ASTB Math Quiz

1. Two trains running in the same direction on the same track travel at the rate of 40 and 45 mph. If the slower train starts out an hour earlier, how long will it take the faster one to catch up with it?

B- 6 hours

A- 5 hours

ANSWER- D
the slower train is 40 miles ahead in one hour. the difference in rate of change is 5 mph, and 40/5= 8 hours

C- 7 hours

The slower train is 40 miles ahead in one hour. The difference in rate of change is 5 mph, and 40/5= 8 hours.

Explanation

The slower train is 40 miles ahead in one hour. The difference in rate of change is 5 mph, and 40/5= 8 hours.

2. A naval detachment has enough rations to feed 15 people for 8 days. If 5 more people join, how many fewer days will the rations last?

A- 1

ANSWER- B
let x=the number of rtion days for 20 people
15 * 8= 20 * x
120= 20x
20x= 120
x= 120/20
x= 6 ration days for 20 persons
so 8 days - 6 days= 2 ration days

D- 4

C- 3

Let x be the number of ration days for 20 people. Using the formula 15 * 8 = 20 * x, we find that x = 6 ration days for 20 persons. Therefore, the rations will last 2 days fewer for the increased group size.

Explanation

Let x be the number of ration days for 20 people. Using the formula 15 * 8 = 20 * x, we find that x = 6 ration days for 20 persons. Therefore, the rations will last 2 days fewer for the increased group size.

3. If a field can be plowed by 12 machines in 7 hours, but 4 machines are broken and cannot be used, how many hours will it take to plow the field with the remaining machines?

C- 9.5 hours

B- 8.5 hours

ANSWER- D
let x equal the numbr of hours to plow with 8 machines.
12 * 7= 8 * x
84= 8x
8x= 84
x= 84/8
x=10.5 hours

A- 7.5 hours

The correct answer can be calculated by setting up the proportion: 12 * 7 = 8 * x, where x is the number of hours it will take with 8 machines. Solving for x gives x = 10.5 hours.

Explanation

The correct answer can be calculated by setting up the proportion: 12 * 7 = 8 * x, where x is the number of hours it will take with 8 machines. Solving for x gives x = 10.5 hours.

4. What is the square root of 4 raised to the 5th power?

D- 64

A- 8

ANSWER- C
SQR of 4=2
2^5= 2*2*2*2*2
=32

B- 16

The square root of 4 is 2. When 2 is raised to the 5th power, it equals 32.

Explanation

The square root of 4 is 2. When 2 is raised to the 5th power, it equals 32.

5. If 3^ (n-2)= 27, what does n equal?

B- 7

C- 8

D- 12

ANSWER- A
3 ^ (n-2)= 27
3^3= 27
n-2= 3
n=5

The correct answer is obtained by solving for n as n=5 in the given equation 3^ (n-2)= 27. Thus, n equals 5 which corresponds to option a.

Explanation

The correct answer is obtained by solving for n as n=5 in the given equation 3^ (n-2)= 27. Thus, n equals 5 which corresponds to option a.

6. If the sum of the edges of a cube is 36 inches, what is the volume of the cube?

24 cubic inches

ANSWER- B
a cube has 12 edges, so each edge is 24/12= 2
volume= L*W*H
v= 2*2*2
v=8

12 cubic inches

64 cubic inches

A cube has 12 edges, so each edge is 36/12 = 3. Volume = L * W * H = 3 * 3 * 3 = 27 cubic inches.

Explanation

A cube has 12 edges, so each edge is 36/12 = 3. Volume = L * W * H = 3 * 3 * 3 = 27 cubic inches.

7. What is the perimeter of a right triangle whose legs are 3 and 4 feet?

ANSWER- B
Using the Pythagorean theorm, the hypotenuse of the right triangle = 5:

3′ + 4′ + 5′ = 12

A. 10 feet

D. 16 feet

C. 14 feet

Using the Pythagorean theorem, the hypotenuse of the right triangle = 5. Therefore, the perimeter would be 3 + 4 + 5 = 12 feet.

