LearnAlgebra foundationsWord problems

Problems requiring one variable

Exercises

Problem set

  1. My age in ten years will be twice my age now. How old am I now?
  2. Rachel’s age now is three times as much as her age was 18 years ago. How old is Rachel now?
  3. Raj’s age 5 years ago was one-third of his age 21 years from now. How old is Raj now?
  4. Adam’s age 3 years ago was half of what his age will be in 9 years. What is his age now?
  5. I bought 10 apples from a vendor. The number of apples he has now is seven-eighth’s of what he had before. How many apples are with the vendor now?

Problem set

  1. The highest score in a test is five less than four times the lowest score. If the highest score is 91, what is the lowest score?
  2. Donald did not have any candies. His brother gave him two-fifths of the candies that he had. Donald also got three candies from his mom. Now, Donald has 25 candies in all. How many candies are left with Donald’ brother now?
  3. Dev had certain number of marbles. His aunt gave him twice as many as he had. But, he found that 5 of the marbles that his aunt gave were broken. He counted all the good marbles that he has now to be 58. How many marbles did his aunt give him?

Problems requiring two variables

Exercises

Problem set

  1. p and q are two numbers whose sum is 134 and whose difference is 88. What are the two numbers?
  2. The sum of two numbers is 112. The difference between the half the first number and seven times the second number is 26. What is the second number?
  3. A and B are two consecutive multiples of 11, and they sum to 187. What are the values of A and B?
  4. Jack and Jill have the same number of marbles. Jack gives 3 of his marbles to Jill. And, Jill gets 2 more marbles from her brother. Now, Jill has three times as many marbles as Jack. How many marbles does Jill have now?
  5. Amy had a fourth as many pencils as Nicole had. Amy lost three of her pencils, while Nicole added two more pencils to her collection. Now, Amy has a sixth as many pencils as Nicole has. How many pencils does Amy have now?

Problem set

  1. I had a third as many candies as my brother. I gave two of my candies to my brother. Now, I have a fourth as many candies as him. How many candies do I have now?
  2. Brian and his brother each maintain a stamp collection. Last year, there were same number of stamps in both the collections. Since then, Brian added 150 stamps to his collection, while his brother added only 20 stamps to his collection. Now, Brian has three times as many stamps as his brother. How many stamps does Brian have now in his collection?
  3. Robert has 21 less pencils than his friend Isabel. They have a new friend Donna to whom Robert gives 5 of his pencils and Isabel gives 10 of her pencils. Now, Robert has half as many pencils as Isabel. How many pencils does Isabel have now?
  4. Amy and Jack together have 80 pebbles. Amy gave 5 of her pebbles to Jack. Now, Jack has a third as many pebbles as Amy does. How many pebbles does Amy have now?

Problem set

  1. Arun is 11 years older than Valeria. When Arun was half his current age, Valeria was half as old as Arun. How old is Valeria now?
  2. Raul has 8 times as many candies as Vladimir. Raul gives 6 of his candies to Vladimir. Now, Raul has twice as many candies as Vladimir. How many candies does Raul have now?
  3. Sum of the two digits of a two digit number is 11. If the digits of the number are reversed, the new two digit number obtained is 27 more than the original number. What is the original number?
  4. Two bikers start biking at the same time from the two ends of a bike trail towards each other. The first biker bikes at 2mph, while the second biker bikes at 3mph. The trail is 20 mile long. How much time after they start do they cross each other?
  5. In the above problem, how much distance does the second biker bike before he crosses the first biker?

Problems involving ratios

Exercises

Problem set

  1. Jon’s dad divided 35 candies in the ratio 3:4 to Jon and his brother. How many candies did Jon get?
  2. Julian and Marco have candies in the ratio 4:7. Marco has 42 candies more than Julian. How many candies does Marco have?
  3. Jaden and Nicolai collect stamps. The numbers of stamps they have are in the ratio 5:7. If Jaden gives 5 of his stamps to Nicolai, Nicolai would have twice as many stamps as Jaden. How many stamps does Jaden have?
  4. Ron, Kannan and Debbie have marbles in the ratio 5:2:7.
    1. If Debbie has 35 marbles more than Kannan, how many marbles does Ron have?
    2. If Debbie and Ron together have 108 marbles, how many marbles does Kannan have?

