, we are asked for the depth of water at 4:00. Since the depth decreased 2 feet, the answer must be (C) 18 feet.
    • There isn't any real shortcut for this; you need to evaluate all three algebraic expressions to determine whether they're true. You might be able to save yourself some time, however, if you are able to see that the expression is symmetrical and therefore I. must be true. Since I. must be true, we can eliminate (B) and (C). For II., the left-hand side is:
      (x − 1)⊙(x + 1)
      = (x − 1)(x + 1) + x − 1 + x + 1
      = x² + 2x − 1
      and the right-hand side is:
      (x⊙x) − 1
      = x² + x + s − 1
      = x² + 2x − 1
      This equals the left-hand side, so II. is true and the answer must be either (D) or (E). Now, for III., the left-hand side is:
      x⊙(y + z)
      = x(y + z) + x + y + z
      = xy + xz + x + y + z
      and the right-hand side is:
      (x⊙y) + (x⊙z)
      = xy + x + y + xz + x + z
      = xy + xz + 2x + y + z
      Ensure that you look at the two closely. They are not the same (note that "2" in front of the x term). Therefore, the answer is (D) I and II only.