Explanation

Using the Pythagorean theorem, the hypotenuse of the right triangle = 5. Therefore, the perimeter would be 3 + 4 + 5 = 12 feet.

8. How much pure acid must be added to 10 ounces of a 55% acid solution in order to produce a 75% acid solution?

ANSWER- D
pure acid:
# of ounces= x, parts pure acid= 100, # of ounces of pure acid= 100x
40% acid solution:
# of ounces= 10, parts pure acid= 44, # of ounces of pure acid= 550
60% acid solution:
# of ounces= 10 + x, parts pure acid= 75, # of ounces of pure acid= 75(10+ x)
SO....
100x + 550= 75 (10 + x)
100x + 550= 750 + 75x
-550 -550
100x= 200+ 75x
-75x - 75x
25x= 200
x= 8

C- 7 ounces

A- 5 ounces

B- 6 ounces

By setting up the equation with the given information of pure acid and the two different solutions, solving for x gives us the correct answer of 8 ounces.

Explanation

By setting up the equation with the given information of pure acid and the two different solutions, solving for x gives us the correct answer of 8 ounces.

9. A farmer who is 6 feet tall wants to determine the height of his barn, he notices that his shadow is 14 feet long and that his barn casts a shadow of 70 feet long. How high is the barn?

D- 45 feet

C- 40 feet

ANSWER- A
6 : 14= x: 70
14x= 70 * 6
14x= 420
x= 30

B- 35 feet

To determine the height of the barn, set up a proportion: 6 : 14 = x : 70. Solve for x to find the height of the barn, which is 30 feet.

Explanation

To determine the height of the barn, set up a proportion: 6 : 14 = x : 70. Solve for x to find the height of the barn, which is 30 feet.

10. If (x - y)^2 = 60 and x^2 + y^2 = 40, then xy = ?

C. -12

A. -40

ANSWER= D
(x-y) ^2= x^2 - 2xy + y^2
60= 40 - 2xy
-2xy= 20
xy= -10

B. -20

The correct way to solve this problem is by expanding (x - y)^2 to get x^2 - 2xy + y^2. By substituting this expression into the first equation and simplifying, we find that xy = -10.

Explanation

The correct way to solve this problem is by expanding (x - y)^2 to get x^2 - 2xy + y^2. By substituting this expression into the first equation and simplifying, we find that xy = -10.

11. What is the measure of angle x?

B- 45 degree

ANSWER- C
arc AB= 80 degrees therefore, AOB = 80 degrees, the two radii are equal.
angle x= (180 degree- 80 degree)
= 1/2 (100 degrees)
= 50 degrees

A- 35 degree

D- 70 degree

arc AB= 80 degrees therefore, AOB = 80 degrees, the two radii are equal. angle x= (180 - 80)/2 = 50 degrees.

Explanation

arc AB= 80 degrees therefore, AOB = 80 degrees, the two radii are equal. angle x= (180 - 80)/2 = 50 degrees.

12. Jane's grades on three tests were: 93, 82, and 72. What must she get on her next test so her average for the four tests is 85?

ANSWER- D
total score of all 3= 247
to get an average of 85 then- 85*4= 340
so.. 340-247= 93

B- 89

C- 90

A- 88

Total score of all 3 tests = 247. To get an average of 85 on 4 tests, the total score needed is 340. Therefore, Jane must score 93 on her next test to achieve an average of 85.

Explanation

Total score of all 3 tests = 247. To get an average of 85 on 4 tests, the total score needed is 340. Therefore, Jane must score 93 on her next test to achieve an average of 85.

13. A family drove from New York to San Francisco, a distance of 3000 miles. They drove 1/8 of the distance the first day and 1/7 of the remaining distance the second day. How many miles were left to be driven?

ANSWER- B
1/8 of 3000= 375
3000- 375= 2625
1/7 of 2625= 375
2625- 375= 2250 miles still to be driven

2400 miles

2200 miles

2500 miles

1/8 of 3000 is 375 miles driven on the first day. Remaining distance is 2625 miles. 1/7 of 2625 is 375 miles driven the second day. Therefore, 2625 - 375 = 2250 miles still to be driven.

