Question 1

In a school fund-raiser, 10 students in class A sold an average (arithmetic mean) of 4 boxes of cookies. In class B, 15 students sold an average of 9 boxes. What was the average number of boxes of cookies sold by the 25 students?
A)13/25
B)1
C)6-1/2
D)7
E)13-6/13

Accepted Answer

The total number of boxes sold by class A is 10 * 4 = 40. The total number of boxes sold by class B is 15 * 9 = 135. The combined total is 40 + 135 = 175. The average for all 25 students is 175 / 25 = 7.

Question 2

The remainder when a number n is divided by 7 is 2. Which of the following is the remainder when 2 n + 1 is divided by 7 ?
A)1
B)2
C)3
D)4
E)5

Accepted Answer

E

Question 3

What are all values of x such that x2 + 7 x + 6 = 0 ?
A)-6 = x = -1
B)-6 = x = 1
C)1 = x = 6
D)x = -6 or x = -1
E)x = 1 or x = 6

Accepted Answer

This quadratic equation can be factored as (x + 6)(x + 1) = 0, which gives the solutions x = -6 and x = -1.

Question 4

A particular flu vaccine is effective for 90 percent of the patients who receive it. What is the probability that it will be effective for both of the next two randomly selected patients?
A) 0.54
B) 0.63
C) 0.72
D) 0.81
E) 0.90

Accepted Answer

The probability that the vaccine is effective for both patients is the product of the individual probabilities: 0.9 * 0.9 = 0.81.

Question 5

The number of bacteria present in a laboratory sample after t days can be represented by 500(2 t). What is the initial number of bacteria present in this sample?
A)250
B)500
C)750
D)1,000
E)2,000

Accepted Answer

The initial number of bacteria is found by evaluating the expression at t = 0, which gives 500(2^0) = 500(1) = 500.

Question 6

Yvonne has studied the cost of tickets over time for her favorite sports team. She has created a model to predict the cost of a ticket in the future. Let C represent the cost of a ticket in dollars and y represent the number of years in the future. Her model is as follows.C = 2.50 y + 13Based on this model, how much will the cost of a ticket increase in two years?
A)$5
B)$8
C)$13
D)$18
E)$26

Accepted Answer

The model shows that the cost of a ticket increases by $2.50 each year. In two years, the increase would be $2.50 * 2 = $5.

Question 7

Cos 2(3 x) + sin 2(3 x) =
A)0
B)1
C)3
D)6
E)9

Accepted Answer

The identity cos^2(x) + sin^2(x) = 1 applies here, with 2(3x) simply being the angle. Regardless of the angle, the sum of the square of cosine and the square of sine equals 1.

Question 8

The population of the United States is approximately 250 million, and the national debt is approximately 4 trillion dollars. If this debt were divided equally among the population, what would be the debt, in dollars, per person?
A)16
B)1,600
C)16,000
D)1,600,000
E)16,000,000

Accepted Answer

To find the debt per person, divide the total debt by the population: 4 trillion dollars / 250 million people = 16,000 dollars per person.

Question 9

A rectangular pool 24 feet long, 8 feet wide, and 4 feet deep is filled with water. Water is leaking from the pool at the rate of 0.40 cubic foot per minute. At this rate, how many hours will it take for the water level to drop 1 foot?
A)4
B)8
C)12
D)16
E)32

Accepted Answer

To find the time it takes for the water level to drop 1 foot, first calculate the volume of water that needs to be leaked out, which is the surface area of the pool (24 feet * 8 feet) times the drop in water level (1 foot), equaling 192 cubic feet. Since the water leaks at a rate of 0.40 cubic foot per minute, divide the total volume by the leak rate (192 cubic feet / 0.40 cubic foot per minute) to get 480 minutes. Converting minutes to hours (480 minutes / 60 minutes per hour) gives 8 hours.

Question 10

It takes 28 minutes for a certain bacteria population to double. If there are 5,241,763 bacteria in this population at 1:00 p.m., which of the following is closest to the number of bacteria in millions at 2:30 p.m. on the same day?
A)80
B)40
C)20
D)15
E)10

Accepted Answer

The answer of It takes 28 minutes for a certain...

Question 11

For what value of x is 8 12 = 16 x?
A)3
B)4
C)8
D)9
E)12

Accepted Answer

The answer of For what value of x is 8...

Question 12

What is the distance between the points (2,10) and (-4,2) in the xy-plane?
A)6
B)8
C)10
D)14
E)18

Accepted Answer

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the xy-plane is given by the formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. For the points (2,10) and (-4,2), this distance is $\sqrt{(-4 - 2)^2 + (2 - 10)^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.

Question 13

In the xy-plane, a line parallel to the x-axis intersects the y-axis at the point (0, 4). This line also intersects a circle in two points. The circle has a radius of 5 and its center is at the origin. What are the coordinates of the two points of intersection?
A)(1, 2) and (2, 1)
B)(2, 1) and (2, -1)
C)(3, 4) and (3, -4)
D)(3, 4) and (-3, 4)
E)(5, 0) and (-5, 0)

Accepted Answer

A line parallel to the x-axis and intersecting the y-axis at (0, 4) is a horizontal line with y=4. The circle's equation is x^2 + y^2 = 25 (since its radius is 5 and it's centered at the origin). Substituting y=4 into the circle's equation gives x^2 + 16 = 25, which simplifies to x^2 = 9. Solving for x gives x = 3 and x = -3, so the points of intersection are (3, 4) and (-3, 4).

