Advanced Mathematical Problem Solving Quiz

1) Ron is half as old as Sam, who is three times as old as Ted. The sum of their ages is 55. How old is Ron?

B

D

A

Let T = Ted's age; S = Sam's age = 3T; R = Ron's age = T/2. The sum of the ages is 55, which means T + 3T + T/2 = 55. Solving for T gives T = 10. Sam is 30 years old and Ron is 15 years old.

Explanation

Let T = Ted's age; S = Sam's age = 3T; R = Ron's age = T/2. The sum of the ages is 55, which means T + 3T + T/2 = 55. Solving for T gives T = 10. Sam is 30 years old and Ron is 15 years old.

2) An Army food supply truck can carry three tons. A breakfast ration weighs 12 ounces, and the other two daily meals weigh 18 ounces each. Assuming each soldier gets three meals per day, on a 10-day trip, how many soldiers can be supplied by one truck?

A. 150 soldiers supplied

B. 175 soldiers supplied

D. 225 soldiers supplied

The correct calculation involves converting tons to pounds, then to ounces and dividing by the total weight per soldier per day to determine the number of soldiers supplied by the truck.

Explanation

The correct calculation involves converting tons to pounds, then to ounces and dividing by the total weight per soldier per day to determine the number of soldiers supplied by the truck.

3) Based on the following information, estimate the weight of a person who is 5'5" tall.

D. 140

B. 130

C. 135

A. 125 B. 130 C. 135 D. 140 E. 145

The estimated weight of a person who is 5'5" tall falls between the weights provided for 5' and 6' heights, which are 110 lbs and 170 lbs respectively. Taking the average, the estimated weight for a person who is 5'5" tall would be around 125 lbs.

Explanation

The estimated weight of a person who is 5'5" tall falls between the weights provided for 5' and 6' heights, which are 110 lbs and 170 lbs respectively. Taking the average, the estimated weight for a person who is 5'5" tall would be around 125 lbs.

4) A certain water pollutant is unsafe at a level of 20 ppm (parts per million). A city's water supply now contains 50 ppm of this pollutant. What percentage of improvement will make the water safe?

The correct answer lies in understanding the difference between the current pollutant level of 50 ppm and the safe level of 20 ppm. The improvement needed is the difference between these two levels, which is 30 ppm. This 30 ppm accounts for 60% of the current 50 ppm pollutant level.

Explanation

The correct answer lies in understanding the difference between the current pollutant level of 50 ppm and the safe level of 20 ppm. The improvement needed is the difference between these two levels, which is 30 ppm. This 30 ppm accounts for 60% of the current 50 ppm pollutant level.

5) A study shows that 600,000 women die each year in pregnancy and childbirth, one-fifth more than scientists previously estimated. How many such deaths did the scientists previously estimate?

E = 700,000

E = 400,000

E = 600,000

The previous estimate E can be calculated by dividing 600,000 by 1.20, which gives E = 500,000.

Explanation

The previous estimate E can be calculated by dividing 600,000 by 1.20, which gives E = 500,000.

6) A woman drives west at 45 miles per hour. After half an hour, a man starts to follow her. How fast must he drive to catch up to her three hours after he starts?

B. The man must travel at 37.5 miles per hour to catch up to the woman.

A. The man must travel at 45 miles per hour to catch up to the woman.

C. The man must travel at 60 miles per hour to catch up to the woman.

The key to solving this problem is understanding the relative distances and times the woman and man have traveled. By calculating the distance the woman has traveled in the total time, you can determine the speed the man needs to catch up to her.

Explanation

The key to solving this problem is understanding the relative distances and times the woman and man have traveled. By calculating the distance the woman has traveled in the total time, you can determine the speed the man needs to catch up to her.

7) All of the rooms in a building are rectangular, with eight-foot ceilings. One room is nine feet wide by 11 feet long. What is the combined area of the four walls, including doors and windows?

A. 152 square feet

D. 288 square feet

B. 210 square feet

C. 260 square feet

Please calculate the area of each wall correctly with the provided dimensions of the room to find the combined area.

Explanation

Please calculate the area of each wall correctly with the provided dimensions of the room to find the combined area.

8) How much water must be added to one liter of a 5% saline solution to get a 2% saline solution?

0.5 liters

2 liters

3 liters

The correct answer is C. Use the equation .05(1) = .02(x), where x is the total amount of water in the resulting 2% solution. Solving for x, you get 2.5. Subtracting the 1 liter of water already present in the 5% solution, you will find that 1.5 liters need to be added.

Explanation

The correct answer is C. Use the equation .05(1) = .02(x), where x is the total amount of water in the resulting 2% solution. Solving for x, you get 2.5. Subtracting the 1 liter of water already present in the 5% solution, you will find that 1.5 liters need to be added.

