The area of circle C in square feet is equal to its circumference in feet. What is the circle’s diameter?

Solution

(4) If area equals circumference than πr² = 2πr. devide both side by πr to obtain r = 2. if radius equal to 2 than diameter = 2 × 2 = 4

If $4^{x^{2}} = 8^{2x}$ , then which of the following could be the value of x?
Indicate all that apply.

A. –2
B. 0
C. 2
D. 2.5
E. 3
F. 4

Solution

A regular polygon is defined as a polygon with all sides congruent and all angles congruent. In a regular polygon with n sides, the measure of one interior angle is equal to 168°. What is the value of n?

A. 12
B. 18
C. 24
D. 30
E. 36

Solution

(D) The sum of the measures of the interior angles on an n-sided polygon is given by the formula Angle Sum = 180(n – 2) degrees. A polygon with n sides also has n angles, and if each angle measures 168°, then the sum of all angles equals 168n. So we can set 180(n – 2) = 168n.
Distribute 180, and 180n – 360 = 168n. Subtract 168n and add 360 to each side to obtain 12n = 360. Divide by 12, and n = 30, (D).

The height, h, of a thrown ball as a function of t, the amount of time it has been in the air, is given by h(t) = –10t²+ 40t. What is the maximum height attained by the ball?

A. 2
B. 20
C. 40
D. 80
E. 160

Solution

The graphs y = x ² + 4 and y = |x| + 4 have how many points in common?

A. none
B. one
C. two
D. three
E. four

Solution

In the rectangular solid above, all edges have integer lengths. If the areas of three of the faces of the solid are 15, 33, and 55, what is its volume?

A. 103
B. 165
C. 495
D. 9,075
E. 27,225

Solution

Question is based on the following table, which shows the vitamin content for three elements—iron, calcium, and zinc—contained in three different vitamin tablets. All units are in milligrams.

What is the minimum number of tablets a person can take, using any combination of the three types of tablets, in order to obtain at least 140 mg of iron and at least 160 mg of calcium?

Solution

(5) We are trying to arrive at 140 mg of iron and 160 mg of calcium efficiently, so we might as well not consume any of Tablet Y, which is highest in zinc, the one element we have no interest in. Notice that 3 of Tablet X and 2 of Tablet Y would contain 3 × 40 + 2 × 20 = 120 + 40 = 160 mg of calcium exactly. This same combination contains 3 × 17 + 2 × 45 = 51 + 90 = 141 mg of iron, which is greater than 140, as required. This combination of 3 + 2 = 5 tablets achieves the desired quantities with very little wastage. No combination of four tablets will have sufficient amounts of both iron and calcium, so 5 is the minimum.

Question is based on the following table, which shows the vitamin content for three elements—iron, calcium, and zinc—contained in three different vitamin tablets. All units are in milligrams.

A person wants to consume at least 120 mg but no more that 150 mg of each of the three elements. Which combination of tablets, considered individually, satisfy this requirement?
Indicate all that apply.

A. two of Tablet X and two of Tablet Z
B. six of Tablet Y
C. three of Tablet Y and three of Tablet Z
D. one of Tablet X, four of Tablet Y, and one of Tablet Z

Solution

(B) Combination (A) has insufficient zinc. Combination (B) contains either 120 or 150 milligrams of each element, as desired. Combination (C) has too much iron. Combination (D) contains too much calcium. Only (B) works.

Question is based on the following table, which shows the vitamin content for three elements—iron, calcium, and zinc—contained in three different vitamin tablets. All units are in milligrams.

Which of the following statements are supported by the data?
Indicate all that apply.

A. The tablet that contains the most iron contains the least zinc.
B. The tablet with the most total iron, calcium, and zinc combined is Tablet Z.
C. The tablet with the most calcium has less iron and less zinc than both of the other tablets.
D. By taking three of each tablet, a person will consume between 140 and 250 mg of all three elements.

Solution

(B, C) Tablet Z has the most iron, but it does not contain less zinc than Tablet X, so (A) is incorrect. Tablet Z has 45 + 20 + 15 = 80 mg of the three elements combined, which is greater than any of the other row totals, so (B) is correct. Tablet X has the most calcium. But it has only 17 units of iron and 14 of zinc, which are less than the other tablets’ contents, so (C) is correct.

For (D), add the values in the middle column to find that one of each tablet would include 40 + 25 + 20 = 85 mg of calcium. Then three of each tablet would contain 3 × 85 = 255 mg of calcium, which is not between 140 and 250, so (D) is not correct. Answer is (B) and (C).

