A Potentially Tricky Concept
In or around the fourth grade, most American students begin learning about the intricacies of dividing numbers. This study is usually combined with lessons regarding fractions and their usefulness in life. However, division is often a difficult concept for students to grasp. It's the opposite of multiplication and can be hard for people to visualize. The other thing that makes division difficult is the fact that many of these types of math problems result in remainders. The idea that one number cannot be evenly, or exactly, divided into another can sometimes leave a youngster's brain crying out, "this division does not compute!"
Interpreting remainders requires a higher level of thinking and is much more than just doing the math and calculating the remaining value. The student must figure out what the question is exacting and decide what the remainder means in terms of that question. In fact, when it comes to division problems, there are four possible ways to interpret the remainder depending on the specific situation where the division operation is being used:
- Leaving the Remainder - This is the most basic form of interpreting the remainder. In this case, the remainder "remains behind" because it is not needed. For example, how many times can 6 go completely into 13? Typically you would write 2 R1 as the answer, but in this case, the solution would be 2. This represents the number of times the whole number, in this case, 6, can go into the number 13 completely. The remainder is discarded because it's not needed, and the solution is only the quotient.
- Finding only the Remainder - In this situation, only the remainder is important to the problem. For example, 13/6 would equal 2 R1, but in certain situations, only the value of the remainder, in this case, 1, is important. Therefore, the solution to these kinds of problems is the remainder itself.
- Sharing the Remainder - In this situation, the remainder is further split into pieces by making it a fraction instead of just leaving the remainder behind. For example, 13/6 would equal 2 R1, but in some cases, the correct answer would be 2 1/6. This version of interpreting the remainder may not appear in some classrooms until future grades or until the students have mastered the basic division.
- Adjusting the Quotient - In this situation, the resulting whole number answer must be adjusted to account for the fact that the remainder cannot simply be discarded for the answer to make sense. For example, 13/6 would equal 2 R1, but in some cases, the correct answer would be "rounded upward" to 3. In other words, the quotient is increased by 1.
These variations are what make interpreting remainders so difficult for many students to grasp.
Nonetheless, understanding division, and therefore remainders, is an important concept to fully grasp. When the division of numbers is fully understood, it makes learning higher mathematics concepts much easier. Furthermore, utilizing fractions will become easier, as well as sharing multiples of things with other people.
As a father of two children, I realized the need for them to get additional practice with division, especially in interpreting remainders. I decided to write up several practice sheets for them and then share these example problems online so that others can benefit from my work. With that said, here are 40 examples of problems where the student needs to interpret the remainder to find the correct answer to the question. If you would like to use them for your student or child, copy and paste them into a word document and print them out.
Ten Example Problems for Leaving the Remainder
- Miles went to the candy store with $20 in his wallet. He sees large rainbow lollipops on sale for $3 each. How many large rainbow lollipops can he buy? Answer: 20/3 = 6 R2 which means that he can only buy 6 large rainbow lollipops.
- Soro was given $100 for his birthday. He wanted to buy Pokemon cards which costs $6 per pack. How many packs of Pokemon cards can Soro buy? Answer: 100/6 = 16 R4 which means that he can only buy 16 packs of Pokemon cards.
- Harry's Chocolate Factory makes candy bars and ships them out to retailers in boxes that contain 36 bars. They do not ship partially full boxes. If Harry's Chocolate Factory made 1,000 candy bars this week, how many full boxes of candy bars can they ship out to retailers? Answer: 1000/36 = 27 R28 which means that Harry's Chocolate Factory can only ship out 27 full boxes this week.
- John was asked to stock the store shelves with boxes of cereal. There were 12 empty shelves that could hold 8 boxes of cereal each. If there were 85 boxes of cereal in the back of the store, how many shelves could John completely fill with cereal boxes? Answer: 85/8 = 10 R5, which means that John only had enough boxes of cereal to completely stock 10 shelves.
- At the park, George saw a salesman selling ice cream cones. If the cones cost $4 each and George has $10, how many ice cream cones can he buy? Answer: 10/4 = 2 R2 which means that George only has enough money to buy 2 ice cream cones.
- Milk is shipped in plastic crates, which each hold 6 1-gallon jugs. If Ken's Dairy only ships milk to retailers in full crates, how many crates of milk did he ship out when his cows produced 75 gallons of milk? Answer: 75/6 = 12 R3, which means that Ken's Dairy shipped out 12 crates of milk.
- A bag of M&M's had 125 candies in it. If Jennifer needs 10 M&M's to fill a treat bag, how many complete treat bags can she make? Answer: 125/10 = 12 R5, which means that Jennifer can make 12 completely filled treat bags.
