Directions: For Question enter your answer in the answer box below the question. Equivalent forms of the correct answer, such as 2.5 and 2.50, are all correct. Fractions do not need to be reduced to lowest terms.

Q. In a triangle with perimeter equal to 24, the side lengths are three consecutive even integers. What is the triangle’s area?

Solution

ANSWER : 24

In the figure above, what is the value of x?

Solution

$\sqrt{2x-3} = x-3$ then what is the value of x?

Solution

If g is the function defined by $g(x) = \frac{(6x-3)(x + 4)}{(x-1)(x+2)(x-3)}$ then g(x) is undefined for which of the following values of x? Indicate all that apply.

A. 1/2

B. 1

C. -1

D. -2

E. 3

F. -4

Solution

(B, D, E) The numbers that make g(x) undefined are numbers that make the denominator equal to 0, because division by 0 is not defined. The product (x − 1)(x + 2)(x − 3) will equal 0 when any of the individual factors equals 0, because zero times anything equals zero. Then either x − 1 = 0, x + 2 = 0, or x − 3 = 0, and x could equal 1, −2, or 3 choices, (B), (D), and (E).

In the rectangular solid above, points A, B, C, and D are vertices, AB = 8, and all edges meet at right angles. Which of the following statements, considered individually, is sufficient to determine the volume of the solid?
Indicate all that apply.

A. The solid is a cube.

B. The area of one face of the solid is 64.

C. BC = CD = AB.

D. BD = 8√2

E. AC = BD

Solution

“A number squared is equal to at least twice the number.” Which of the following numbers satisfy this statement?
Indicate all such numbers.

A. 10 ⁻²
B. 0.9
C. 1
D. 2
E. 3
F. 10

Solution

Questions are based on the following data, which show the employment breakdown of the 80 million person workforce in Country X in the year 2011 and the projected employment breakdown of the 120 million person workforce predicted for the country in the year 2050.

How many more people are projected to be employed in the technology sector in 2050 than in 2011?

4 million
8 million
12 million
16 million
20 million

Solution

(D) In 2011, the technology sector consists of 25% of 80 million people, which is 20 million people. In 2050, technology represents 30% of 120 million people, which is .30 × 120 million, or 36 million people. The increase is 36 − 20 = 16 million people, (D).

In 2011, for every five people who worked in the health sector, n people worked in the service sector. What is the value of n?

Solution

ANSWER : 8

For how many of the six employment categories shown will the total number of employees increase from 2011 to 2050?

two
three
four
five
six

Solution

(C) Be careful. The total number of employees increases from 80 million to 120 million between 2011 and 2050, so a decrease in percentage for a category could correspond to an increase in actual workers. For example, the service sector in 2011 contains 20% of 80 million employees, which is 16 million employees, while in 2050, the percentage drops to 16.67%, but .1667 × 120 million ≈ 20 million employees, a net increase. In addition to the service sector, the health, technology, and education sectors also see an increase in the number of employees (a higher percentage of a higher total), so there are four such categories in all, (C).

The total number of people employed in education and manufacturing in the year 2050 will exceed by how many the total number of people employed in education and other in 2011?

4 million
5 million
6 million
7 million
8 million

Solution

(C) In 2050, the education and manufacturing sectors account for 15% + 8.33% = 23.33% of the 120 million total employees. This equals .2333 × 120 million ≈ 28 million employees. In 2011, education and other account for 10% + 17.5% = 27.5% of the 80 million total employees. This equals .275 × 80 million = 22 million employees. The first figure exceeds the second by 28 − 22 = 6 million (C).

A sheet of rectangular paper measures 12 cm by 20 cm. At the edge of the paper, a margin exactly 1-cm wide is painted all the way around. What percentage of the area of the original rectangle remains unpainted?

A. 95%
B. 90%
C. 85%
D. 80%
E. 75%

Solution

Sam’s bowling scores for seven games were, in order played, 54, 35, 89, 76, 48, x, and y. Sam’s mode score for the seven games was 76, and his median score was 54. If x < y, which of the following could be the value of x?

A. 50
B. 55
C. 70
D. 76
E. 80

Solution

(A) The mode of a list is the most frequently recurring number on the list. Since 76 is the mode, 76 must appear at least twice on the list. So at least one of the unknown scores must equal 76.

The median, or middle value on a list, is found by first rewriting the given numbers in order from least to greatest. Here, after inserting the second 76, we know these 6 scores: 35, 48,54 76, 76, 89.

Since the median is given as 54, draw a triangle below the 54, and think of that as the balancing point. There are three values to the right of the triangle but only two to the left. Therefore, the final missing score must be less than 54, to ensure that 54 is in the middle. Only choice (A), 50, works. Thus, the two missing scores are 50 and 76, and since the problem asks for the smaller of these, (A) is correct.

Ann’s tree is 76 inches tall and is growing at a rate of 3.6 inches per year. Bob’s tree is 58 inches tall and is growing at a rate of 4.8 inches per year. In how many years will Bob’s tree reach Ann’s tree in height?

A. 12
B. 14
C. 15
D. 20
E. 25

Solution

$400 is placed in an account earning an annual interest rate of 6%. If the account is compounded annually, how much more interest is earned in the second year than in the first?

A. $1.22
B. $1.33
C. $1.43
D. $1.44
E. $1.54

Solution

(D) In year one, the account earned 6% of $400 in interest. That is, .06 × 400, or $24. When the interest is “compounded,” this $24 is reinvested in the account, which will then hold 400 + 24 = $424. In year two, the interest is 6% of $424, or .06 × 424, which equals $25.44. The year two interest is greater by 25.44 − 24 = $1.44, (D).

Q. In a mixture of nuts, the ratio of almonds to cashews to peanuts by weight is 5:8:11. When 156 pounds of the mixture are made, how many pounds of peanuts are used?

lbs.

Solution

ANSWER: 71.5

Quantity A	Quantity B
The circumference of the circle.	27

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

The revenue, R, that a company earns from offering x promotional discounts is given by the function
R(x) = −2x ² + 32x +160.

Quantity A	Quantity B
The revenue earned from offering five discounts	The revenue earned from offering 11 discounts

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 + 2b = 3, and a − b = 12

Quantity A	Quantity B
a	b

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) Line up the two given equations vertically, and then subtract to eliminate the variable a, remembering to distribute the negative sign across the second equation:

a + 2b = 3 ;

$\frac{- (a-b) = 12)}{3b = -9}$

Divide this last equation, 3b = −9, by 3 on each side, and b = −3. Now substitute this value of b into either given equation, say the second, to obtain the value of a. So a − (−3) = 12, which means a + 3 = 12, yielding a = 9. Because 9 > −3, (A) is greater.

xy = √99

Quantity A	Quantity B
x ²	99 / y ²

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) Start with the given equation, xy = √99 and divide both sides by y to obtain x = ( √99 / y ) Now square both sides:

x² = ( √99 ) ² / y ²

= 99 / y ²

Thus, the two quantities are equal, (C).

s and t are positive integers greater than 1.

Quantity A	Quantity B
S ^t/4	S ^{t – 4}

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. Think of s as a constant, such as 2 or 3, and replace t with various values. If t = 100, then Column A reads s ²⁵ , while Column B reads s 96 , and Quantity (B) is much greater. If t = 4, then Column A becomes s 1 , which equals s, while Column B reads s ⁰ , which equals 1, and this time Quantity (A) is greater. Therefore, (D) is correct, since it is possible to make either quantity greater.

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