ASTB Math Mastery Challenge: Test Your Skills

1) A field can be plowed by 12 machines in 7 hours. If 4 machines are broken and cannot be used, how many hours will it take to plow the field?

X = (12 * 7) / 6 = 14 hours

Let x equal the number of hours to plow with 8 machines. So,

X = (12 * 7) / 10 = 8.4 hours

X = (12 * 7) / 4 = 21 hours

The correct answer can be found by using the formula 'work = rate * time'. Since 12 machines can plow the field in 7 hours, that means the rate of plowing is 12/7. When 4 machines are broken, the number of working machines reduces to 8. So, to find the new time it takes to plow the field with 8 machines, the formula becomes (12 * 7) / 8 = 10.5 hours.

Explanation

The correct answer can be found by using the formula 'work = rate * time'. Since 12 machines can plow the field in 7 hours, that means the rate of plowing is 12/7. When 4 machines are broken, the number of working machines reduces to 8. So, to find the new time it takes to plow the field with 8 machines, the formula becomes (12 * 7) / 8 = 10.5 hours.

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

7 ounces

3 ounces

10 ounces

To solve this problem, we set up an equation based on the amount of pure acid in the initial 55% acid solution and the final 75% acid solution. By solving this equation, we find that 5 ounces of pure acid must be added to achieve the desired concentration.

Explanation

To solve this problem, we set up an equation based on the amount of pure acid in the initial 55% acid solution and the final 75% acid solution. By solving this equation, we find that 5 ounces of pure acid must be added to achieve the desired concentration.

3) Jane received grades of 93, 82, and 72 on three tests. What grade must she receive on the next test so that her average for these four tests is 85?

Jane must earn 79 on the next test to maintain an average of 85.

Jane must earn 88 on the next test to maintain an average of 85.

Jane must earn 100 on the next test to maintain an average of 85.

Jane's current test grades total 247: 93 + 82 + 72 = 247. To average 85, Jane's grades must total 340: 85 * 4 = 340. To raise her average grade, Jane must earn 93 on the next test: 340 - 247 = 93.

Explanation

Jane's current test grades total 247: 93 + 82 + 72 = 247. To average 85, Jane's grades must total 340: 85 * 4 = 340. To raise her average grade, Jane must earn 93 on the next test: 340 - 247 = 93.

4) What is the equivalent representation of 1,000,000?

1,000,000 / 1

100 x 10,000

1,000 x 1,000

10 × 10 × 10 × 10 × 10 × 10 = 1,000,000, or 10 raised to the 6th power.

1,000,000 can be represented as 10 multiplied by itself 6 times, resulting in 10^6.

Explanation

1,000,000 can be represented as 10 multiplied by itself 6 times, resulting in 10^6.

5) If a = 4b and 8b = 30c, then a =

20c

10c

35c

A = 4b 2a = 8b = 30c 2a = 30ca = 15c.

To find the value of a, we first use the fact that a = 4b. Substituting 4b in place of a in the second equation 8b = 30c gives us 8(4b) = 30c. Simplifying this gives us 32b = 30c, which further simplifies to 16b = 15c. Finally, substituting back the value of b in a = 4b gives us a = 4(15c) = 15c.

Explanation

To find the value of a, we first use the fact that a = 4b. Substituting 4b in place of a in the second equation 8b = 30c gives us 8(4b) = 30c. Simplifying this gives us 32b = 30c, which further simplifies to 16b = 15c. Finally, substituting back the value of b in a = 4b gives us a = 4(15c) = 15c.

6) 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?

48 quarters

81 quarters

72 quarters

In this question, the solution involves setting up an equation based on the given information and solving for the unknown quantity, which is the number of quarters in the cash box. The correct answer is derived by carefully analyzing the relationship between the different types of coins present in the bank and the probability of selecting a quarter.

Explanation

In this question, the solution involves setting up an equation based on the given information and solving for the unknown quantity, which is the number of quarters in the cash box. The correct answer is derived by carefully analyzing the relationship between the different types of coins present in the bank and the probability of selecting a quarter.

7) A CD system originally priced at $1,000 is first discounted by 20%, then later by another 10%. If a 5% tax is added to the purchase price, how much would a customer buying the system at its lowest price pay for it to the nearest dollar, including tax?

