# Solutions for Practice Test 9, The Official SAT Study Guide, Section 8

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SAT Practice Test Solutions:
2014–15 SAT Practice Test
2013–14 SAT Practice Test
The Official SAT Study Guide, second edition
• Practice Test 1: Sections 3, 7, 8.
• Practice Test 2: Sections 2, 5, 8.
• Practice Test 3: Sections 2, 5, 8.
• Practice Test 4: Sections 3, 6, 9.
• Practice Test 5: Sections 2, 4, 8.
• Practice Test 6: Sections 2, 4, 8.
• Practice Test 7: Sections 3, 7, 9.
• Practice Test 8: Sections 3, 7, 9.
• Practice Test 9: Sections 2, 5, 8.
• Practice Test 10: Sections 2, 5, 8.
SAT Math Tips

Here are solutions for section 8 of practice test #9 in The Official SAT Study Guide, second edition, found on pages 914–919. 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 6,700 = 100(6k + 7), then 6,700 = 600k + 700, or 6,000 = 600k. So, k = 10. Select (C) 10.
• Convert the sentence into an equation: Convert the first half of the sentence into an equation (to be precise, inequality):  3 3 more than + n n is a negative number < 0
• Convert the sentence into an equation: Similarly, convert the second half of the sentence into an inequality:  5 5 more than + n n is a positive number > 0
• So, we now have, 3 + n < 0 and 5 + n > 0. So, n has to be between −3 and −5 (but not −3 or −5).
• Look at the answer choices: The only answer between −3 and −5 is (B) −4. Select that answer.
2. Since the dotted lines bisect the angles with measures x° and y°, then z = ½x + ½y = ½(70) + ½(40) = 55. Select (E) 55.
3. Guess and check: Try each answer until you find one that does not fit:
• For (A), if there are 20 pieces of fruit in the basket, then there are 8 apples, so eliminate that answer.
• For (B), if there are 35 pieces of fruit, then there are 14 apples. Eliminate that answer.
• For (C), if there are 52 pieces of fruit, then there are 20.4 apples. It doesn't make sense to have a fractional number of apples, so this must be our answer. Select (C) 52.
• Draw a diagram: You may find it helpful to draw a square and an equilateral triangle to picture the problem.
• Estimate the answer: If the two shapes have equal perimeters, and the square has more sides thant the triangle, then the sides of the triangle must be a bit bigger than 3, perhaps 4 or so.
• Look at the answer choices: We can definitely eliminate answers (A) and (B). (C) seems the best guess, but it's probably a good idea to spend a bit more time confirming it.
• If the square has sides of length 3, its perimeter is 12. If the equilateral triangle has a perimeter of 12, then each side must be 12 ÷ 3 = 4. Select (C) 4.
• If x = −1, then x, x³, and x5 will equal −1, so the expressions containing these terms will all be negative. We can eliminate answers (A), (C), and (E).
• Answer (B) is equal to 4k(−1)² = 4k. Answer (D) is equal to 8k(−1)4 = 8k. As k is positive, (D) has the greater value. Select (D) 8kx4.
• If Josephine cycles the fastest, then the steepest part of the graph should be in the middle. Using that, we can eliminate choices (A), (C), and (D).
• Since Josephine runs faster than she swims, the third part of the graph should be steeper than the first part of the graph. We can eliminate choice (B). Select (E).
• Draw a diagram: You may find it useful to draw a graph of the situation.
• When x = √6, then, from the first equation, y = 6 − 7 = −1. So, k = −1.
• Substitute x = √6, y = −1 into the second equation:
−1 = −6 + j
j = 5
Select (A) 5.
4. Guess and check: Starting with the smallest value, check to see whether it makes the equation true. For (A), |2 − 4| = 2, which is < 3. Select (A) 4. Note: If it turned out the smallest value didn't work, you should try the largest value next.
• Solution 1:
• Estimate the answer: The angle labelled x° appears to definitely be more than 60° and less than 90°. Perhaps it it somewhere around 75° or so.
• Look at the answer choices: There are a few answers close to 75°, including (C) 72° and (D) 80°. We can definitely eliminate answer (A), though.
• Draw a diagram: Draw lines from one vertex to two opposite vertices.
• The lines that were drawn divide the pentagon into three triangles. Since, from the Reference Information, the sum of the measures in degrees of the angles of a triangle is 180, and the pentagon is made up of three triangles, the sum of the measures of degrees of the angles of the pentagon is 540°.
• As the pentagon has five interior angles, and the question tells us that all interior angles of that pentagon are congruent, then each angle measures 540/5 = 108°.
• Since x is the supplement of a 108° angle, it measures 180 − 108 = 72°. Select (C) 72.
• Solution 2: You might recall that the sum of the interior angles of a polygon with n sides is 180(n − 2). So, the sum of the interior angles of the pentagon in the diagram, which has 5 sides, is 180(5 − 2) = 180 · 3 = 540. Since the question states that all interior angles of the polygon are congruent, then each angle must be 540/5 = 108 degrees. Now, since the angle labelled x° is supplementary with one of the angles of the pentagon, the measure of x is 180 − 108 = 72. Select (C) 72.
• Estimate the answer: If the length of the drawing of the tool is 3/8 of its actual length, and the drawing is 6 inches long, then the actual length of the tool will be more than twice that length and less than three times that length (keep in mind that 3/8 is between ½ and ⅓). So, the length will be somewhere between 12 and 18 inches.
• Look at the answer choices: The only answer between 12 and 18 is (C) 16. Select that answer.
• Solution 1: Say that (x + 3)/2 = y, where y is an integer. Then:
x + 3 = 2y
x = 3 + 2y
Now, since y is an integer, 2y is an even integer. So, x must be 3 more than an even integer. An integer 3 more than an even integer is an odd integer. Select (E) an odd integer.
• Solution 2: Guess and check: Select values for x such that (x + 3)/2 is not an integer, and eliminate all answers that correspond to the value chosen. To start with, if x = 2, (x + 3)/2 is not an integer. So, we can eliminate (B) a positive integer and (D) an even integer. (x + 3)/2 is not an integer if x = 6, so we can eliminate (C) a multiple of 3. Finally, (x + 3)/2 is not an integer if x = −2, so we can eliminate (A) a negative integer. Select (E) an odd integer.
• Draw a diagram: Draw line QS on the diagram.
• Estimate the answer: For comparison purposes, draw line PR on the diagram. Line QS appears to have half the slop of line PR. The slope of PR is (10 − 6)/(11 − 3) = ½, so the slope of QS appears to be around ¼.
• Look at the answer choices: The only answer that makes sense based on our estimate is (B) ¼. Select that answer.
5. If p is a factor of n + 3, then kp = n + 3, for some k. Now, n + 10 = n + 3 + 7 = kp + 7. Now, since p is a factor of n + 10, p is a factor of kp + 7. Since p is a factor of kp, then p must be a factor of 7 if p is a factor of kp + 7. If p is a factor of 7, then p = 1 or p = 7, since 7 is prime. We are told that p is greater than 1, so p = 7. Select (B) 7.
• Draw a diagram: Note that YB < YA < YC < YD and YC = YE. Mentally rotate all of the angles onto a plane and draw a diagram as follows:
• From the diagram, it's not too hard to see that angle D is smallest. Select (D) ∠XDY.
6. When you see a problem like this, it's usually easiest to try to turn the expression that you have to evaluate into something resembling the givens. We can do that by factoring the expression:
x²yxy²
= xy(xy)
= 7(5)
= 35
Select (D) 35.