SAT Practice Math Questions (Open-Ended)

66, 78, 75, 69, 78, 77, 70 The table above shows the temperatures, in degrees Fahrenheit, in a city in Hawaii over a one-week period. If m represents the median temperature, f represents the temperature that occurs most often, and a represents the average (arithmetic mean) of the seven temperatures, which of the following is the correct order of m, f, and a?

(A) a < m < f (B) a < f < m (C) m < a < f (D) m < f < a (E) a = m < f

The positive integer, n, is not divisible by 7. The remainder when n^2 and the remainder when n is divided by 7 are each equal to k. What is k? a) 1 b) 2 c) 4 d) 6 e) cannot be determined

1 Since the positive integer n leaves nonzero remainder k when divided by 7, it can be written as n = (7 times a) + k, where a is a nonnegative integer and k is equal to one of the values 1, 2, 3, 4, 5, or 6. Since n^2 leaves the same nonzero remainder k when divided by 7, it can be written as n = (7 times b) + k, where b is a nonnegative integer and k has the same value. It is also true that n^2 = ((7 times a) + k)^2 = (49 times a^2) + (14 times a times k) + k^2, which can be written as 7 times ((7 times a^2) + 2 times a times k) + k^2. Since n^2 = (7 times b) + k = 7 times ((7 times a^2) + 2 times a times k) + k^2, it follows that k and k^2 leave the same remainder when divided by 7. We know that k is equal to one of the values 1, 2, 3, 4, 5, or 6. We can see which of these values for k has the property that k and k^2 leave the same remainder when divided by 7. If k = 1, then k^2 = 1, which leaves remainder 1 when divided by 7. Thus, 1 is a possible value for k. But we are not done yet; since one of the answer choices is (E), It cannot be determined from the information given, we must continue and check 2, 3, 4, 5, and 6 as possible values for k. If k = 2, then k^2 = 4, which leaves remainder 4 not equal to 2 when divided by 7. If k = 3, then k^2 = 9, which leaves remainder 2 not equal to 3 when divided by 7. If k = 4, then k^2 = 16, which leaves remainder 2 not equal to 4 when divided by 7. If k = 5, then k^2 = 25, which leaves remainder 4 not equal to 5 when divided by 7. If k = 6, then k^2 = 36, which leaves remainder 1 not equal to 6 when divided by 7. Therefore, the only possible value of k is 1, which is choice (A).

What is the result when 436,921 is rounded to the nearest thousand and then expressed in scientific notation?

4.37 x 10^5

In the x y-coordinate plane above, line l contains the points (0 comma 0) and (1 comma 2). If line m (not shown) contains the point (0 comma 0) and is perpendicular to l, what is an equation of m? (A) y = -1/2x (B) y = -1/2x + 1 (C) y = -x (D) y = -x +2 (E) y = -2x

A) -1/2x

At the beginning of 2006, both Alan and Dave were taller than Boris, and Boris was taller than Charles. During the year, Alan grew 2 inches, Boris and Dave each grew 4 inches, and Charles grew 3 inches. Of the following, which could NOT have been true at the beginning of 2007? (A) Alan was shorter than Boris. (B) Alan was shorter than Charles. (C) Boris was shorter than Dave. (D) Dave was shorter than Alan. (E) Dave was shorter than Charles.

E) Consider the choices in turn. At the beginning of 2006, Alan was taller than Boris; during the year, Alan grew 2 inches and Boris grew 4 inches. Since Alan grew less than Boris, it is possible that Alan was shorter than Boris at the beginning of 2007. So choice (A) is not the correct answer. At the beginning of 2006, Alan was taller than Boris, who was taller than Charles; thus, Alan was taller than Charles. During the year, Alan grew 2 inches and Charles grew 3 inches. Since Alan grew less than Charles, it is possible that Alan was shorter than Charles at the beginning of 2007. So choice (B) is not the correct answer. At the beginning of 2006, Dave was taller than Boris; during the year, Boris and Dave each grew 4 inches. Thus Boris was still shorter than Dave at the beginning of 2007. So choice (C) is not the correct answer. At the beginning of 2006, Alan and Dave were each taller than Boris, but we cannot determine whether Alan was shorter than Dave or Dave was shorter than Alan, nor what the difference in their heights was. So even though Dave grew 4 inches during the year while Alan grew only 2 inches, it is possible that Dave was shorter than Alan at the beginning of 2007. So choice (D) is not the correct answer. At the beginning of 2006, Dave was taller than Boris, who was taller than Charles; thus, Dave was taller than Charles. During the year, Dave grew 4 inches and Charles grew only 3 inches, so Dave remained taller than Charles. Thus it could not be true that Dave was shorter than Charles at the beginning of 2007. Therefore, choice (E) is the correct answer.