9) Newly hired nurses have to buy duty shoes at the full price of $84.50, but nurses who have served at least a year get a 15% discount. Nurses who have served at least three years get an additional 10% off the discounted price. How much does a nurse who has served at least three years have to pay for shoes?

D. $75.00

C. $70.00

A. $63.75

You can't just take 25% off the original price, because the 10% discount after three years of service is taken off the price that has already been reduced by 15%. After the 15% discount, the price is $71.83. 90% of that (subtracting 10%) is $64.65.

Explanation

You can't just take 25% off the original price, because the 10% discount after three years of service is taken off the price that has already been reduced by 15%. After the 15% discount, the price is $71.83. 90% of that (subtracting 10%) is $64.65.

10) Jason is six times as old as Kate. In two years, Jason will be twice as old as Kate is then. How old is Jason now?

Let J = Jason's age and K = Kate's age. By setting up the equations based on the information provided, we can solve for the ages of Jason and Kate. The correct answer is J = 6K, which means Jason is 6 times older than Kate. Kate's age turns out to be 1, making Jason's age 6.

Explanation

Let J = Jason's age and K = Kate's age. By setting up the equations based on the information provided, we can solve for the ages of Jason and Kate. The correct answer is J = 6K, which means Jason is 6 times older than Kate. Kate's age turns out to be 1, making Jason's age 6.

11) A patient's hospice stay cost one-fourth as much as his visit to the emergency room. His home nursing cost twice as much as his hospice stay. If his total healthcare bill was $140,000, how much did his home nursing cost?

B. N = $25,000

D. N = $50,000

A. N = $30,000

Let E = emergency room cost; H = hospice cost, which is E; N = home nursing cost, which is 2H or 2(E). The total bill is E + H + N, which is E + E + E = $140,000. Therefore, home nursing cost N would be $40,000.

Explanation

Let E = emergency room cost; H = hospice cost, which is E; N = home nursing cost, which is 2H or 2(E). The total bill is E + H + N, which is E + E + E = $140,000. Therefore, home nursing cost N would be $40,000.

12) The first digit of the square root of 112,092 is?

D

A

B

When a number is marked off in groups of two digits each, starting at the decimal point, the square root of the largest square in the left-hand group, whether one or two digits, is the first digit of the square root of the number. In this case (11-20-92), 9 is the largest square in 11, and 3 is the square root of 9.

Explanation

When a number is marked off in groups of two digits each, starting at the decimal point, the square root of the largest square in the left-hand group, whether one or two digits, is the first digit of the square root of the number. In this case (11-20-92), 9 is the largest square in 11, and 3 is the square root of 9.

13) Roberta draws two similar pentagons. The perimeter of the larger pentagon is 93 feet; one of its sides measures 24 feet. If the perimeter of the smaller pentagon equals 31 feet, then the corresponding side of the smaller pentagon measures.

A. The corresponding side of the smaller pentagon measures 20 feet.

C. The corresponding side of the smaller pentagon measures 27 feet.

D. The corresponding side of the smaller pentagon measures 30 feet.

The correct answer can be found using proportions and cross-multiplication to determine the length of the corresponding side of the smaller pentagon based on the relationship between the perimeters and known side lengths of the larger pentagon.

Explanation

The correct answer can be found using proportions and cross-multiplication to determine the length of the corresponding side of the smaller pentagon based on the relationship between the perimeters and known side lengths of the larger pentagon.

14) Isadora wants to know the perimeter of the face of a building; however, she does not have a ladder. She knows that the building's rectangular façade casts a 36-foot shadow at noon, while a nearby mailbox casts a 12-foot shadow at noon. The mailbox is 4.5 feet tall. If the length of the façade is 54 feet, what is the measure of the façade's perimeter?

B. The perimeter of the façade is 108 feet.

A. The perimeter of the façade is 67.5 feet.

C. The perimeter of the façade is 40.5 feet.

Explanation provided in correct answer.

Explanation

Explanation provided in correct answer.

15) Find the circumference, in meters, of a circle with a radius of 25 centimeters.

D. 50 cm

C. 314 cm

B. 78.5 cm

To find the circumference of a circle, you use the formula C = 2πr, where r is the radius of the circle. In this case, the radius is 25 cm, so the correct calculation is (2)(3.14)(25) = 157 cm, which equals 1.57 meters.

Explanation

To find the circumference of a circle, you use the formula C = 2πr, where r is the radius of the circle. In this case, the radius is 25 cm, so the correct calculation is (2)(3.14)(25) = 157 cm, which equals 1.57 meters.

