Problems requiring one variable

Exercises

Problem set

1. In ten years, I will be twice as old as I am now. How old am I now?
2. Rachel’s age now is three times as much as her age was years ago. How old is Rachel now?
3. Raj’s age years ago would be one-third of his age years from now. How old is Raj now?
4. I bought 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?
5. The highest score in a test is five less than four times the lowest score. If the highest score is , what is the lowest score?

Problem set

1. Adam’s age years ago is half of what his age will be in years. What is his age now?
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 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 of the marbles that his aunt gave were broken. He counted all the good marbles that he has now to be . How many marbles did his aunt give him?

Problems requiring two variables

Exercises

Problem set

1. and are two numbers whose sum is and whose difference is . What are the two numbers?
2. The sum of two numbers is . The difference between the half the first number and seven times the second number is . What is the second number?
3. and are two consecutive multiples of , and they sum to . What are the values of and ?
4. Jack and Jill have the same number of marbles. Jack gives of his marbles to Jill. And, Jill gets 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 stamps to his collection, while his brother added only 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 less pencils than his friend Isabel. They have a new friend Donna to whom Robert gives of his pencils and Isabel gives 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 pebbles. Amy gave 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 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 times as many candies as Vladimir. Raul gives 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 . If the digits of the number are reversed, the new two digit number obtained is 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 mph, while the second biker bikes at mph. The trail is 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 candies in the ratio to Jon and his brother. How many candies did Jon get?
2. Julian and Marco have candies in the ratio . Marco has 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 . If Jaden gives 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 .
   1. If Debbie has marbles more than Kannan, how many marbles does Ron have?
   2. If Debbie and Ron together have 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 . Xenia gave of her shells to Saloni. Xenia now has more shells than Saloni. How many shells does Saloni have now?
2. The ages of Nancy and Bruce are in the ratio of . Seven years ago, their ages summed to . 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 . Because of demand for green cars, the dealership added more green cars to its inventory. The ratio of green cars to gray cars is now . What is the number of cars in the dealership now?
4. A basket has apples and oranges. The ratio of apples to oranges is . Six more apples and six more oranges were added to the basket. Now, the ratio of apples to oranges is . What is the number of apples in the basket now?
5. In 2017, the ratio of teachers to students in a school was . In 2018, in addition to retaining all its students and teachers from 2017, the school expanded to recruit two more teachers and enroll additional students. The total number of students and teachers in the school now is .
   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 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 , and the coins amount to . How many quarters does Julia have?
3. Noah has some quarters and nickels amounting to . He has 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 . The ratio of number of dimes to number of quarters is . 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 to buy two different kinds of pens. Each pen of the first kind costed , while each pen of the second kind costed . Altogether, John bought pens. How many pens of the first kind did John buy?
2. A class of has boy and girl kids. Each boy kid brings to class green colored pencils, while each girl kid brings blue colored pencils. No other pencils are brought to class. The ratio of total number of green pencils to blue pencils is to . How many girl students are there in the class?
3. Qin has in the form of nickels, dimes and quarters. The ratio of number of nickels to number of dimes is , while the ratio of number of dimes to number of quarters is . How many nickels does she have?
4. Kabir has in quarters, dimes and nickels. The ratio of the number of quarters to the number of dimes that he has is . The total number of coins he has is . How many nickels does he have?
5. Amy, Sam and Raj together have . If Amy had 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, tonnes of grain were required to support an affected neighborhood with families. How many tonnes of grain are required to support a neighborhood of families?
2. A truck consumes gallons of gas to drive miles. How many miles can the truck drive on gallons of gas?
3. men are able to paint square feet of walls in a day. How many men are required to paint square feet of walls in a day?
4. high school students are able to arrange books in bookshelves in one hour. How many bookshelves can students arrange in one hour?
5. To pave a m long road in a day, it takes machines. How many machines does it take to pave a m 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. high school students are able to arrange books in a library in hours. How many hours do students need to arrange the library books?
2. It takes machines days to pave a road. How many machines are required to pave a similar road in days?
3. A commuter covers the distance between his home and work in minutes if he drives his car at mph. On a particular day, he needs to drive his car on a replacement tire, and he needs to drive his car at 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 units more than that of the original rectangle. And, the area of the stretched rectangle is square units more than that of the original rectangle. What is the width of the rectangle?
2. Average of three numbers is . Two of the numbers are and . What is the third number?
3. Anthony runs faster than Lisa. They are both running the same distance. If Lisa takes hour minutes to run the distance, how much time does Anthony take?