Solutions for Practice Test 3, The Official SAT Study Guide, Section 8

Here are solutions for section 8 of the third practice test in The Official SAT Study Guide, second edition, found on pages 543–548. 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:

     ¾	¾
     of	×
     a number	x
     is	=
     18	18

     This gives:
     ¾x = 18
   • Divide both sides of the equation by 3:
     ¼x = 6
     Therefore, ¼ of the number is 6. Select (C) 6.
2. • Estimate the answer: The * operator seems to create a number somewhat less than the square of the number. So, 5* will be somewhat less than 25.
   • Look at the answer choices: (B) 20 and (C) 24 correspond with our estimate. The other three answer choices can be eliminated.
   • Substitute k = 5 in k* = k(k − 1):
     5* = 5(5 − 1)
     5* = 5(4)
     5* = 20
     Select (B) 20.
3. • Read the question and understand it carefully: The question asks for the age at which the diameter of the eye pupil is the same during the day and night. At this age, the two lines in the graph will cross.
   • The two lines in the graph cross at 45. Select (D) 45.
4. If Toni spends two hours each day commuting, she will spend 2/24 = 1/12 of the total hours in the day commuting. Select (A) 1/12.
5. Square both sides of the equation:
   (√3)² = (x + 1)²
   3 = (x + 1)²
   Select (C) 3.
6. Solution 1:
   • Estimate the answer: We can assume the diagram to be drawn to scale, so we can examine it closely for clues. Looking very carefully at the diagram, t appears to be the smallest angle and x the largest angle. So, t < r < x.
   • Look at the answer choices: t < r < x is one of the answer choices. Select (A) t < r < x.
   Solution 2: In triangles, the smallest angle is always opposite the smallest side, and the largest angle opposite the largest side. So, t < r < x. Select (A) t < r < x.
7. Solution 1: Guess and check: Try each answer until one works. When using guess and check for an inequality, it's usually best to start with an extreme value. So, we'll start with the largest value, (E) 5:
   |6 − 5(5)|
   =|−19|
   =19
   Nope. Let's try the smallest value, (A) −3:
   |6 − 5(−3)|
   =|21|
   =21
   This choice works. Select (A) −3:
   
   Solution 2:
   • Solve the inequality, being very careful about which way the inequality sign points:
     |6 − 5y| > 20
     6 − 5y > 20 or 6 − 5y < −20
     −5y > 14 or −5y < −26
     y < −14/5 or y > 26/5
   • Look at the answer choices: (A) −3 is < −14/5. Select (A) −3.
8. Solution 1:
   • Draw a diagram: It may help to draw a diagram:
     
   • Estimate the answer: We are not given the length of the hypotenuse, but it must be the longest side, so it must be at least 4. However, it must be less than 7, due to the triangle inequality. So, the perimeter must be somewhere between 11 and 17. Doubling each side, the perimeter would be somewhere between 22 and 34.
   • Look at the answer choices: The only answer choice between 22 and 34 is (D) 24. Select that answer.
   Solution 2:
   • Draw a diagram: Draw a diagram as above.
   • First, find the length of the hypotenuse. We can use the Pythagorean Theorem (given in the Reference Information) to do so:
     3² + 4² = h²
     25 = h²
     h = 5
     So, the lengths of the sides are 3, 4, and 5.
   • If the length of each side were doubled, the side lengths would be 6, 8, and 10. 6 + 8 + 10 = 24. Select (D) 24.
9. Estimate the answer: Looking at the table, C appears to have about 3.5 times foreign locations than United States locations. No other company appears to have more than 3 times more foreign locations. Select (C) C.
10. • Estimate the answer: Line n seems to be fairly steep, having a slope of about 2 or 3.
    • Look at the answer choices: Both (D) 2 and (E) 3 correspond to our answer choices. The other three answers can be eliminated.
    • Draw a diagram: Draw, as well as you can, a straight line up and down along x = 1 and straight lines across along y = 2 and y = 3. Doing this will allow you to see that (3, 1) is on line n. Now, since n goes through (0, 0), the slope is 3/1 = 3. Select (E) 3.
11. Solution 1: If 2x + 5 = 3kx + 5 for all values of x, then the x-coefficients on both sides of the equation must be equal. So:
    2 = 3k
    k = ⅔
    Select (E) ⅔.
    