16) An automobile gasoline gauge reads 1/8 full. After the gas tank is filled with 15 gallons, the gauge reads 7/8 full. What is the capacity of the tank?

30

25

10

Subtract the amount of gas already in the tank from the amount filled. 7/8 - 1/8 = 6/8 or 3/4. 3/4 of the capacity is equal to 15 gallons. Solving for x... x = 20.

Explanation

Subtract the amount of gas already in the tank from the amount filled. 7/8 - 1/8 = 6/8 or 3/4. 3/4 of the capacity is equal to 15 gallons. Solving for x... x = 20.

17) An automobile gasoline gauge reads 1/8 full. After the gas tank is filled with 15 gallons, the gauge reads 7/8 full. What is the capacity of the tank?

22 gallons

24 gallons

18 gallons

The correct calculation is: Subtract the amount of gas already in the tank from the amount filled. 7/8 – 1/8 = 6/8 or 3/4. ¾ of the capacity is equal to 15 gallons. 3/4 x = 15. Solving for x… x = 20.

Explanation

The correct calculation is: Subtract the amount of gas already in the tank from the amount filled. 7/8 – 1/8 = 6/8 or 3/4. ¾ of the capacity is equal to 15 gallons. 3/4 x = 15. Solving for x… x = 20.

18) A student spends ¾ of his money on tuition and 1/3 of what remains on books. If he has $50 dollars left, how much money did he start with?

$200

$250

$150

The correct answer is determined by subtracting the amount spent from 1 (total amount), which gives 1/6. Then, solve for the total amount starting with $50 left.

Explanation

The correct answer is determined by subtracting the amount spent from 1 (total amount), which gives 1/6. Then, solve for the total amount starting with $50 left.

19) A family budgets 1/3 of its income for rent, 1/4 for food and 1/6 for clothing. If the family has $420 left, what is their monthly income?

1960

1150

1420

The correct way to solve this question is to add up 1/3, 1/4, and 1/6 to get 3/4. Then subtract 3/4 from 1 to get 1/4. Finally, multiply 1/4 by $420 to find the monthly income.

Explanation

The correct way to solve this question is to add up 1/3, 1/4, and 1/6 to get 3/4. Then subtract 3/4 from 1 to get 1/4. Finally, multiply 1/4 by $420 to find the monthly income.

20) Two cars travel toward each other at the same time from cities 416 miles apart. The average rate of one car is 10 m.p.h. faster than the other car. If they pass each other in 4 hours, what is the average rate of each car?

50 mph and 60 mph

40 mph and 50 mph

45 mph and 55 mph

To find the average rate of each car, we first use the formula D = R x T. By setting up the distances travelled by each car (slower car = 4s and faster car = 4s + 40) and adding them together to equal the total distance of 416 miles, we can solve for s. Therefore, the average rate of the slower car is 47 mph and the faster car is 57 mph.

Explanation

To find the average rate of each car, we first use the formula D = R x T. By setting up the distances travelled by each car (slower car = 4s and faster car = 4s + 40) and adding them together to equal the total distance of 416 miles, we can solve for s. Therefore, the average rate of the slower car is 47 mph and the faster car is 57 mph.

21) A man leaves his home and jogs at 8 m.p.h. Fifteen minutes later, his wife leaves home and bikes after him at 11m.p.h. How long after the wife leaves does she catch up with her husband?

1 hour

20 minutes

30 minutes

The correct answer is found by setting the distances traveled by both the husband and wife equal to each other. The correct answer is 40 minutes, as calculated by solving 11h = 8h + 2 for h.

Explanation

The correct answer is found by setting the distances traveled by both the husband and wife equal to each other. The correct answer is 40 minutes, as calculated by solving 11h = 8h + 2 for h.

22) A plane makes a round trip from San Francisco airport, which lasts 5 hours. If the average rate out is 200 m.p.h. and the average rate back is 300 m.p.h., what is the plane’s greatest distance from the airport?

500 miles

400 miles

700 miles

The correct answer is obtained by calculating the distance flown out and back at their respective speeds. The plane's greatest distance from the airport is 600 miles based on the given average speeds and total trip duration.

Explanation

The correct answer is obtained by calculating the distance flown out and back at their respective speeds. The plane's greatest distance from the airport is 600 miles based on the given average speeds and total trip duration.

23) Travis makes a 255-mile trip by car. He drives the first 2 hours at 45 miles per hour. At what speed must he travel for the remainder of the trip in order to arrive at his destination 5 hours after he started the trip?

60 miles per hour

50 miles per hour

40 miles per hour

Travis needs to travel at 55 miles per hour for the remaining distance of 165 miles to reach his destination in 3 hours, totaling 5 hours for the trip.