- Each pizza requires exactly 10 ounces of cheese to perfectly cover the sauce. If Zoe had 96 ounces of cheese in her fridge, how many pizzas would she have enough cheese to make? Answer: 96/10 = 9 R6 which means that Zoe has enough cheese to make 9 pizzas.
- An art project requires 30 inches of ribbon to complete. If Jane has 500 inches of ribbon in her drawer, how many complete art projects can she make? Answer: 500/30 = 16 R20, which means that Jane has enough ribbon to make 16 art projects.
- A one-mile roadway paving project requires an average of 453 gallons of paint to mark all of the lane lines. If a contractor has 11,650 gallons of paint in his warehouse, how many one-mile roadway paving projects can the contractor complete with the paint he has on hand? Answer: 11,650/453 = 25 R325 which means that the contractor has enough paint to complete 25 one-mile roadway paving projects.
Ten Example Problems for Finding Only the Remainder
- Joan is collecting eggs from her chickens and groups them into cartons by the dozen. She can only sell cartons that have 12 eggs in them. If her hens lay 59 eggs, how many eggs will there be in the last partially filled carton? Answer: 59/12 = 4 R11 which means that 11 eggs will partially fill the last carton.
- Grandma's famous cookie recipe requires 2 cups of flour for each batch. If there are approximately 9 cups of flour in the bag, how much flour would be left if Grandma made as many batches of cookies as possible? Answer: 9/2 = 4 R1 which means that 1 cup of flour will remain in the bag after all of the cookies are baked.
- Jason was wrapping presents for a Christmas party. He has a total of 950ft of tape available to wrap presents. If each present needs 15ft of tape to seal properly, how much tape will be left if Jason wraps as many presents as he can with this tape? Answer: 950/15 = 63 R5 which means that 5ft of tape will be left when the present wrapping is complete.
- After a hard day's work, Mary had finished baking 33 apple pies. She gave an equal number of pies to each of 10 families and saved the rest for herself. How many pies did she save for herself? Answer: 33/10 = 3 R3 which means that she saved 3 pies for herself.
- Draco produced 52 songs last year. If one album can hold 15 songs, how many songs will not be included on an album if Draco releases the most amount of complete albums that he can? Answer: 52/15 = 3 R7 which means that 7 songs won’t get put onto a new album.
- Sherry is a carpenter that makes wooden furniture. A wooden picnic table requires 19 pieces of standard-sized boards to construct. If sherry has as a stock of 450 boards on hand, how many boards would be left if she made as many picnic tables as possible? Answer: 450/19 = 23 R13 which means that Sherry would have 13 boards left in her stock.
- Bonnie sells honey in 6-ounce containers. After harvest, she fills as many containers as possible to sell at the market and keeps the remaining honey for herself. If Bonnie’s bees produced 95 ounces of pure, delicious natural honey, how much would she keep for herself? Answer: 95/6 = 15 R5 which means Bonnie would have 5 ounces of honey left for herself.
- Dan’s dogs eat a lot of food. However, in order to keep the dogs healthy, Dan only feeds them exactly 7 cups of food per day. If one bag of dog food has 144 cups of food in it, how much dog food will be left after feeding them exactly 7 cups a day for as many days as possible? Answer: 144/7 = 20 R4 which means that after 20 days of feeding, 4 cups of food will be left in the bag.
- A business market analysis report requires 32 sheets of paper to be considered complete. If the copy machine has 359 sheets of paper left in the tray, how many sheets of paper will remain after printing out as many copies of the report as possible? Answer: 359/32 = 11 R7 which means that after printing as many copies of the report as possible, there will be 7 sheets of paper left in the machine.
- A pool filter can be used for 3 months before it needs to be replaced. If Jack only replaced the pool filter when required and is never late nor early, how many months would remain on the last pool filter after using his pool for 28 months? Answer: 28/3 = 9R 1, which means that after 28 months, the current filter would have only 1 month left before it would need to be replaced.
Ten Example Problems for Sharing the Remainder
- Josh, James, Jordan, and Johnny worked hard cleaning up Mr. McGregor's backyard. If Mr. McGregor gave the children a total of $50 for their hard work, how much money would each child get? Answer: 50/4 = 12 R2 which means that each child would get $12 and then there would be $2 left over. However, the remainder can be further split by simply writing a fraction since surely no one would leave the remaining $2 behind: $12 and $2/4 becomes $12.50 each.