Explanation

1/8 of 3000 is 375 miles driven on the first day. Remaining distance is 2625 miles. 1/7 of 2625 is 375 miles driven the second day. Therefore, 2625 - 375 = 2250 miles still to be driven.

14. In a 4-hour exam of 420 questions, there are 60 math problems. If twice as much time should be allowed for each math problem as for each of the other questions, how many minutes should be spent on the math problems?

D

B

C

ANSWER- A
let x= min to be spent on each math prob
x * 60 + x/2 * 360= 240
60 x + 180 x= 240
240 x= 240
x=1
so, 60x= 60 minutes to be spent on the 60 math probs

The correct answer is calculated based on providing twice as much time for math problems compared to others. By solving the equation provided, it is determined that 60 minutes should be spent on the 60 math problems.

Explanation

The correct answer is calculated based on providing twice as much time for math problems compared to others. By solving the equation provided, it is determined that 60 minutes should be spent on the 60 math problems.

15. Which of the following represents 1,000,000?

10^7

ANSWER- C

10^4

10^5

In mathematics, the notation 10^6 represents 1,000,000 as it means 10 multiplied by itself 6 times, which equals 1,000,000.

Explanation

In mathematics, the notation 10^6 represents 1,000,000 as it means 10 multiplied by itself 6 times, which equals 1,000,000.

16. If a= 4b and 8b= 30c, then a=

ANSWER- D
a= 4b
2a= 8b= 30c
2a=30c
a= 15c

B- 9c

C- 12c

A- 6c

To find the value of a, we start with a= 4b and then substitute 4b from the first equation into the second equation, which gives us 2a= 30c. Solving for a, we get a= 15c.

Explanation

To find the value of a, we start with a= 4b and then substitute 4b from the first equation into the second equation, which gives us 2a= 30c. Solving for a, we get a= 15c.

17. A cash box contains a certain number of coins, of which 63 are dimes and 33 are nickels. The rest of the coins in the box are quarters. If the probability of selecting a quarter from this bank is one in five, how many quarters does the bank contain?

27

35

30

ANSWER- A
let x= the number of quarters ( the numerator of the formulas fraction), and let x + 96= the total number of coins ( the fractions denominator). then solve for x:
1/5= x/ x+96
5x= x+96
4x= 96
x= 24

To solve this problem, we set up the equation: 1/5 = x / (x+96), where x is the number of quarters. By solving for x, we find that x = 24. Therefore, there are 24 quarters in the bank.

Explanation

To solve this problem, we set up the equation: 1/5 = x / (x+96), where x is the number of quarters. By solving for x, we find that x = 24. Therefore, there are 24 quarters in the bank.

18. A CD system originally priced at $1000 is first discounted by 20%, then later by another 10%. If a 5% tax is added to the purchase price, how would a customer buying the system at its lowest price pay for it to the nearest dollar, including tax?

D- $800

A- $720

ANSWER- C
after the 20% discount= $800 (1000 * .20= 200.. 1000- 200= 800)
after 10% off $800= $720 (800* .10= 80.. 800-80=720)
after 5% tax= $756 (720*.05= 36.. 720+36=756)

B- $750

The correct calculation is: after the 20% discount= $800 (1000 * .20= 200, 1000- 200= 800). After 10% off $800= $720 (800* .10= 80, 800-80=720). After 5% tax= $756 (720*.05= 36, 720+36=756).

Explanation

The correct calculation is: after the 20% discount= $800 (1000 * .20= 200, 1000- 200= 800). After 10% off $800= $720 (800* .10= 80, 800-80=720). After 5% tax= $756 (720*.05= 36, 720+36=756).

19. A closed rectangular box with a square base is 3 inches high. If the volume of the box is 48 cubic inches, what is the surface area of the box in square inches?