$740

$725

$770

Explanation for the multiple-choice question will be provided once the user selects an answer to explain why it is incorrect.

Explanation

Explanation for the multiple-choice question will be provided once the user selects an answer to explain why it is incorrect.

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

36 square inches

16 square inches

40 square inches

First, determine the dimensions of the square base. The box's height is given as 3. Accordingly, knowing the box's volume to be 48: Because the base is square, each side must be 4 inches (4 x 4 = 16). Now you can calculate the total surface area: 2(lw + lh + wh) = 2(16 + 12 + 12) = 2(40) = 80 square inches. Therefore, the correct surface area is 80 square inches, not 16, 36, or 40 square inches.

Explanation

First, determine the dimensions of the square base. The box's height is given as 3. Accordingly, knowing the box's volume to be 48: Because the base is square, each side must be 4 inches (4 x 4 = 16). Now you can calculate the total surface area: 2(lw + lh + wh) = 2(16 + 12 + 12) = 2(40) = 80 square inches. Therefore, the correct surface area is 80 square inches, not 16, 36, or 40 square inches.

9) 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?

35 mph

50 mph

30 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.

10) If x is less than 0 and y is greater than 0, then what is the result when x*y is calculated?

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

The product is zero.

The product is undefined.

The product is positive.

When multiplying a negative number with a positive number, the result will always be negative due to the rules of multiplication with signed numbers.

Explanation

When multiplying a negative number with a positive number, the result will always be negative due to the rules of multiplication with signed numbers.

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

50°

Let x equal the number of degrees in angle E.

70°

60°

To find the correct answer, we need to consider the relationship between an angle and its complement, and set up an equation based on the information provided in the question.

Explanation

To find the correct answer, we need to consider the relationship between an angle and its complement, and set up an equation based on the information provided in the question.

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

540 miles

620 miles

750 miles

The correct way to calculate the distance is to find out how far each ship has traveled in 9 days, then subtract this total from the initial 2,000 miles to get the remaining distance between them.

Explanation

The correct way to calculate the distance is to find out how far each ship has traveled in 9 days, then subtract this total from the initial 2,000 miles to get the remaining distance between them.

13) Successive discounts of 20 percent and 10 percent are equivalent to a single discount of:

30%

25%

35%

Successive discounts of 20% and 10% are calculated as follows: 100 * 0.20 = 20; 100 - 20 = 80; 80 * 0.10 = 8; 80 - 8 = 72; Therefore, the equivalent single discount is 100 - 72 = 28%.

Explanation

Successive discounts of 20% and 10% are calculated as follows: 100 * 0.20 = 20; 100 - 20 = 80; 80 * 0.10 = 8; 80 - 8 = 72; Therefore, the equivalent single discount is 100 - 72 = 28%.

14) A bridge crosses a river that is 1,400 feet wide. One bank of the river holds 1/3 of the bridge, while the other holds 1/5 of it. How long is the bridge?

1200 feet

2

1600 feet

1800 feet

To find the total length of the bridge, you need to add the lengths represented by '1/3' and '1/5' of the bridge. This results in 1/3 + 1/5 = 5/15 + 3/15 = 8/15 of the bridge being accounted for. Since 8/15 represents 1,400 feet, dividing 1,400 by 8/15 gives you 2100 feet as the total length of the bridge.

Explanation

To find the total length of the bridge, you need to add the lengths represented by '1/3' and '1/5' of the bridge. This results in 1/3 + 1/5 = 5/15 + 3/15 = 8/15 of the bridge being accounted for. Since 8/15 represents 1,400 feet, dividing 1,400 by 8/15 gives you 2100 feet as the total length of the bridge.

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

Using the Pythagorean theorem, the hypotenuse of the right triangle = 5: 3′ + 4′ + 5′ = 12

8 feet

15 feet

10 feet

The perimeter of a right triangle is the sum of all its sides. In this case, the legs of the right triangle are given as 3 feet and 4 feet. By using the Pythagorean theorem, we find the hypotenuse to be 5 feet. Therefore, the perimeter would be 3 feet + 4 feet + 5 feet = 12 feet.

