# SAT Math Medium Practice Quiz (A) 3 (B) 4 (C) 5 (D) 6 (E) 7

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1 SAT Math Medium Practice Quiz Numbers and Operations 1. How many positive integers less than 100 are divisible by 3, 5, and 7? (A) None (B) One (C) Two (D) Three (E) Four 2. Twice an integer is added to the square of the integer. Net, the result is increased by 1. Which of the following could be the value of the final sum? 5. A positive integer is subtracted from its square. The units digit of the result could be which of the following? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7 6. A three-digit number, XYZ, is formed of three non-zero digits X, Y, and Z. If the value of X is twice the value of Z, and the value of Y is three times the value of X, what is the number XYZ? (A) 36 (B) 41 (C) 45 (D) 48 (E) percent of 2 12 is equal to which of the following? (A) What percent of the non-negative integers less than 30 are prime? 4. In a particular card game, the minimum score a player can achieve in a single game is 20, and the maimum score possible in a single game is 52. If a player plays three games and scores a total of 141 points, what is the least number of points that the player could have scored in one of the games? (A) 35 (B) 36 (C) 37 (D) 38 (E) 39 (B) 6 12 (C) (D) (E) Amy can walk 3600 feet in 10 minutes. Walking at the same rate, how many yards can Amy walk in 10 seconds? (1 yard = 3 feet.) erikthered.com/tutor pg. 1

2 SAT Math Medium Practice Quiz 9. A standard die is a cube with one to si dots on each face. The first face has one dot, the second face has two dots, and so forth. The dots are arranged so that the total number of dots on each pair of opposite faces is 7. All of the following could be views of a standard die EXCEPT 11. When a particular non-zero integer is subtracted from its square, the result is neither negative nor positive. What is the integer? (A) (B) 12. If p is a prime number, which of the following could also be a prime number? (A) p 2 (B) 2p (C) (C) p 2 (D) p+2 (E) 2p+2 (D) m, 2m, 4m,... (E) 13. The first term in the sequence above is m, and each term thereafter is equal to twice the previous term. If m is an integer, which of the following could NOT be the sum of the first four terms of this sequence? 10. How many even integers greater than two and less then twenty are equal in value to the sum of two prime numbers? (A) 25 (B) 15 (C) 45 (D) 75 (E) 120 (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 erikthered.com/tutor pg. 2

3 SAT Math Medium Practice Quiz Algebra and Functions 5 y g 1. Suppose that n is an integer such that n is 12 greater 3 than n. Which of the following is the value of n? f (A) 72 (B) 36 (C) 24 (D) 24 (E) O If k is a negative number and (2k)2 8 value of k? (A) 4 (B) 3 (C) 2 (D) 1 (E) 0 + k 2 = 1, what is the 5. The graphs of the functions f and g are lines, as shown above. If a = f(4), what is the value of g(a)? (A) 1.5 (B) 2 (C) 3 (D) 4 (E) If =, where 0, then which of the following 3. Let f() = (+3)( 3) for any value of. What is the least possible value of f()? could be the value of? (A) 1 (A) 9 (B) 3 (C) 0 (D) 3 (E) 9 (B) 1 2 (C) 1 2 (D) 1 (E) 2 4. Let a, b, and c be non-zero numbers such that c is 24 greater than b, and b is 24 greater than a. If c = 3, then a what is the value of b? (A) 48 (B) 24 (C) 24 (D) 48 (E) = 2 7. In the equation above, what is the value of? erikthered.com/tutor pg. 3

4 SAT Math Medium Practice Quiz 8. If, y, and z are positive numbers, and is y% of z, then which of the following must be equal in value to y? (A) (B) (C) (D) (E) z 100z 100 z z 100z 11. If y = 18, where and y are positive integers, 0.2 then what is the value of? (A) 1 (B) 2 (C) 3 (D) 4 (E) When 1 is divided by the reciprocal of a particular number, the result is 8 less than the number. The number 3 is which of the following? 9. If = a+ b,thenwhichofthefollowingisequivalent to? (A) a+b+2 ab (B) a+b+2ab (C) a+b+ab (D) a+b (E) 2a+2b (A) 3 (B) 6 (C) 8 (D) 9 (E) If (+1) 2 ( 1) 2 = 32, what is the value of? = 4m 2 y = (2m+1)(2m 1) 10. Given the two equations above, what is y in terms of? (A) 1 (B) +1 (C) (A) 4 (B) 6 (C) 8 (D) 10 (E) If y < 82 and y is a positive multiple of 3, what is the greatest possible integer value of? (A) 9 (B) 13 (C) 27 (D) 30 (E) 41 (D) 2 1 (E) 2 +1 erikthered.com/tutor pg. 4