The price of milk increased to $2.52 per gallon. If this represented a 12% increase in the price, what was the cost of buying nine gallons of milk before the increase?

Solution

($20.25) Let p = the original price of milk. Then p plus 12% of p becomes p + .12p = 1.12p. So 1.12p = $2.52, and by division p = $2.25. The cost of nine gallons at this price is 9 × 2.25 = $20.25

The first five prime numbers in order are as follows: 2, 3, 5, 7, 11. The sum of the next four prime numbers in order is how much greater than the sum of the first five prime numbers?

Solution

(44) Prime numbers are numbers with exactly two positive integer factors, namely 1 and the number itself. The sum of the prime numbers given is 2 + 3 + 5 + 7 + 11 = 28. The next four prime numbers in order are 13, 17, 19, 23. Their sum is 13 + 17 + 19 + 23 = 72. This is 72 – 28 = 44 greater.

On a test, the boys’ average score is 70, and the girls’ average score is 80. If there are 18 boys and 12 girls in the class, what is the test average for the entire class?

Solution

(74) For 18 boys at 70 points per boy, there are 18 × 70 = 1,260 total points. For 12 girls at 80 points per girl, there are 12 × 80 = 960 total points. Then, the combined sum of test scores for the entire class is 1,260 + 960 = 2,220 points. Divide this by 30 to obtain the average score per student, because there are 18 + 12 = 30 students in the class. 2,220 ÷ 30 = 74.

*Divide the number of boys and girls by 6, the greatest common factor of 18 and 12, in order to work with an equivalent ratio of children involving smaller numbers. The ratio of 18 boys to 12 girls is thus reduced to 3:2. For three boys at 70 points per boy, there are 3 × 70 = 210 total points. For two girls at 80 points per girl, there are 2 × 80 = 160 total points. This yields 210 + 160 = 370 total points for all five students together (3 boys + 2 girls). Divide 370 by 5 to obtain the average score per student. 370 ÷ 5 = 74.

Adult tickets to a movie cost $9 each and children’s tickets cost $5 each. When 44 tickets were bought, the total cost was $312. How many of the tickets purchased were adult tickets?

Solution

A man will choose two different flavors of ice cream for his sundae from the six flavors listed.

Quantity A	Quantity B
The number of different combinations of flavors that the man could possibly choose	15

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

Quantity A	Quantity B
The volume of a right circular cylinder with height equal to 15x and diameter equal to 1y	The volume of a right circular cylinder with height equal to 1x and diameter equal to 4y

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

1,498 students are being assigned to classrooms that hold exactly 35 students each. N is the number of students left over when as many classrooms as possible are filled.

Quantity A	Quantity B
N	25

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

Ann earned a 25% raise to bring her salary to $20,000.
Bob received a 20% decrease in salary to bring his salary to $20,000.

Quantity A	Quantity B
The difference between Bob and Ann’s salaries before these changes were made	$9,000

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

(C) If Ann’s original salary was a, then a + .25a = 20,000, so 1.25a = 20,000, and a = 16,000. If Bob’s original salary was b, then 1b – .20b = 20,000, so .80b = 20,000, and b = 25,000, after dividing both sides by .8. The difference in their original salaries was 25,000 – 16,000 = 9,000, (C).

Quantity A	Quantity B
The slope of the line	-17/6

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

Question is based on the following information: Maya flipped 24 coins, each of which landed with either a head face up or a tail face up, hereafter referred to as an outcome of “heads” or “tails.”

If all the coins are fair, in that the probability of “heads” = probability of “tails” = 1/2 for every toss, which of the following events has the greatest probability?

A. Of the 24 tosses, the number of “heads” equals 11.
B. Of the 24 tosses, the number of “heads” equals 12.
C. Of the 24 tosses, the number of “tails” equals 14.
D. Of the 24 tosses, the number of “tails” is at most 3.
E. Of the 24 tosses, the number of “heads” is it least 22.

Solution

Which of the following could not be the ratio of “heads” to “tails?”

Indicate all that apply.

A. 1:1
B. 2:1
C. 3:1
D. 4:1
E. 5:1

Solution

(D) Test the answer choices. If the ratio of heads to tails were 1:1, choice (A), then for some integer x, 1x + 1x = 24, so 2x = 24, and x = 12. This would mean 12 heads and 12 tails, which is permissible. The choice that could not be the ratio of heads to tails is (D), 4:1, because then 4x + 1x would equal 24. If 5x = 24, x is not an integer. Specifically, we could not have 4.8 tosses that resulted in “tails.”

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