SAT Timed Section*: Math

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1 SAT Timed Section*: Math *These practice questions are designed to be taken within the specified time period without interruption in order to simulate an actual SAT section as much as possible.

2 Time -- 5 Minutes 0 Questions Directions: For this section, solve each problem and decide which is the best of the choices given. Fill in the corresponding circle on the answer sheet. You may use any of the available space for scratchwork. Notes: 1. The use of a calculator is permitted.. All numbers used are real numbers. 3. Figures that accompany problems in this test are intended to provide information useful in solving the problems. They are drawn as accurately as possible EXCEPT when it is stated in a specific problem that a figure is not drawn to scale. All figures lie in a plane unless otherwise indicated. 4. Unless otherwise specified, the domain of any function f is assumed to be the set of all real numbers x for which f(x) is a real number. 1. If 3x + 7 = 1, what is the value of 6x 5? (A) 5 (B) 10 (C) 1 (D) 17 (E) 19. There are eight sections of seats in an auditorium. Each section contains at least 300 seats but fewer than 400 seats. Which of the following could be the number of seats in the auditorium? (A) 1,600 (B),000 (C),00 (D),600 (E) 3,00 3. The points A, B, C, D, and E lie at ( 4, 0), (, 0), (, 0), (0, 4), and (0, 5) respectively. Which of the following line segments has the greatest length? (A) AD (B) BD (C) AE (D) AC (E) CE 4. In the figure, MO LN, LO =, MO = ON, and LM = 4. What is MN? N O L (A) 6 (B) 3 (C) 3 (D) 3 3 (E) 5 M 4

3 5. The average (arithmetic mean) of x and y is 5, the average of x and z is 8, and the average of y and z is 11. What is the value of z? (A) (B) 5 (C) 7 (D) 14 (E) 8 8. During the game, the green team scored one-eighth of its points in the first quarter, one-third in the second quarter, one-fourth in the third quarter, and the remaining points in the fourth quarter. If its total score for the game was 48, how many points did the green team score in the fourth quarter? (A) 18 (B) 14 (C) 1 (D) 10 (E) 7 9. If 3 3x = 81 x 4, what is the value of x? 6. In the figure above, each square is tangent to the containing circle at only one point. If the area of each square is x, what is the area of the shaded region in terms of x? (A) (π )x (B) (π 4)x (C) (4 π)x (D) (π 1)x (E) (π )x (A) -4 (B) - (C) 4 (D) 1 (E) If 7 less than 4 times a certain number is 8 more than the number, what is the number? (A) -11 (B) -5 (C) 3 (D) 5 (E) 5 7. If rstv = 1 and stuv < 0, which of the following must be true? (A) r > 0 (B) s < 1 (C) t < 0 (D) u 0 (E) v 1

4 3, 7, 7 x The first term in the sequence of numbers shown above is 3. Each even numbered term is 4 more than the previous term, and each odd-numbered term after the first is -1 times the previous term. For example, the second term is 3 + 4, and the third term is ( 1) 7. What is the 155 th term of the sequence? (A) -7 (B) -3 (C) 1 (D) 3 (E) In the figure above, what is x? (A) 40 (B) 50 (C) 60 (D) 75 (E) 90 x f(x) -a b a -b a c 1. The table above shows some of the values for the function ƒ. If ƒ is a linear function, what is the value of the x- intercept in terms of a, b, and c? (A) a (B) a c (C) a b (D) b c (E) In the xy-plane, the equation of the line l is y = 3(x + ) + 4. If the line m is the reflection of line l in the y-axis, what is the equation of the line m? (A) y = 3(x ) 4 (B) y = 3(x ) + 4 (C) y = 3(x ) + 4 (D) y = 3(x + ) 4 (E) y = 3( x) The number x + 8 is how much greater than x? (A) 6 (B) 10 (C) x 10 (D) x 6 (E) x + 6

5 A C B (8, k) 16. In the figure above, if AB = 10, what is the value of k? (A) 6 (B) 8 (C) 10 (D) 1 (E) If b + (x 4) = s, what is x + in terms of s and b? (A) s b+1 (B) x b+6 (C) 1 s+b (D) s b (E) b s 19. Twice the larger of two numbers is three more than five times the smaller, and the sum of four times the larger and three times the smaller is 71. What is the larger number? (A) 5 (B) 7 (C) 9 (D) 14 (E) The set B consists of all even integers between 34 and m. If the sum of these integers is 74, what is the value of m? (A) 19 (B) 36 (C) 37 (D) 38 (E) 40 STOP 18. Sarah has at least one quarter, one dime, one nickel, and one penny. If she has three times as many pennies as nickels, the same number of nickels as dimes, and twice as many dimes as quarters, what is the least amount of money she could have? (A) $0.41 (B) $0.48 (C) $0.58 (D) $0.61 (E) $0.71