5 SAT Math Medium Practice Quiz 15. For two particular numbers and y, the values of y and 2 y are consecutive odd integers, respectively. What is the value of? (A) 1 f() = 2+1 g() = The two equations above define the functions f and g. If f(c) = g(c) for some number c, what is the value of c? (B) 1 2 (C) 0 (D) 1 (E) 2 erikthered.com/tutor pg. 5

6 SAT Math Medium Practice Quiz Geometry 1. The length of a particular rectangle is 4 greater than the width of the rectangle. If the perimeter of the rectangle is 16, what is the area of the rectangle? A O B (A) 2 (B) 6 (C) 8 (D) 12 (E) Semicircular arcs ÂO and ÔB divide the circle above with center O into two regions. If the length of diameter AB is 12, what is the area of the shaded region? 2. The area of square B is 25% greater than the area of square A. If the area of square B is 80 square inches, then what is the length in inches of the perimeter of square A? (A) 6π (B) 9π (C) 18π (D) 36π (E) 72π (A) 8 (B) 12 (C) 16 (D) 32 (E) 64 y 70 l Note: Figure not drawn to scale. B C 5. In the figure above, two line segments meet at a point on line l. If the value of y is equal to the square of the value of, what is the value of y? 2 A 30 D 3. In the figure above, ABCD is a rectangle. What is the area of ABCD? 6. A particular square has a side of length s. If the area of the square is numerically equal to the perimeter of the square, then what is the value of s? (A) 2 (B) 3 (C) 6 (D) 2 2 (E) 2 3 erikthered.com/tutor pg. 6

7 7. The volume of a particular cube is numerically equal to the total area of its faces. What is the length of one edge of the cube? SAT Math Medium Practice Quiz A B C 8. In the -y plane, the line 3 + 2y = 72 intersects the -ais at = b, for some number b. What is the value of b? (A) 0 (B) 6 (C) 12 (D) 24 (E) In the figure above, AC is a diameter of the large circle and B lies on AC so that AB is a diameter of the small circle. If AB = 1 and BC = 2, what is the area of the shaded region? (A) π 4 (B) π B C (C) 2π (D) 9π 4 (E) 9π E 120 A D 9. In thefigureabove,abcd isasquareandtriangleaed is isosceles. If AE = 4, what is the area of the square? (A) 4 3 (B) 16 (C) 16 3 (D) 32 (E) 48 erikthered.com/tutor pg. 7

8 SAT Math Medium Practice Quiz Data, Statistics, and Probability { 1, 2, 3, 4, 5, 6, 7 } 1. Set M is shown above. Each number in set N is generated by dividing each number in set M by 2. What is the average (arithmetic mean) of the numbers in set N? (A) 1 (B) (C) The median of a particular set of 10 integers is itself an integer. Which of the following could be true? I. The integers of the set are all identical. II. The integers of the set are consecutive. III. The 5 th largest and 6 th largest integers of the set are odd. (A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III (D) 2 (E) 4 2. A family is going to choose two pets at random from among a group of four animals: a cat, a dog, a bird, and a lizard. What is the probability that the pets that the family chooses will include the lizard? 3. If is chosen at random from the set {2, 3, 4, 5, 6}, and y is chosen at random from the set {11, 13, 15}, what is the probability that y is even? erikthered.com/tutor pg. 8

11 SAT Math Medium Practice Quiz Answers Algebra and Functions 1. A (Estimated Difficulty Level: 3) First, convert the words into an equation: n 3 = n From here, you could work with the answers by substituting them into the equation until it works. Or, use algebra to solve the equation: n 3 n 2 = 2n 6 3n 6 = n 6 = 12 so that n = 72. (You could also solve the equation by multiplying through by 6 first to get rid of the fractions.) 3. A (Estimated Difficulty Level: 3) It is really helpful to recognize that ( + 3)( 3) = 2 9. The equation 2 y 2 = (+y)( y) is called the difference of two squares and is useful to memorize for the SAT. Since f() = 2 9 and the least possible value of 2 is zero, then the least possible value of f() is 0 9 = 9, so answer A is correct. 4. D (Estimated Difficulty Level: 3) Since c is 24 greater than b, and b is 24 greater than a, c must be 48 greater than a (c = a + 48). Also, c/a = 3, which means that c = 3a. We substitute to get 3a = a + 48 so that a = 24, which means that b = a+24 = 48. If the algebra was too tricky, you also could have used the answers by plugging them in for b, solving for a and c, and then checking to see if c/a = C (Estimated Difficulty Level: 3) Work with the answers, plugging them in for k until the equation works. You will see that when k = 2, the left-hand side of the equation becomes: ( 4) 2 8 so that answer C is correct = = 2 1 = 1 Or, use algebra to solve the equation: (2k) k 2 = k2 2 + k 2 = 1. Multiplybothsidesby2andputeverythingononeside: Factor this to get: k 2 +k 2 = 0. (k +2)(k 1) = 0, so that k = 2 since we need the negative solution. Whew! Working with the answers may have been easier. 5. C (Estimated Difficulty Level: 3) Use the graph to find f(4) as follows. First, go to = 4 on the -ais. Net, go vertically up until you reach the line for the function f. Finally, go over horizontally to find the y-value for that point (see the graph below) O y You should see that when = 4, the y-value of function f is 2. Since a = f(4), we know that a = 2 and we need to find g(2). Repeat the process, now using the line for the function g. You will see that g(2) = 3. g f erikthered.com/tutor pg. 11

