There are at least three times as many boys as girls in a class. On a test, the boys’ average score is 78, and the girls’ average score is 94. Which of the following could be the average test score for the entire class?

Indicate all such values.

A. 88
B. 86
C. 84
D. 82
E. 80

Solution

The function f is defined by f (x) = $\sqrt{x^{2}}$

For which of the following values of x is f (x) = x ?

Indicate all such values.

A. −10,000
B. −1
C. −.00001
D. 0
E. .00001
F. 10,000

Solution

The positive integers k, m, and n have the property that k is a factor of m, and m is a factor of n. Which of the following must be true?
Indicate all that apply.

A. k is a factor of n.
B. m is a factor of kn.
C. n is a factor of km.
D. (n/k) and (n/m) are both integers.

Solution

In right triangle ABC above, BC = 12, and AD = 9. Which of the following statements, considered individually, is sufficient to determine the triangle’s area?
Indicate all that apply.

A. Triangle ABD is equilateral
B. DC = 6
C. $\overline{BD}$ is not perpendicular to $\overline{AC}$ .
D. AB = (3/4) BC
E. AB ≠ AD

Solution

For which of the following integers, n, is the number 60n equal to the square of another integer?
Indicate all such values of n.

A. 2 ²⁸
B. 2 ⁹ × 3 ⁵
C. 3 ³ × 5 ⁵
D. 2 ¹⁰ × 3 ⁹ × 5 ⁸
E. 3 × 5 × 7 ²⁸

Solution

Questions are based on the following data, which shows the high and low dollar values of five tech company’s stocks in the year 2012.

If Joe bought 800 shares of PalFind at the stock’s lowest value during the year and sold the shares at the stock’s highest value during the year, what was Joe’s profit?

A. $1,600
B. $8,000
C. $16,000
D. $32,000
E. $46,400

Solution

(C) Joe bought the stock at just under $40, and sold it at just under $60. His profit is 60 − 40 = $20 per share. If he bought 800 shares, his profit was 800 × 20 = $16,000, (C)

Questions are based on the following data, which shows the high and low dollar values of five tech company’s stocks in the year 2012.

For the company whose stock value changed by the highest percentage during the year, what percent of the stock’s maximum value was its minimum value?

A. 30%
B. 35%
C. 40%
D. 45%
E. 50%

Solution

Questions are based on the following data, which shows the high and low dollar values of five tech company’s stocks in the year 2012.

Ann held 90 shares of the company whose stock value changed by the least amount during the year. If she sold these shares at their maximum value and used the money to buy shares of the least expensive stock at its lowest value of the year, how many shares was she able to buy?

Solution

(630) Microtech, whose stock values spanned less than one horizontal line on the graph, saw the least change in value. The highest value of the year for Microtech stocks was $70. The least expensive stock, LinkUp, had a value of $10 at its lowest price of the year. This means that Ann could purchase seven stocks of the latter for every one stock sold of the former. Then if she sold 90 shares of Microtech, she could purchase 90 × 7 = 630 shares of LinkUp.

A vintage clothing dealer bought x shirts at c dollars each and sold them at r dollars each. Which of the following expressions represents the profit the dealer made?

A. x(c − r)
B. x(r − c)
C. c(x − r)
D. r(x − c)
E. x(c + r)

Solution

(B) The profit per shirt is given by the price at which a shirt is sold minus the price at which it was bought. Here, this quantity is r − c. The total profit will equal the number of shirts sold × the profit per shirt, which is simply x(r − c), choice (B).

What is the ratio of $\frac{3}{5}$ to $\frac{6}{11}$ ?

10/11
11/10
18/55
63/55
55/18

Solution

To earn a joint degree, a student must take any two math classes from the eight math classes that are offered and any two physics classes from the nine physics classes that are offered. How many different combinations of classes can a student take if he or she wishes to earn a joint degree?

A. 68
B. 288
C. 988
D. 1,008
E. 4,032

Solution

If 5b = 6c = 7d = 120, then what is the value of $\frac{7bd}{c}$ ?

A. 100
B. 120
C. 144
D. 148
E. 196

Solution

The diagonal length of a rectangular TV screen measuring 70 × 240 is how much greater than the diagonal length of a rectangular TV screen measuring 80 × 150?

Solution

One Chinese Yuan is worth approximately how many Thai Baht?

A. 0.2
B. 0.3
C. 4.4
D. 4.6
E. 5.0

Solution

(D) Since 6.5 Yuan = $1 U.S. = 30 Baht, then 6.5 Yuan = 30 Baht. Divide both sides by 6.5 to obtain 1 Yuan ≈ 4.6 Baht, (D).

Which of the following is worth the most?

A. One U.S. Dollar
B. One Baht
C. One Yuan
D. One Rupee
E. One Swiss Franc

Solution

(E) One U.S. Dollar buys more than one Yuan, Baht, or Rupee, so those currencies are worth less than $1. The U.S. Dollar buys fewer than one Swiss Franc, so the Franc is worth more than $1, the only such currency, making the Swiss Franc, (E), greatest in value.

Line L in the xy-plane is given by the equation 6x + 8y = 72.

Quantity A	Quantity B
The x-coordinate of the x-intercept of line L	The y-coordinate of the y-intercept of line L

A. Quantity A is greater.
B. Quantity B is greater
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.

Solution

(A) Find the x-intercept of a graph by setting y equal to 0. Then 6x = 72, and x = 12. Find the y-intercept of a graph by setting x equal to 0. Then 8y = 72, and y = 9. Quantity (A) is greater.

On a list of 19 real numbers, ten numbers are greater than 60, and nine numbers are less than 60.

Quantity A	Quantity B
The median of the list.	60.1

A. Quantity A is greater.
B. Quantity B is greater
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.

Solution

(D) The median of a list of numbers is the middle value. If there are 19 numbers, the median will be the 10th smallest and the 10th largest. In other words, there will be nine values greater than the median, and nine values less than the median. Since there are also exactly nine numbers given as less than 60, these must be the nine smallest values, so the median will be the least of the numbers greater than 60. This number could be just barely greater than 60, say 60.0001, which is less than (B), or much greater than 60, say 1,000, which is greater than (B). Therefore, (D) is correct.

n = 3 ³

Quantity A	Quantity B
n ⁿ	3 ⁸¹

A. Quantity A is greater.
B. Quantity B is greater
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.

Solution

10 < |k| < 20

Quantity A	Quantity B
The greatest negative integer, k, that makes the above statement true.	The greatest negative integer, k, that makes the above statement false.

A. Quantity A is greater.
B. Quantity B is greater
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.

Solution

(B) Recall that the absolute value of a number equals the distance from 0 to that number on a number line. The integers that satisfy 10 < |k| < 20 are as follows: 11, 12, 13, 14, 15, 16, 17, 18, 19, −11, −12, −13, −14, −15, −16, −17, −18, −19. Of the negative integers on this list, the greatest is −11, so Quantity (A) is −11. The greatest of all negative integers, −1, makes the absolute value inequality false because |−1| equals 1, which is not between 10 and 20. So Quantity (B) is −1. Since −1 > −11, (B) is correct.

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