6 SAT Math Timed Section: Answers and Explanations Answers 1. A. D 3. C 4. A 5. D 6. E 7. D 8. B 9. E 10. D 11. E 1. E 13. A 14. B 15. B 16. A 17. A 18. D 19. D 0. A Explanations 1. A. Use the first equation to solve for x: 3x + 7 = 1. Thus, 3x = 5 and x = 5. Substitute this value 3 for x into the second equation: 6x 5 = 6( 5 ) 5 = 10 5 = D. There are 8 sections with at least 300 seats, therefore, at minimum there are seats, or,400 seats. Based on this, you can eliminate choices A, B and C. The question stem also tells us that each section must have fewer than 400 seats, so at most, each section could have 399 seats. Thus, 8 399, or 3,19 seats is the maximum. Choice D is the only one that provides an option between,400 and 3,19, inclusive. Written as an inequality, this would look something like this:,400 x 3,19. Choice E, 3,00 seats would be an option if the question was worded slightly differently, allowing each section to have a maximum of 400 seats. 3. C. To solve this question, use the distance formula: (x x 1 ) + (y y 1 ). Applying it to all the answer choices, you can determine the lengths of all the line segments and find that segment AE (choice C) is the longest. 4. A. First, use the Pythagorean Theorem to find OM: 4 = + x 16 = 4 + x 1 = x x = 1 = 3 Since OM = ON, ON = 3. From here, find MN using the Pythagorean Theorem: MN = ( 3) + ( 3) MN = = 4 MN = 4 = 6 5. D. This question requires a basic understanding of averages. First, do some translation. If the average of x and y is 5, you can translate this into the following equation: x+y = 5 Eliminate the fraction by multiplying both sides by : x + y = 10 Then, do the same for the other two pieces of information you are given: 6

7 x+z y+z = 8 x + z = 16 = 11 y + z = Now that you have 3 simple expressions and 3 variables (x, y and z), you can isolate the variables and solve for z. The fastest way is to put y and z in terms of x: y = 10 x z = 16 x y + z = (10 x) + (16 x) = 6 x = 4 = x x = Now that we have x, we can solve for z: z = 16 = 14 Thus, the answer is D. But, notice that they also include the value of x (choice A), so despite this being a relatively straightforward question, it can still be easy to miss it if you forget what it is you are looking for! 6. E. This question is all about figuring out what information you need to solve the problem. First, the question wants to know the area of the shaded region in terms of x. The area of the shaded region is equal to the area of the circle minus the area of the big square (the area created by all four smaller squares). We already know the second part of the puzzle (the area of the big square), since they tell us the area of each individual square = x. Thus, the area of the big square is 4x. Now we just need to figure out the area of the circle. The area of a circle is calculated by the formula πr. So, we need to figure out the radius of the circle! The radius is the length of any line drawn from any point on the outside of the circle to the center. If you draw the diagonal of any of the squares, you ll see that it is equal to the radius of the circle. Let s take a look: When you draw a diagonal inside a square you divide it into two right triangles. You can use the Pythagorean Theorem r x to find the length of the hypotenuse (the diagonal, which is also the circle s radius). This tells us that x + x = r. x From there, we can plug in (x + x OR x ) into the formula for the area of a circle in place of r : πr = πx Going back to the original formula we put together (the area of the circle minus the area of the larger square), we figure out that the area of the shaded region: πx 4x OR x(π ) 7. D. The only thing we know for sure about these two relationships is that they do not equal zero. Therefore, r, s, t, u and v cannot equal zero. The correct choice is D. 8. B. We know the total number of points scored in the game (48). Thus, if they scored one-eighth in the first quarter, they scored 6 points in the first quarter ( = 6). Likewise, if they scored onethird of their points in the second quarter, they scored 16 points in the second quarter. And if one-