Explanation

The perimeter of a right triangle is the sum of all its sides. In this case, the legs of the right triangle are given as 3 feet and 4 feet. By using the Pythagorean theorem, we find the hypotenuse to be 5 feet. Therefore, the perimeter would be 3 feet + 4 feet + 5 feet = 12 feet.

16) What is the third root of 64?

4 × 4 × 4 = 64.

8

5

6

The third root of a number is the value that when multiplied by itself three times equals the original number. In this case, 4 x 4 x 4 = 64.

Explanation

The third root of a number is the value that when multiplied by itself three times equals the original number. In this case, 4 x 4 x 4 = 64.

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

B) It will take the faster train 2 hours to catch up with the slower train.

The slower train is 40 miles ahead in one hour. The difference in rate is 5 mph, and

A) It will take the faster train 5 hours to catch up with the slower train.

C) It will take the faster train 10 hours to catch up with the slower train.

The correct answer can be found by dividing the initial distance difference of 40 miles by the difference in speed of 5 mph. This results in 8 hours for the faster train to catch up with the slower train.

Explanation

The correct answer can be found by dividing the initial distance difference of 40 miles by the difference in speed of 5 mph. This results in 8 hours for the faster train to catch up with the slower train.

18) A naval detachment has enough rations to feed fifteen people for 8 days. If five more people join the detachment, for how many fewer days will the rations last?

Let x = the number of ration days for 20 persons.

19) What is the square root of 4 raised to the fifth power?

64

16

8

To find the answer, we first calculate 4^5 which is 1024. Then, find the square root of 1024 which is 32.

Explanation

To find the answer, we first calculate 4^5 which is 1024. Then, find the square root of 1024 which is 32.

20) If the sum of the edges of a cube is 24 inches, what is the volume of the cube?

Each edge is 4 inches and the volume of the cube is 64 cubic inches.

A cube has 12 edges, so each edge is

Each edge is 3 inches and the volume of the cube is 27 cubic inches.

Each edge is 6 inches and the volume of the cube is 216 cubic inches.

For a cube, the sum of the edges can be found by multiplying the number of edges (12 for a cube) with the length of each edge. Therefore, if the sum of edges is 24 inches, each edge is 2 inches (24 / 12 = 2 inches). The volume of a cube is calculated by cubing the length of one of its edges (V = edge³). So, with each edge being 2 inches, the volume of the cube is 2³ = 8 cubic inches.

Explanation

For a cube, the sum of the edges can be found by multiplying the number of edges (12 for a cube) with the length of each edge. Therefore, if the sum of edges is 24 inches, each edge is 2 inches (24 / 12 = 2 inches). The volume of a cube is calculated by cubing the length of one of its edges (V = edge³). So, with each edge being 2 inches, the volume of the cube is 2³ = 8 cubic inches.

21) 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 70 feet long. How high is the barn?

42 feet

48 feet

36 feet

4

This is a proportionality problem where the ratio of the height of the object to its shadow length is equal for both the farmer and the barn. So, in this case, the barn is 30 feet tall.

Explanation

This is a proportionality problem where the ratio of the height of the object to its shadow length is equal for both the farmer and the barn. So, in this case, the barn is 30 feet tall.

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

15

5

3

-5

To solve this question, we need to use the formulas for the sum and product of roots in a quadratic equation. By expanding (x - y)^2 = 60, we get x^2 - 2xy + y^2 = 60. Combining this with x^2 + y^2 = 40, we get -2xy = 20, which means xy = -10. Therefore, the correct answer is -10.

Explanation

To solve this question, we need to use the formulas for the sum and product of roots in a quadratic equation. By expanding (x - y)^2 = 60, we get x^2 - 2xy + y^2 = 60. Combining this with x^2 + y^2 = 40, we get -2xy = 20, which means xy = -10. Therefore, the correct answer is -10.

23) In the figure shown below, what is the measure of angle x?

Arc AB = 90° therefore, AOB = 90°. The two radii are equal.

Arc AB = 60° therefore, AOB = 60°. The two radii are equal.

Arc AB = 80° therefore, AOB = 80°. The two radii are equal.

Arc AB = 100° therefore, AOB = 100°. The two radii are equal.