Question 14

Suppose 4 r = 3 s = 10 t, where r, s, and t are positive integers. What is the sum of the least values of r, s, and t for which this equality is true?
A)7
B)17
C)41
D)82
E)120

Accepted Answer

To find the least values of r, s, and t, we need to find a common multiple of the coefficients 4, 3, and 10. The least common multiple (LCM) of 4, 3, and 10 is 60. Setting 4r = 60, 3s = 60, and 10t = 60 gives r = 15, s = 20, and t = 6. The sum of these values is 15 + 20 + 6 = 41.

Question 15

3 3 + 4(8-5) ?=
A) 6.5
B) 11
C) 27.5
D)29
E)34.16

Accepted Answer

The expression simplifies to 3.3 + 4(3) = 3.3 + 12 = 15.3, which is not listed. However, if interpreted as 33 + 4(3), it equals 33 + 12 = 45, which is also not listed. The closest match to the given options is 29, assuming a typo or misinterpretation in the problem.

Question 16

In a group of 1,200 adults, there are 300 vegetarians. What is the ratio of nonvegetarians to vegetarians in the group?
A)1 to 3
B)1 to 4
C)3 to 1
D)4 to 1
E)4 to 3

Accepted Answer

There are 900 nonvegetarians (1200 total - 300 vegetarians). The ratio of nonvegetarians to vegetarians is 900:300, which simplifies to 3:1.

Question 17

A certain company keeps a list of 50 employees and their annual salaries. When the salary of the very highly paid president is added to this list, which of the following statistics is most likely to be approximately the same or nearly the same for the original list and the new list?
A)The highest salary
B)The range
C)The mean
D)The median
E)The standard deviation

Accepted Answer

Adding the president's salary, which is very high, will significantly affect the highest salary, range, mean, and potentially the standard deviation. However, the median, which is the middle value, will be least affected because it depends on the order of the numbers rather than their values. With 50 employees, the median is the average of the 25th and 26th highest salaries. Adding one more salary (the president's) shifts the median to the new 26th salary, which should be very close to the original median value.

Question 18

The least common multiple of 8, 12, and a third number is 120. Which of the following could be the third number?
A)15
B)16
C)24
D)32
E)48

Accepted Answer

The least common multiple (LCM) of 8 and 12 is 24. To get an LCM of 120 with a third number, the third number must introduce a factor that 24 does not have. The number 15 introduces the factor 5, which is needed to reach 120 (since 24 * 5 = 120).

Question 19

The length of a rectangle is 3 more than its width. If L represents the length, what is an expression for the width? A)3 -]>

Accepted Answer

L = W + 3
To find an expression for the width (W), we need to solve this equation for W:
W = L - 3
Therefore, the expression for the width in terms of the length is:
C) L - 3

Question 20

A circle with diameter 10 centimeters is to be cut from a square of paper 10 centimeters on a side. Of the following, which is closest to the amount of paper left over after the circle is cut out?
A)9 square centimeters
B)21 square centimeters
C)24 square centimeters
D)69 square centimeters
E)84 square centimeters

Accepted Answer

The area of the square is $10 \times 10 = 100$ square centimeters. The radius of the circle is half the diameter, so $5$ centimeters. The area of the circle is $\pi r^2 = \pi \times 5^2 = 25\pi$ square centimeters. Since $\pi$ is approximately $3.14$, the area of the circle is about $78.5$ square centimeters. Subtracting this from the area of the square gives $100 - 78.5 = 21.5$ square centimeters, which is closest to 21 square centimeters.

Quiz 4: Math: Mathematics Intermediate

In a school fund-raiser, 10 students in class A sold an average (arithmetic mean) of 4 boxes of cookies. In class B, 15 students sold an average of 9 boxes. What was the average number of boxes of cookies sold by the 25 students?

The remainder when a number n is divided by 7 is 2. Which of the following is the remainder when 2 n + 1 is divided by 7 ?

What are all values of x such that x2 + 7 x + 6 = 0 ?

A particular flu vaccine is effective for 90 percent of the patients who receive it. What is the probability that it will be effective for both of the next two randomly selected patients?

The number of bacteria present in a laboratory sample after t days can be represented by 500(2 t) . What is the initial number of bacteria present in this sample?

Cos 2(3 x) + sin 2(3 x) =

The population of the United States is approximately 250 million, and the national debt is approximately 4 trillion dollars. If this debt were divided equally among the population, what would be the debt, in dollars, per person?

A rectangular pool 24 feet long, 8 feet wide, and 4 feet deep is filled with water. Water is leaking from the pool at the rate of 0.40 cubic foot per minute. At this rate, how many hours will it take for the water level to drop 1 foot?

It takes 28 minutes for a certain bacteria population to double. If there are 5,241,763 bacteria in this population at 1:00 p.m., which of the following is closest to the number of bacteria in millions at 2:30 p.m. on the same day?

For what value of x is 8 12 = 16 x?

What is the distance between the points (2,10) and (-4,2) in the xy-plane?

In the xy-plane, a line parallel to the x-axis intersects the y-axis at the point (0, 4) . This line also intersects a circle in two points. The circle has a radius of 5 and its center is at the origin. What are the coordinates of the two points of intersection?

Suppose 4 r = 3 s = 10 t, where r, s, and t are positive integers. What is the sum of the least values of r, s, and t for which this equality is true?

3 3 + 4(8-5) ?=

In a group of 1,200 adults, there are 300 vegetarians. What is the ratio of nonvegetarians to vegetarians in the group?

A certain company keeps a list of 50 employees and their annual salaries. When the salary of the very highly paid president is added to this list, which of the following statistics is most likely to be approximately the same or nearly the same for the original list and the new list?

The least common multiple of 8, 12, and a third number is 120. Which of the following could be the third number?

The length of a rectangle is 3 more than its width. If L represents the length, what is an expression for the width?

A circle with diameter 10 centimeters is to be cut from a square of paper 10 centimeters on a side. Of the following, which is closest to the amount of paper left over after the circle is cut out?