    Solution 2: Solve the equation for k:
    2x + 5 = 3kx + 5
    2x = 3kx
    2 = 3k
    k = ⅔
    Select (E) ⅔.
12. Solution 1:
    • Read the question carefully and determine what it is asking: Note particularly the words "different" and "least."
    • Guess and check: Look at each answer, starting with the smallest since the question asked for the least, to see whether it is possible. Is (A) None possible? No, you can't have eleven negative integers sum to zero. Is (B) One possible? Well, it could be if you had a really big positive number and 10 smaller numbers. So, (B) One is possible. Select that answer.
    Solution 2:
    • Read the question carefully and determine what it is asking, as above.
    • Try a smaller case: If 11 numbers is too many to handle, try 2, 3, 4, etc. and see if you can find a pattern:
      • For two integers, you could have 1 and −1, so the answer would be One.
      • For three integers, you could have 1, 0, and −1, so the answer would be One.
      • For four integers, you could have 3, 0, −1, and −2, so the answer would be One.
      You can try further cases if you like; the answer will always be one. So, you can infer that the answer has to be (B) one.
13. Solution 1:Break the problem up: 17 = 10 + 5 + 2. Find out how many ways there are to get 10, 5, and 2 points, respectively:
    • 10: 3 ways (1 10-point token, 2 5-point tokens, or 10 1-point tokens).
    • 5: 2 ways (1 5-point token or 5 1-point tokens).
    • 2 1 way (2 1-point tokens).
    To find the total number of combinations, multiply the possibilities together: 3 × 2 × 1 = 6. If you think about it, none of these 6 combinations are duplicates of any other, so the answer is (E) Six.
    
    Solution 2: List each possibility out:
    • 1 × 10 + 1 × 5 + 2 × 1
    • 3 × 5 + 2 × 1
    • 1 × 5 + 12 × 1
    • 1 × 10 + 7 × 1
    • 2 × 5 + 7 × 1
    • 17 × 1
    The largest answer choice is (E) Six, and we've found six. Select (E) Six.
14. The graph of y = 2f(x) is the graph of y = f(x), stretched out vertically by a factor of 2. The only graph that is stretched out vertically is (D). Select that answer.
15. • Estimate the answer: The answer has to be at least 25, since all of the negative terms will be less than 100. However, the positive terms grow by a factor of 4 each time, so only 3 or 4 of them can be positive. So, the answer should be either 28 or 29.
    • Look at the answer choices: The only choice that is either 28 or 29 is (C) 28. Select that answer.
16. • Draw a diagram: It isn't easy to draw a cube inscribed in a sphere, but a rough sketch will help you visualize the problem.
    • Work backwards: We need to find the diameter of the sphere. The diameter of a circle is equal to the largest distance between two points on the circumference; similarly, the diameter of a sphere is equal to the largest distance between two points on the surface. We know that two opposite corners of the cube both touch the surface, and it seems reasonable to think that the cube's diagonal runs through the center, so it seems reasonable to believe that the diagonal of the cube is the length of the diameter.
    • Draw another diagram: It may help to draw another diagram of the cube, especially if you haven't drawn anything yet:
      
    • Continue working backwards: Looking at the above diagram, we could find the diagonal of the cube using the Pythagorean theorem if we knew the length of the cube and the diagonal of the top face.
    • Since the volume of the cube is 8, the length of each side must be the cube root of 8, or 2.
    • The diagonal of the top face, by the Pythagorean theorem must be
      √2² + 2²
      = √8
    • The diagonal of the cube is:
      √(√8)² + 2²
      = √8 + 4
      = √12
      = 2√3
      This must be the length of the diameter. Select (D) 2√3 (approximately 3.46).