Explanation

Travis needs to travel at 55 miles per hour for the remaining distance of 165 miles to reach his destination in 3 hours, totaling 5 hours for the trip.

24) Three pipes can fill a tank in 8, 12 and 24 minutes respectively. How long would it take all 3 pipes, working together to fill the tank?

6 minutes

15 minutes

10 minutes

§ H/8 + H/12 + H/24 = 1§ Solving for H…H =4 minutes.

The correct answer is obtained by using the formula H/8 + H/12 + H/24 = 1 and solving for H, which gives H = 4 minutes. This means all three pipes working together can fill the tank in 4 minutes.

Explanation

The correct answer is obtained by using the formula H/8 + H/12 + H/24 = 1 and solving for H, which gives H = 4 minutes. This means all three pipes working together can fill the tank in 4 minutes.

25) Fritz can build a bookcase in 8 hours and Holly can build it in 10 hours. After working together for 3 hours, Fritz leaves. How long will it take Holly to finish the bookcase by herself?

6.5 hours

4 hours

2.5 hours

In 3 hours time, Fritz has completed 3/8 of the bookcase and Holly has completed 3/10. Fritz leaves, so Holly works alone at x/10. 3/8 + 3/10 + x/10 = 1. Solving for x, we get x = 3.25 hours.

Explanation

In 3 hours time, Fritz has completed 3/8 of the bookcase and Holly has completed 3/10. Fritz leaves, so Holly works alone at x/10. 3/8 + 3/10 + x/10 = 1. Solving for x, we get x = 3.25 hours.

26) Working alone, Allen can build a fence in 8 hours, and Danny can build the same fence in 16 hours. If after working alone for 2 hours, Allen is joined by Danny, how many more hours will it take the two of them to complete the fence?

6 hours

2 hours

8 hours

Allen has worked two hours, so he has completed 2/8 of the job. Danny joins him, so we will get: 2/8 + x/8 + x/16 = 1. Solving for x gives x = 4 hours.

Explanation

Allen has worked two hours, so he has completed 2/8 of the job. Danny joins him, so we will get: 2/8 + x/8 + x/16 = 1. Solving for x gives x = 4 hours.

27) A lawn is 21 feet wide and 39 feet long. How much will it cost to weed and feed it if a gardening service charges $0.40 per square yard for this treatment?

$61.80

$86.00

$42.30

First convert feet to yards. 3ft = 1yd. 21/3 = 7, 39/3 = 13. Area of a rectangle is l x w. 13 x 7 = 91. 91 x 0.4 = $36.40.

Explanation

First convert feet to yards. 3ft = 1yd. 21/3 = 7, 39/3 = 13. Area of a rectangle is l x w. 13 x 7 = 91. 91 x 0.4 = $36.40.

28) In a clothing factory, 5 workers finish production of 6 garments each per day, 3 others turn out 4 garments each per day, and one worker turns out 12 per day. What is the average number of garments produced per worker per day?

4

8

10

To find the average number of garments produced per worker per day, you need to calculate the total number of garments produced by all workers and then divide it by the total number of workers. In this case, the total number of garments produced is 54 and the total number of workers is 9, leading to an average of 6 garments per worker per day.

Explanation

To find the average number of garments produced per worker per day, you need to calculate the total number of garments produced by all workers and then divide it by the total number of workers. In this case, the total number of garments produced is 54 and the total number of workers is 9, leading to an average of 6 garments per worker per day.

29) How many degrees are in an angle that is 20 degrees less than 4 times its supplement?

X = 160 degrees

X = 120 degrees

X = 180 degrees

To find the answer, set up the equation X = 4(180 - X) - 20 and solve for X. In this case, X = 140 degrees.

Explanation

To find the answer, set up the equation X = 4(180 - X) - 20 and solve for X. In this case, X = 140 degrees.

30) The perimeter of a rectangle is 50 ft. If the length of the rectangle is 4 more than twice its width, what are the dimensions of the rectangle?

Width is 8 and length is 20.

Width is 5 and length is 14.

Width is 11 and length is 22.

The correct way to solve this problem is to set up the equation 50 = 2(2x + 4) + 2x, which represents the perimeter of the rectangle. By solving for x, we find that the width is 7 and the length is 18. The incorrect answers show different dimensions that do not satisfy the given conditions of the problem.

Explanation

The correct way to solve this problem is to set up the equation 50 = 2(2x + 4) + 2x, which represents the perimeter of the rectangle. By solving for x, we find that the width is 7 and the length is 18. The incorrect answers show different dimensions that do not satisfy the given conditions of the problem.