The measure of angle x is determined by the arc AB in the given figure, as the two radii are equal the angle formed at the center is also equal to the arc's measure.

Explanation

The measure of angle x is determined by the arc AB in the given figure, as the two radii are equal the angle formed at the center is also equal to the arc's measure.

24) A family drove from New York to San Francisco, a distance of 3,000 miles. They drove 1/3 of the distance the first day and 1/2 of the remaining distance the second day. How many miles were left to be driven?

1,250 miles

1,500 miles

3

1,000 miles

On the first day, they drove 1/3 of 3,000 miles, which is 1,000 miles. The remaining distance was 2,000 miles. On the second day, they drove 1/2 of the remaining distance, which is 1,000 miles. Therefore, the total distance driven was 1,000 miles + 1,000 miles = 2,000 miles. The miles left to be driven are 3,000 miles - 2,000 miles = 1,000 miles.

Explanation

On the first day, they drove 1/3 of 3,000 miles, which is 1,000 miles. The remaining distance was 2,000 miles. On the second day, they drove 1/2 of the remaining distance, which is 1,000 miles. Therefore, the total distance driven was 1,000 miles + 1,000 miles = 2,000 miles. The miles left to be driven are 3,000 miles - 2,000 miles = 1,000 miles.

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

The correct approach is to assign 'x' as the minutes to be spent on each math problem and then calculate 60 * x to get the total time to be spent on all 60 math problems.

Explanation

The correct approach is to assign 'x' as the minutes to be spent on each math problem and then calculate 60 * x to get the total time to be spent on all 60 math problems.

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

When x = 8, y = 40

When x = 8, y = 30

When x = 8, y = 20

4

To find the value of y when x = 8, we use the direct variation formula x^2 = ky. Given x = 2 when y = 10, we can plug in these values to find k. Once we have k, we can then find y when x = 8 by solving for y using the same formula.

Explanation

To find the value of y when x = 8, we use the direct variation formula x^2 = ky. Given x = 2 when y = 10, we can plug in these values to find k. Once we have k, we can then find y when x = 8 by solving for y using the same formula.

27) An island is defended by a battery of coastal guns placed at the easternmost point of the island and having a maximum range of 5 miles. A ship, sailing due north at 24 mph along a course that will bring it within 4 miles of these guns, is approaching the position where it will be 5 miles from these guns. Assuming that the ship will maintain a straight course and the same speed, for approximately how long will the ship be within range of the coastal guns?

The ship will be within range of the coastal guns for the entire duration of its journey.

The ship will never be within range of the coastal guns due to its speed and course.

The ship will only be within range of the coastal guns briefly before moving out of range.

The correct answer provides a step-by-step explanation of how to calculate the time the ship will be within range of the coastal guns based on its position and speed. The incorrect answers provide common misconceptions that can be easily refuted based on the information given in the question.

Explanation

The correct answer provides a step-by-step explanation of how to calculate the time the ship will be within range of the coastal guns based on its position and speed. The incorrect answers provide common misconceptions that can be easily refuted based on the information given in the question.

28) 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?

The tank will never fill with both pipes open as the emptying rate is higher than the filling rate.

It will take 37.5 minutes to fill the tank with both pipes open.

The key to solving this question is understanding the concept of filling and emptying rates when pipes are working together.

Explanation

The key to solving this question is understanding the concept of filling and emptying rates when pipes are working together.

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

108

90

3

72

The total number of recruits is 216. If 36 are black and 54 are Hispanic, then the total number of black and Hispanic recruits is 36 + 54 = 90. Subtracting this from the total gives us 216 - 90 = 126 white recruits. Therefore, the correct answer is 144.

Explanation

The total number of recruits is 216. If 36 are black and 54 are Hispanic, then the total number of black and Hispanic recruits is 36 + 54 = 90. Subtracting this from the total gives us 216 - 90 = 126 white recruits. Therefore, the correct answer is 144.

30) If 3n - 2 = 27, what does n equal?

N = 13

N = 11

N = 7

4

To solve the equation, first add 2 to both sides: 3n = 29. Then divide by 3 to solve for n, giving n = 9.

Explanation

To solve the equation, first add 2 to both sides: 3n = 29. Then divide by 3 to solve for n, giving n = 9.