If n is an integer and if n^2 is a positive integer, which of the following must also be a positive integer? (A) (n^2) + n (B) (2 times (n^2)) minus n (C) (n^2) minus (n^3) (D) (n^3) + n (E) (2 times (n^3)) + n

B) If n is an integer and if n^2 is a positive integer, then (2 times (n^2)) minus n = (2 times (n^2)) + (negative n) is the sum of two integers, and must therefore be an integer. Since n^2 is a positive integer, it follows that absolute value of n is greater than or equal to 1. The integer (2 times (n^2)) minus n will be positive if 2 times (n^2) greater than n. Since absolute value of n is greater than or equal to 1, it follows that n^2 greater than n, and so 2 times (n^2) greater than n. Therefore, if n^2 is a positive integer, (2 times (n^2)) minus n must also be a positive integer. If n^2 is a positive integer, then the expressions in the other choices must be integers, but they need not be positive integers: For (n^2) + n, if n = negative 1, then (n^2) + n = (negative 1)^2 + (negative 1) = 1 minus 1 = 0, which is not positive. For (n^2) minus (n^3), if n=2, then (n^2) minus (n^3) = (2^2) minus (2^3) = 4 minus 8 = negative 4, which is not positive. For (n^3) + n, if n = negative 2, then (n^3) + n = ((negative 2)^3) + (negative 2) = negative 8 minus 2 = negative 10, which is not positive. For (2 times (n^3)) + n, if n = negative 2, then (2 times (n^3)) + n = 2 times ((negative 2)^3) + (negative 2) = negative 16 minus 2 = negative 18, which is not positive. Therefore, the correct answer is (B).

If two sides of the triangle above have lengths 5 and 6, the perimeter of the triangle could be which of the following? I. 11 II. 15 III. 24 (A) Roman numeral I only (B) Roman numeral II only (C) Roman numeral III only (D) Roman numeral II and Roman numeral III only (E) Roman numeral I, Roman numeral II, and Roman numeral III

B) II only In questions of this type, statements Roman numeral 1, Roman numeral 2, and Roman numeral 3 should each be considered independently of the others. You must determine which of those statements could be true. Statement Roman numeral 1 cannot be true. The perimeter of the triangle cannot be 11, since the sum of the two given sides is 11 without even considering the third side of the triangle. Continuing to work the problem, you see that in Roman numeral 2, if the perimeter were 15, then the third side of the triangle would be 15 minus (6 + 5), or 4. A triangle can have side lengths of 4, 5, and 6. So the perimeter of the triangle could be 15. Finally, consider whether it is possible for the triangle to have a perimeter of 24. In this case, the third side of the triangle would be 24 minus (6 + 5) = 13. The third side of this triangle cannot be 13, since the sum of the other two sides is not greater than 13. By the Triangle Inequality, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. So the correct answer is Roman numeral 2 only.

The projected sales volume of a video game cartridge is given by the function s of p = 3000 over ((2 times p) + a) where s is the number of cartridges sold, in thousands; p is the price per cartridge, in dollars; and a is a constant. If according to the projections, 100000 cartridges are sold at 10 dollars per cartridge, how many cartridges will be sold at 20 dollars per cartridge? (A) 20000 (B) 50000 (C) 60000 (D) 150000 (E) 200000

C)

If x greater than 1 and square root x over x^3 = x^m, what is the value of m? (A) -7/2 (B) -3 (C) -5/2 (D) -2 (E) -3/2

C) -5/2 Since square root x can be written as x^(1/2) and 1/(x^3) can be written as x^-3, the left side of the equation is x^(1/2) times x^-3) = x^((1/2) - 3) = x^(-5/2). Since x^(-5/2) = x^m, the value of m is -5/2.

The set S consists of all multiples of 6. Which of the following sets are contained within S? I. The set of all multiples of 3 II. The set of all multiples of 9 III. The set of all multiples of 12 (A) I only (B) II only (C) III only (D) I and III only (E) II and III only

C) III only

A special lottery is to be held to select the student who will live in the only deluxe room in a dormitory. There are 100 seniors, 150 juniors, and 200 sophomores who applied. Each senior's name is placed in the lottery 3 times; each junior's name, 2 times; and each sophomore's name, 1 time. What is the probability that a senior's name will be chosen? (A) 1/8 (B) 2/9 (C) 2/7 (D) 3/8 (E) 1/2

D) 3/8

If k is divisible by 2, 3, and 15, which of the following is also divisible by these numbers? (A) k + 5 (B) k + 15 (C) k + 20 (D) k + 30 (E) k + 45

D) k +30

There c cars in a car service use a total of g gallons of gasoline per week. If each of the cars uses the same amount of gasoline, then, at this rate, which of the following represents the number of gallons used by 5 of the cars in 2 weeks? a) 10cg b) 2g/5c c) 5g/2c d) g/10c e) 10g/c

E) 10g/c