8 fourth were scored in the third quarter, 1 points were scored in the third quarter. This means that 34 points ( = 34) were scored in the first three quarters. Since we want to know how many points the team scored in the fourth quarter, we just subtract 34 from 48, the total number of points scored in the game = 14, choice B. 9. E. To solve for x, you ll need to manipulate the expression so that the bases are the same. Since 81 = 3 4, you can replace 81 in the equation, making all the bases the same (3). Thus the equation becomes: 3 3x = (3 4 ) x 4 3 3x = 3 4x 16 At this point, you can drop the bases and just work with what s in the exponent part: 3x = 4x 16 x = D. This is really a direct translation problem. Say your number is x, we just translate everything in the problem. Seven less than 4 times a certain number becomes: 4x 7. And 8 more than the number x + 8. Is translates to equal to so we set these two expressions equal to each other: 4x 7 = x + 8 3x = 15 x = E. For this problem, it s best to just fill in the diagram with all the information you can infer from the information you are given. In this case, the 80 angle measurement you are given is a bit of a red herring. You don t need it to solve the problem. If you recognize the triangle shape we ve outlined in red below, you can see that third angle in the triangle must equal 90 degrees since all three interior angles in a triangle must add up to 180. This also means that x = 90 because it is x and the 90 degree angle of the triangle are vertical angles. x

9 1. E. Since f(a) = b and f( a) = b, the graph of the function passes through the origin (0, 0) which happens to be the midpoint between the points (a, b) and ( a, b). Thus, the x-intercept is the value where f(x) = 0. That value is x = 0, since it satisfies the function. 13. A. When you are given a series like this, it s always a good idea to follow the directions and write out the first few numbers in it to see if you can recognize a pattern. In this case, if you write out the first 5 terms, you end up with: 1 st term: 3 nd term: 7 3 rd term: -7 4 th term: -3 5 th term: 3 At this point, you are back where you started from. Therefore, you have a loop consisting of 4 terms that repeats over and over. To find the 155 th term, divide 155 by 4 and you ll see that this will be loops and the 155 th term would be the 3 rd term in the loop, or B. Based on the definition for a reflection about the y-axis, for all points, (x, f(x)), the reflected graph will contain points ( x, f(x)). Thus, a reflection about the y-axis is equivalent to a translation of a point by x units where the line y = 0 is the bisector of the distance between the original graph, the vertex is at y = 4 and the graph is shifted right units. In the reflected graph, the vertex occurs at y = 4 but the graph is shifted left units to give the equation y = 3(x ) + 4, which is choice B. 15. B. To determine the difference, subtract x from x + 8: (x + 8) (x ) = A. If you draw in a perpendicular line from point B to the x-axis, you will create a right triangle whose hypotenuse is 10 (formed by AB) and whose base is 8. To find k, which is the length of the last leg, use the Pythagorean Theorem: 8 + k = 10. k = 36 and k = 36 = 6. B (8, k) 10 k A 8 C 17. A. b + (x 4) = s. First, solve for x: b + x 8 = s x = s b + 8 x = s b+8 x + = s b+8 + = s b = s b+1

10 18. D. Let Q = D, D = N and N = 3P where Q is the number of quarters, D is the number of dimes, N is the number of nickels and P is the number of pennies. If Sarah has one quarter, then she has dimes, nickels and 6 pennies. The total of the least amount of money Sarah can have is $0.61: 1 Q = $0.5 D = $0.0 N = $ P = $ D. Let x is the larger number and let y equal the small number. The relationships translated from the text are: x = 5y + 3 and 4x + 3y = 71. Solve for x in the first equation, then plug that value into the second equation: x = 5y + 3 x = 5y+3 4 5y+3 + 3y = 71 10y y = 71 13y = 65 y = 5 Now plug y = 5 into the first equation: x = 5(5) + 3 = 8 x = A. For the sum to add up to +74, you have to move from -34 in the positive direction. Thus, the sum of all the negative even integers will be negative. You must have the same values on the positive end to cancel out the negative values. Thus, for a -34, there must be a +34. As a result, from -34 to +34, the sum is zero. The next two even positive integers are 36 and 38, whose sum is 74. Therefore, m = 38 and m = 19.

Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics

Math 2 Coordinate Geometry Part 3 Inequalities & Quadratics 1 DISTANCE BETWEEN TWO POINTS - REVIEW To find the distance between two points, use the Pythagorean theorem. The difference between x 1 and x