Solutions for Practice Test 7, The Official SAT Study Guide, Section 9

Here are solutions for section 9 of practice test #7 in The Official SAT Study Guide, second edition, found on pages 795–800. The solutions below demonstrate faster, more informal methods that might work better for you 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 6 out of 10 cars are red, the probability that a car selected at random is red is 6/10 = 3/5. Select (B) 3/5.
2. • Estimate the answer: We can't assume the figure is drawn to scale here, but from eyeballing the diagram, (B), (C), (D), and (E) appear to be true and (A) appears to be false, so (A) seems a likely possibility.
   • If BD bisects AC, then AD = DC, so we can eliminate (D). This also means that the three sides of triangles ADB and CDB are equal, so the triangles are congruent, from which we can conclude that w = z, x = y, and angle D is a right angle, so BD is perpendicular to AC. So, we can eliminate (B), (C), and (E). Select (A) w = x.
3. 15 percent of m would equal half of 30 percent of m, or 40/2 = 20. Select (B) 20.
4. Try a special case: Say that n = −5. Looking at the answers, (A) evaluates to −2.5, (B) evaluates to −10, (C) evaluates to −3, (D) evaluates to −7, and (E) evaluates to 7. The only one that is positive is (E) 2 − n. Select that answer.
5. • Estimate the answer: Since 1.2 is just a bit more than 1, the correct answer must be such that the first number is just a bit more than the second.
   • Look at the answer choices: Of the five answer choices, the only such answer choice is (D) 6 to 5. Select that answer.
6. • Estimate the answer: Since each symbol represents 5 million homes, the answer must be more than 15 million and less than 20 million. Since the fourth symbol appears to be half a home, the answer is probably half way in between, or 17.5 million.
   • Look at the answer choices: The only reasonable-looking answer is (D) 17.5 million. Select that answer.
7. Guess and check: Don't try to solve the equation. Instead, try each of the answers as a value for a until you find one that results in a positive integer value for b. If a = 3, then b² = 2, so b is √2, which is not a positive integer. If a = 4, then b² = 9, so b = 3. Select (B) 4.
8. • Estimate the answer: Since u and w are close together on the number line, then u − w will be close to zero. Because we are asked to find |u − w|, the value will also be positive.
   • Look at the answer choices: The only possibility that is small and positive is (C) x. Select that answer.
9. Convert the sentence into an equation:

   	(
   	(
   	(
   A number n	n
   is increased by	+
   5	5
   and	[ignore]
   the result is	)
   multiplied by	×
   5	5
   This result	)
   is decreased by	−
   5	5
   Finally	[ignore]
   that result	)
   is divided by	÷
   5	5

   Putting this all together, we get:
     (((n + 5) × 5) − 5) ÷ 5
   = ((5n + 25) − 5) ÷ 5
   =(5n + 20) ÷ 5
   =n + 4
   Select (D) n + 4.
10. • Try a special case: Say that n = 1. Phillip would use four six-inch pieces of masking tape to put up that one poster. So, he would use 24 inches, or (since 12 inches = 1 foot) 2 feet of tape, and there would be 298 feet left on the roll.
    • Look at the answer choices: See which answer corresponds to 298 when n = 1. You should be able to see what 300 − 2 = 298, so select (B) 300 − 2n.
11. • Draw a diagram: It can help to draw a diagram to picture the situation.
    • Try a special case: Say that the slope of line m is 0 (and draw a line picturing this special case in your diagram). Then, its reflection, line l, would also have a slope of 0.
    • Try a special case: Try another special case. Say that the slope of line m is a small negative number such as −0.1. Then, line m would be falling slightly to the right, and line l would be rising at the same rate towards the right, so its slope would be 0.1.
    • Therefore, the slope of line l would be the slope of line m with the opposite sign, or 4/5. Select (B) 4/5.
12. Look at the answer choices: Check whether n = p for any of the values of p given in choices (A) through (D). For (A), when p = 0, n = 0. Select (A) 0.
13. • Estimate the answer: The diagram is not drawn to scale, but it does not appear to be highly inaccurate, so we might be able to get a rough idea of the answer by looking at it. Both y° and x° appear to be definitely over 90°, possibly around 120°, so our estimate of x + y is around 240 or so.
    • Look at the answer choices: Answer (D) seems to be the only reasonable choice, being the only one that's over 180° and close to our estimate. However, since the diagram wasn't drawn to scale, it's probably a good idea to investigate further.
    • Try a special case: We know that z = 30. Say that the other two angles of the triangle are 75° and 75°. Then, both x and y are equal to 180 − 75 = 105, so x + y = 210. Select (D) 210.
14. • Try a special case: When x = 0, f(0) = c. We are told that c is a positive number, so f(0) will be positive.
    • Look at the answer choices: See which answers show the graph having a positive value at x = 0. The only such graph is (E). Select that one.
15. • Estimate the answer: It might be difficult to estimate an exact value for AB, but we can find a range. If we travelled from A to B along the edges of the cube, instead of in a straight line, we would travel a distance of 4 units, so AB must be less than 4. However, AB is definitely longer than the length of the edge of the cube, 2. So, the answer is between 2 and 4.
    • Look at the answer choices: We can eliminate (A) and (B), as they are less than 2. The other three answer choices are still possible.
    • Draw a diagram: Draw AB on the diagram.
    • Work backwards: We could find AB if we knew the distance from A to the vertex of the cube below B (let's call that vertex C, so the distance is AC), because AB would be the hypotenuse of a right triangle with legs AC and BC, and BC is 1. Label vertex C, and draw AC on the diagram. Now, we can find AC, because it in turn is the hypotenuse of a right triangle with legs 2 and 1. So, the length of AC is √1² + 2² = √5. Now, the length of AB is √1² + 5 = √6. Select (D) √6.
16. Don't get too anxious about the weird symbols (in this answer I've used {x} to denote them), just treat them as if they were functions and solve the equation:
    {a} = {a − 2}
    a² − a = (a − 2)² − (a − 2)
    a² − a = a² − 4a + 4 − a + 2
    4a = 6
    a = 3/2
    Select (C) 3/2.