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
- Practice Test 2: Sections
- Practice Test 3: Sections
- Practice Test 4: Sections
- Practice Test 5: Sections
- Practice Test 6: Sections
- Practice Test 7: Sections
- Practice Test 8: Sections
- Practice Test 9: Sections
- Practice Test 10: Sections
Here are solutions for section 6 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.
- Solution 1: List the first 12 terms of the sequence out:
4, 11, 18, 25, 32, 39, 46, 53, 60, 67, 74, 81
Select (B) 81.
Solution 2: Look for a pattern: To get the second term of the sequence, you would start with 4 and add 7. To get the third term of the sequence, you would start with 4 and add 7 twice. To get the fourth term of the sequence, you would start with 4 and add 7 three times. Carrying on this pattern, to find the twelfth term you would start with 4 and add 7 11 times. So, the twelfth term would be 4 + 7 × 11 = 81. Select (B) 81.
Solution 3: You might know that the formula for the nth term of an arithmetic sequence is tn = a + (n − 1)d, where a is the first term and d is the difference between the terms. In the given sequence, the first term is 4 and the difference between terms is 7. So:
t12 = 4 + 11 × 7 = 81
Select (B) 81.
- Solution 1:
Solution 2: Guess and check: Try each of the answer choices as a value for x until you find one that makes the equation true. You will eventually find that the only one that works is (E) 9. Select that answer.
- Draw a diagram: On your graphing calculator, plot
y = (x − 2)
- Examine the graph that you've drawn to see where it meets the x-axis. It does so at −5 and 9. −5 is not one of the answer choices, but 9 is. Select (E) 9.
- Solution 1: If you're reasonably good at math, your instincts may tell you that the answer's obvious: It's 15. Since this is question 3, it's probably reasonable to trust your instincts. Select (B) 15 and move on without spending more time.
Solution 2: Try a special case: Say that each of t and y is 15 (so their average is 15) and that each of w and x is 15 (so their average is 15). The average of t, y, w, and x is ¼(15 + 15 + 15 + 15) = 15. Select (B) 15.
- Solution 1: Look at the answer choices: Look at the answers, read them through carefully, and eliminate any that can't be correct:
The only remaining choice is (A). Select (A) If Fred cannot swim, then he is not Kay's brother.
- There doesn't seem to be anything obviously wrong with (A).
- If Dave can swim, he might possibly be Kay's brother, so (B) is incorrect.
- As for (C), there are people in the world who can swim who are not Kay's brother, so this is incorrect.
- (D) is obviously incorrect; it contradicts the given statement.
- The given statement doesn't tell us anything about people who are not Kay's brothers; there are billions of them, and some can swim and some can't, so (E) is incorrect.
- Estimate the answer: We can assume that the diagram is drawn to scale. Since AC is the diameter of the circle, ∠ABC is a right angle according to Thales' theorem (or, if you don't remember that, it's fine for our estimate to eyeball angle ABC and see that it appears to be a right angle, or fairly close). Now, ∠ABO is definitely less than ∠ABC, but it's definitely more than half of ∠ABC. So, our answer must be greater than 45° and less than 90°.
- Look at the answer choices: The only answer choice greater than 45° and less than 90° is (D) 60°. Select that answer.
Solution 3: We are told that AB = AO. Now, AO = BO, since both line segments are radii of the circle. So, all three sides of the triangle are equal; it must be equilateral. So, each angle of the triangle, including ∠ABO must measure 180° ÷ 3 = 60°. Select (D) 60°.
- Estimate the answer: We can assume that the diagram is drawn to scale. Triangle ABO looks like an equilateral triangle. Each angle of an equilateral triangle is 60°, so the answer must lie somewhere around 60°.
- Look at the answer choices: The only answer choice that is close to 60° is (D) 60°. Select that answer.
- Look at the answer choices: See whether any choices stand out. Looking at (A) and (B), it isn't too hard to tell that they are different expressions, so one of the two must be the answer. Eliminate (C), (D), and (E).
- Expand the given expression and both remaining answer choices:
Choice (A) doesn't match. Select it.
- The answer choice expands to ac + ak/b.
- Choice (A) expands to ac/b + ak/b.
- Choice (B) expands to ac + ak/b.
- Try a special case: Say that a = 1, b = 2, c = 3, and k = 4. Then (a/b)(bc + k) = ½(6 + 4) = 5.
- Look at the answer choices: Evaluate each answer choice when a = 1, b = 2, c = 3, and k = 4. If one answer does not evaluate to 5, that is the answer. (A) evaluates to 1((3 + 4)/2) = 7/2 ≠ 5. Select (A).
- Draw a diagram: We're already given a diagram, but it isn't labelled well. We are given the values of r, s, t, u, and w, so write those in in the correct spaces in the diagram.
- Work backwards:
- We would know the value of x if we knew the other two angles in the triangle because "The sum of the measures in degrees of the angles of a triangle is 180" (see the Reference Information if you've forgot).
- We already have one of the other two angles in the triangle; we're just missing ∠EPC.
- We could find ∠EPC if we knew ∠DPF, because those two angles are vertical angles and so are equal.