90

ANSWER- B
First, determine the dimension of the square base. the box's height is given as 3. box's volume to be is 48:
48= 3/w
48/3= w
w= 16
becasue the base is square, each side must be 4 inches ( 4*4=16). now you can calculate the total surface area:
2w+ 2wh+2w= 2*12+ 2*16+ 2*12
= 24+ 32+24
= 80

66

81

To find the surface area, determine the dimensions of the square base using the volume provided. Once the base side length is found to be 4 inches, calculate the total surface area using the formula. The correct answer is 80 square inches.

Explanation

To find the surface area, determine the dimensions of the square base using the volume provided. Once the base side length is found to be 4 inches, calculate the total surface area using the formula. The correct answer is 80 square inches.

20. If x^2 varies directly as y and if x=2 when y=10, what is the value of y when x=8?

B- 130

A- 32

ANSWER- C
4:10= 64:y
4y= 64 *10
y= 640/4
y=160

D- 168

When x=2, y=10. To find y when x=8, use the direct variation formula. (2^2):10 = (8^2):y. Simplifying gives 4:10 = 64:y, hence y = 160.

Explanation

When x=2, y=10. To find y when x=8, use the direct variation formula. (2^2):10 = (8^2):y. Simplifying gives 4:10 = 64:y, hence y = 160.

21. An island is defended by a battery of coastal guns with a maximum range of 5 miles. A ship sailing due north at 24 mph will be within 4 miles of these guns on a straight course. How long will the ship be within range of the coastal guns?

C- 25 min

D- 30 min

B- 20 min

ANSWER- A
Referring to the sketch above, the ship will be within range of the coastal guns from point A to point C. Triangle OBA and triangle OBC are right triangles with one leg of 4 miles and a hypotenuse of 5 miles.

The time for the ship to travel from point A to point C must be determined. Let x = the distance from A to B; 2x = the distance from A to C.

Let t = time required to travel from A to C (in minutes).

The ship will be within range of the coastal guns from point A to point C based on the provided diagram and calculations.

Explanation

The ship will be within range of the coastal guns from point A to point C based on the provided diagram and calculations.

22. A tank that holds 500 gallons of water can be filled by one pipe in 25 minutes and emptied by another in 50 minutes. How long would it take to fill the tank if both pipes are open?

C. 55 minutes

D. 56 minutes

ANSWER- A
The filling rate is gallons/minute. The emptying rate is gallons/minute.

The difference, 20 –10 = 10, is how many gallons are added per minute with both pipes open, so it takes minutes to fill the tank.

B. 52 minutes

The filling rate is 20 gallons/minute. The emptying rate is 10 gallons/minute. The difference, 20 - 10 = 10, is how many gallons are added per minute with both pipes open, so it takes 50 minutes to fill the tank.

Explanation

The filling rate is 20 gallons/minute. The emptying rate is 10 gallons/minute. The difference, 20 - 10 = 10, is how many gallons are added per minute with both pipes open, so it takes 50 minutes to fill the tank.

23. A class of 216 recruits consists of three racial and ethnic groups. If 72 are black, 54 are Hispanic, and the remaining recruits are white, how many of the recruits in the class are white?

ANSWER- B
1/3 * 216= 72 blacks
1/4 * 216= 54 hispanics
72+54= 126
216- 126= 90 whites

94

75

68

To find the number of white recruits, subtract the total of black and Hispanic recruits from the total recruits in the class. 72 + 54 = 126. 216 - 126 = 90.

Explanation

To find the number of white recruits, subtract the total of black and Hispanic recruits from the total recruits in the class. 72 + 54 = 126. 216 - 126 = 90.

24. If x is less than 0 and y is greater than 0, then

D. xy is greater than 0.

A. x is greater than y.

B. y is greater than x.

ANSWER- C
When a negative number and a positive number are multiplied, the product is negative.

When a negative number and a positive number are multiplied, the product is negative.

Explanation

When a negative number and a positive number are multiplied, the product is negative.

25. If a driver completes a trip of 240 miles at the rate of 30 mph, at what rate would the driver have to travel on the return trip in order to average 40 mph for the round trip?