Problem set

  1. Xenia and Saloni collect shells. The ratio of the number of shells that Xenia has to the number of shells that Saloni has is 8:5. Xenia gave 6 of her shells to Saloni. Xenia now has 9 more shells than Saloni. How many shells does Saloni have now?
  2. The ages of Nancy and Bruce are in the ratio of 4:7. Seven years ago, their ages summed to 52. How old is Bruce now?
  3. In a dealership that has green cars and gray cars in its inventory, the ratio of number of green cars to the number of gray cars is 2:3. Because of demand for green cars, the dealership added 8 more green cars to its inventory. The ratio of green cars to gray cars is now 5:6. What is the number of cars in the dealership now?
  4. A basket has apples and oranges. The ratio of apples to oranges is 10:13. Six more apples and six more oranges were added to the basket. Now, the ratio of apples to oranges is 4:5. What is the number of apples in the basket now?
  5. In 2017, the ratio of teachers to students in a school was 2:15. In 2018, in addition to retaining all its students and teachers from 2017, the school expanded to recruit two more teachers and enroll 40 additional students. The total number of students and teachers in the school now is 280.
    1. What is the teacher to student ratio in the school in 2018?
    2. Did the teacher to student ratio become better or worse in 2018 compared to 2017?

Problems involving dollar amounts

Exercises

Problem set

  1. Yomi has same number of dimes as Monica has quarters. If Monica has \$4.50 more money than Yomi, how many dimes does Yomi have?
  2. Julia has some quarters and some dimes. The total number of coins she has is 31, and the coins amount to \$4.75. How many quarters does Julia have?
  3. Noah has some quarters and nickels amounting to \$9.50. He has 10 more nickels than he has quarters. How many nickels does Noah have?
  4. Ralph has quarters and nickels. The dollar amount he has from his quarters is twice the dollar amount he has from his nickels. What is the ratio of the number of quarters to the number of nickels that he has?
  5. Naveen has dimes and quarters amounting to \$10.50. The ratio of number of dimes to number of quarters is 5:4. How many quarters does he have?
  6. (Problem incomplete) Joey has some money in quarters and nickels. The ratio of number of quarters to number of nickels is . Joey gives a certain number of his quarters and an equal number of his nickels to his brother Chandler.

Problem set

  1. John spent \$10.00 to buy two different kinds of pens. Each pen of the first kind costed \$1.70, while each pen of the second kind costed 60c. Altogether, John bought 13 pens. How many pens of the first kind did John buy?
  2. A class of 49 has boy and girl kids. Each boy kid brings to class 2 green colored pencils, while each girl kid brings 3 blue colored pencils. No other pencils are brought to class. The ratio of total number of green pencils to blue pencils is 8 to 9. How many girl students are there in the class?
  3. Qin has \$7.00 in the form of nickels, dimes and quarters. The ratio of number of nickels to number of dimes is 4:3, while the ratio of number of dimes to number of quarters is 2:1. How many nickels does she have?
  4. Kabir has \$6.10 in quarters, dimes and nickels. The ratio of the number of quarters to the number of dimes that he has is 3:4. The total number of coins he has is 42. How many nickels does he have?
  5. Amy, Sam and Raj together have \$98. If Amy had \$4 more, she would have had twice as much as what Sam has. Raj has half as much as Amy has now. How much does Sam have now?

Problems involving proportions

Exercises

Problem set

For each of the following questions, in addition to what is being asked, determine the constant of proportionality, and also indicate what the constant of proportionality means for the problem.

  1. During a hurricane relief, 33 tonnes of grain were required to support an affected neighborhood with 18000 families. How many tonnes of grain are required to support a neighborhood of 42000 families?
  2. A truck consumes 28 gallons of gas to drive 350 miles. How many miles can the truck drive on 100 gallons of gas?
  3. 12 men are able to paint 3900 square feet of walls in a day. How many men are required to paint 900 square feet of walls in a day?
  4. 16 high school students are able to arrange books in 30 bookshelves in one hour. How many bookshelves can 30 students arrange in one hour?
  5. To pave a 5600m long road in a day, it takes 24 machines. How many machines does it take to pave a 11900m long road in the same amount of time?

Problem set

For each of the following questions, in addition to what is being asked, determine the constant of proportionality, and also indicate what the constant of proportionality means for the problem.

  1. 16 high school students are able to arrange books in a library in 5 hours. How many hours do 28 students need to arrange the library books?
  2. It takes 24 machines 5 days to pave a road. How many machines are required to pave a similar road in 3 days?
  3. A commuter covers the distance between his home and work in 35 minutes if he drives his car at 70 mph. On a particular day, he needs to drive his car on a replacement tire, and he needs to drive his car at 30 mph. How much time would he take from his home to work on that day?

Miscellaneous problems

Exercises

Problem set

  1. A rectangle was stretched along its length, while its width remained unchanged. The length of the stretched rectangle is 7 units more than that of the original rectangle. And, the area of the stretched rectangle is 35 square units more than that of the original rectangle. What is the width of the rectangle?
  2. Average of three numbers is 211. Two of the numbers are 100 and 200. What is the third number?
  3. Anthony runs 10\% faster than Lisa. They are both running the same distance. If Lisa takes 1 hour 17 minutes to run the distance, how much time does Anthony take?