- We could find ∠DPF if we knew ∠APD and ∠BPF. These three angles add to 180° because ∠APB is a straight angle.
- We can find both ∠APD and ∠BPF right now, because we know the other two angles in the triangles these angles are in, and the angles of a triangle add to 180°.
- Work forwards: Now that we know how to find the answer, we'll now work in the opposite direction, forwards, from what we know:
- The missing angle in triangle BPF must be 180 − 50 − 60 = 70°. The missing angle in triangle APD must be 180 − 45 − 50 = 85°. Fill these in on the diagram as well.
- Now, ∠APB is a straight angle, so it measures 180°. So, ∠DPF = 180° − ∠APD − ∠FPD = 180 − 85 − 70 = 25°.
- Now, ∠DPF = ∠EPC = 25° since they are vertical angles.
- Finally, x = 180 − 90 − 25 = 65. Select (C) 65.
- Look at the diagram to find where the curve is higher than the straight line. The only places it is higher are between −3 and 0 (but not including the endpoints). Select (B) −3 < x < 0 only.
- This question isn't too difficult if you tackle it in an organized fashion:
|4 magazines × 12 issues per year =||48
|2 magazines × 4 issues per year =||8
|1 magazine × 52 issues per year =||52
- Convert the sentence into an equation:
This results in:
|is equal to||=
3 + 2x = 4
- Solving the equation, we get:
3 + 2x = 4
2x = 1
x = ½
- Read the question carefully and determine what it is asking: You are given a table with total sales figures at the end of each week. You are asked to find how many copies are sold during the third week. You can find this by subtracting the total sales after the third week from the total sales after the second week.
- Substituting the appropriate numbers from the table, the answer is 6800-5500=1300.
This question isn't too hard, but all of the fractions can be confusing. You may find it easier to write the first equation as
j ÷ k = 32
Since k = 3/2:
j ÷ (3/2) = 32
Multiply both sides by 3/2:
j = 48
Multiply both sides by ½:
½j = 24
You don't have to do this, but you might want to substitute k for x + y and then solve for k:
k + 3z = 600
To get rid of z, subtract 3 × the second equation from the first equation:
k + z = 400
And so k = 300. Alternately, you can skip the substitution and solve for x + y directly using the exact same method as above.
|3k + 3z = 1200
|−||k + 3z = 600
|2k = 600
- Draw a diagram: Draw 25 trays:
- Draw a cup on 15 of the trays, starting from the left:
- Now, draw a plate on 21 of the trays. Since every tray must have something on it, start from the right this time. This gives you the following:
- Eleven of the trays have both a cup and a plate. Enter 11.
- Solution 1:
Solution 2: You've forgotten that the slope of the perpendicular is the negative reciprocal? Or do you hate coordinate geometry but love geometry? No sweat. Here's what to do:
- Draw a diagram: Draw line m perpendicular to line l and passing through the origin.
- Estimate the answer: The slope of m doesn't appear to be too high. The answer is probably around 1, or maybe a bit less.
- Find the slope of l. Slope = rise/run. The rise is −3 and the run is 2, so the slope is −3/2.
- The slope of a line perpendicular to l is the negative reciprocal, or 2/3.
- Convert the problem into a geometry problem: Line l makes a right triangle with the coordinate axes. Draw that, and then draw line m. You'll get the following:
- If you look at the diagram, you'll notice that the big right triangle and the two smaller right triangles have the same three angles, so they are all similar to each other. So, the legs of each of the three right triangles in the diagram are in the ratio 2:3.
- Now, the question asks us to find the slope, which equals rise/run. Draw lines representing the rise and the run in on the diagram. I've labelled these lines a (rise) and b (run):
- The triangle formed with lines a, b, and m is a right triangle. Not only that, but its angles are the same as those of the other triangles, so it is similar to them. So, a:b = 2:3. Enter 2/3.
- First, find the largest prime number less than 50. 49 is not a prime number, because 49 = 7 × 7. 47 is a prime number though. Next, find the smallest prime number larger than 50. 51 isn't prime; 51 = 3 × 17. 53 is prime, though. 53×47 = 2491.
- Estimate the answer: The diagram is drawn to scale, so use it to make an estimate. The area of triangle RST is 7. If we draw a vertical line from T to a point, say X, on QR, we have a rectangle, RSTX, whose area must be 2 × 7 = 14. Looking at the diagram, RSTX is obviously more than half the figure, so the figure's area must be less than 2 × 14 = 28, but it must be more than 14. You might guess that the answer is somewhere around 23.
- Try a special case: Say that PS = 5. Then, PT = 2 and TS = 3. Since the area of RST = 7 = ½bh, and the base is 3, the height RS must be 14/3. So, the two sides of the rectangle are 5 and 14/3, so the area must be 5 × 14/3 = 70/3. You don't have to convert this to a decimal, just enter 70/3.
- Look back: You might want to convert 70/3 to a decimal to ensure that it's reasonably close to your estimate. As it turns out, 70/3 = 23.333..., so you can be confident that it is the correct answer.