ANSWER- C
Traveling at 30 mph for 8 hours, the driver travels 240 miles; the round trip was 480 miles. Coming back, 480 miles at 40 mph would take 12 hours:

If the first 240 miles took 8 hours, the return 240 miles must be covered in 4 hours, which equals a speed of 60 mph.

A. 50 mph

D. 65 mph

B. 55 mph

Traveling at 30 mph for 8 hours, the driver travels 240 miles; the round trip was 480 miles. Coming back, 480 miles at 40 mph would take 12 hours. If the first 240 miles took 8 hours, the return 240 miles must be covered in 4 hours, which equals a speed of 60 mph.

Explanation

Traveling at 30 mph for 8 hours, the driver travels 240 miles; the round trip was 480 miles. Coming back, 480 miles at 40 mph would take 12 hours. If the first 240 miles took 8 hours, the return 240 miles must be covered in 4 hours, which equals a speed of 60 mph.

26. Angle E is 40° smaller than its complement. The number of degrees in angle E is

B. 45°

D. 90°

C. 60°

ANSWER- A
Let x equal the number of degrees in angle E.

Since angle E is 40° smaller than its complement, we can set up the equation x + (x + 40) = 90 where x represents angle E. Solving this equation gives us x = 25°, so the correct answer is a. 25°.

Explanation

Since angle E is 40° smaller than its complement, we can set up the equation x + (x + 40) = 90 where x represents angle E. Solving this equation gives us x = 25°, so the correct answer is a. 25°.

27. Two ships are 2,000 miles apart and sailing toward each other. One sails at the rate of 80 miles per day and the other at the rate of 100 miles per day. How far apart will they be at the end of 9 days?

ANSWER- B
80 × 9 days = 720 miles; 100 × 9 days = 900 miles; 720 + 900 = 1620 miles; 2000 – 1620 = 380 miles

C. 440 miles

D. 500 miles

A. 364 miles

80 x 9 days = 720 miles; 100 x 9 days = 900 miles; 720 + 900 = 1620 miles; 2000 - 1620 = 380 miles.

Explanation

80 x 9 days = 720 miles; 100 x 9 days = 900 miles; 720 + 900 = 1620 miles; 2000 - 1620 = 380 miles.

28. Successive discounts of 20 percent and 10 percent are equivalent to a single discount of

B. 29 percent.

C. 30 percent.

ANSWER- A
The correct answer is (A). 100 × .20 = 20; 100 – 20 = 80; 80 × .10 = 8; 80 – 8 = 72; 100 – 72 = 28.

D. 31 percent.

To find the equivalent single discount, we start with 100 (original price). With a 20% discount, the price becomes 80. Then, with a 10% discount, the price becomes 80 * 0.90 = 72. So, the equivalent single discount would be 100 - 72 = 28 percent.

Explanation

To find the equivalent single discount, we start with 100 (original price). With a 20% discount, the price becomes 80. Then, with a 10% discount, the price becomes 80 * 0.90 = 72. So, the equivalent single discount would be 100 - 72 = 28 percent.

29. A bridge crosses a river that is 1,400 feet wide. One bank of the river holds 1/5 of the bridge, while the other holds 1/10 of it. How long is the bridge?

A. 1,700 feet

B. 1,800 feet

ANSWER- C
1/ 5 X+ 1/10 + 1400= X
2X+ X + 14000= 10X
14000= 7X
2000= X

D. 2,100 feet

By solving for the length of the bridge using the given fractions and the width of the river, it is determined that the bridge is 2,000 feet long.

Explanation

By solving for the length of the bridge using the given fractions and the width of the river, it is determined that the bridge is 2,000 feet long.

30. What is the third root of 64?

2

8

ANSWER- C
4*4*4= 64

3

To find the third root of a number, you need to find a number that when multiplied by itself three times equals the original number. In this case, 4 * 4 * 4 = 64.

Explanation

To find the third root of a number, you need to find a number that when multiplied by itself three times equals the original number. In this case, 4 * 4 * 4 = 64.

Astb Math Quiz