13 SAT Math Medium Practice Quiz Answers 12. E (Estimated Difficulty Level: 3) Make an equation from the words. If is the number, then the reciprocal is 1/, so the equation is 1/3 1/ = C (Estimated Difficulty Level: 3) Since y is less than 82, to get the greatest possible integer value of, we need the least possible value of y. The least positive multiple of 3 is 3, so y = 3. The greatest integer value of such that 3 < 82 is = 27, so answer C is correct. This is the same as 3 = E (Estimated Difficulty Level: 3) Solving this equation gives = C (Estimated Difficulty Level: 3) Since (+1) 2 = and ( 1) 2 = 2 2+1, the equation becomes ( ) ( 2 2+1)= 32 so that 4 = 32 or = 8. Or, work with the answers by plugging them in for until the equation works. Consecutive odd integers always differ by 2. For eample, 3 and 5 are consecutive, odd integers and 5 3 = 2. Since y and 2 y are consecutive odd integers, we know that (2 y) ( y) = 2. Simplifying, you will get 2 y +y = 2 so that = 2, making answer E correct (Estimated Difficulty Level: 3) Since f(c) = 2c + 1 and g(c) = 3c 2, f(c) = g(c) is the same as 2c+1 = 3c 2, which means that c = 3. erikthered.com/tutor pg. 13

16 SAT Math Medium Practice Quiz Answers Data, Statistics, and Probability 1. D (Estimated Difficulty Level: 3) One way to do this is to find all the numbers in set N, and then calculate the average of those numbers. (Your calculator may be useful here.) Set N is: { 0.5, 1, 1.5, 2, 2.5, 3, 3.5 } so that the averageof the numbers in set N is 14/7 = 2. It may be easier and less error-prone to notice that if you divide a set of numbers by 2, the average of the set will get divided by 2 as well. Since the average of set M is 4, the average of set N is /5 or 0.6 (Estimated Difficulty Level: 3) There are 15 possible values of y: 2 11 = = = = andsoforth. Youshouldfindthat9oftheseareeven, so that the probability that y is even is 9/15 = 3/5 = 0.6. You may also notice that an even product will result only when you multiply by one of 2, 4, or 6, so that there will be 3 3 = 9 even numbers out of 3 5 = 15 numbers total. 4. D (Estimated Difficulty Level: 3+) 2. 1/2 or 0.5 (Estimated Difficulty Level: 3) Often, a good way to start probability problems on the SAT is to list all the possibilities, and then count the number that match the given criteria. Using c for cat, d for dog, and so forth, all the possible combinations of two pets are: cd cb cl db dl bl Since three of the si possible combinations include the lizard, the probability that the two pets chosen by the family include the lizard is 3/6 = 1/2 = 0.5. For choice I, suppose that the set is a group of ten 2 s. Themedian ofthisset is2, whichis aninteger, sochoice I could be true. For choice II, note that for a set of ten numbers, there is no single middle number, so the median is the average of the two middle numbers (the 5 th largest and 6 th largest). But the average of any two consecutive integers is never an integer, so choice II can never be true. (As an eample, suppose that the set is the integers from 1 to 10. The two middle numbers are 5 and 6 and the median is (5+6)/2 = 5.5.) This reasoning helps with choice III: suppose that the middle twonumbers(the 5 th largestand 6 th largest)are 7 and 11. Then the median is (7+11)/2 = 9, which is an integer, so choice III could be true, making answer D correct. erikthered.com/tutor pg. 16

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