- Mom baked a batch of 12 cookies. The dog ate 2 leaving 10 on the tray. If four kids were to split the remaining cookies equally (leaving the tray clean), how many cookies would each child get? Answer: 10/4 = 2 R2 the remainder can be further divided by converting it to a fraction, 2/4. This reduces to 1/2. Therefore, each child would get 2 ½ cookies.
- Moe, Joe, and Larry are hired to mow lawns around the neighborhood. If 10 yards need to be mowed, how many yards would each person be expected to mow? Answer: 10/3 = 3 R3 which results in 3 and 1/3 yards each.
- A pack of 6 hungry lions is about to be fed. If the zookeeper dumps a bag containing 63 pounds of meat into the den, how much meat would each lion eat assuming they each consume the same amount? Answer: 63/6 = 10 R3 which converts to 10 and 3/6 and reduces to 10 ½ pounds of meat each.
- A team of 45 scientists wins a prize of $1,125,009 (after taxes) for discovering a new material that can stay solid at temperatures exceeding 5000 degrees. If the prize is split equally amongst the 45 scientists, how much money do they each get? Answer: 1,125,009/45 = 25,000 R9 which converts to $25,000 and $9/45 = $25,000 and $1/5 each which is $25,000.20.
- Six children were making slime. They had a 64oz bottle of glue and poured it equally into six bowls. How much glue did each child get? Answer: 64/6 = 10 R4. The remaining 4oz can be divided into 6 equal parts by using a fraction which results in 4/6oz. This reduces to 2/3oz. Therefore, each child received 10 and 2/3 ounces of glue to make slime with.
- In the nursery were 9 hungry babies. A tired mom warmed up 75 ounces of formula for them to drink. If each baby received the same amount of formula (and none was wasted) how much formula did each baby get to drink? Answer: 75/9 = 8 R3. The remaining 3oz can be divided into 9 equal parts by using a fraction which results in 3/9. This reduces to 1/3. Therefore, each baby received 8 and 1/3 ounces of formula to drink.
- My three brothers and I sold our Nintendo 64 as well as all the games and accessories to a dealer for $425. If the money was split equally amongst the four of us, how much money did we each get? Answer: 425/4 = 106 R1. The remaining $1 can be split into 4 quarters of $0.25 each. Therefore, each of got to keep $106.25.
- A fuel shortage hit southern Tucson and the gas station only had 500gallons of gas left. There were 60 customers waiting for gas. If the gas station owner rationed the fuel and split it equally amongst the 60 customers, how many gallons of gas would each customer get? Answer: 500/60 = 8 R20. The remaining 20 gallons can be divided into 60 equal parts by using a fraction which results in 20/60. This reduces to 1/3. Therefore, each customer received 8 and 1/3 gallons of gas.
- Charles was preparing to take 19 people on a three-day camping adventure. He packed 95 gallons of water for the trip. If each camper (including Charles) gets an equal amount of water for their needs, how much water does everyone get? Answer: 95/20 = 4 R15. The remaining 15 gallons can be divided into 20 equal parts by using a fraction which results in 15/20. This reduces to 3/4. Therefore, each camper will get 4 and 3/4 gallons of water to use.
Ten Example Problems for Adjusting the Quotient
- Charles has 38 books that he wants to put on shelves. Each shelf in his bookcase can hold 8 books. How many shelves does Charles need to hold his books? Answer: 38/8 = 4 R6 which means that 5 shelves would be needed to hold all of the books.
- 28 students are planning to go on the class field trip to the zoo. If the school has to rent vans that hold 8 students each to transport them to the zoo, how many vans must they rent? Answer: 28/8 = 3 R4 which means that 4 vans will be needed to make sure that each student has a ride to the zoo.
- Shelly sells seashells on eBay. Someone ordered sixty seashells from Shelly. If Shelly can pack 8 Seashells in each box, how many boxes does Shelly need to ship out her seashells? Answer: 60/8 = 7 R4 which means that 8 boxes will be needed to make sure that Shelly can fit all the seashells in her shipment.
- Batteries come in packs of 6. If Mitchell needs to put batteries in 20 batteries to power 10 TV remotes, how many packs of batteries does Mitchell need to buy? Answer: 20/6 = 3 R2 which means that 4 packs of batteries will be needed to power 10 TV remote controls.
- Ten kids are going camping this winter. If each tent can hold up to three kids, how many tents will be needed so that all of the kids have a place to sleep? Answer: 10/3 = 3 R1, which means that at least 4 tents are needed so that all the kids can enjoy the camping experience.
- Janice needed to bake 90 cupcakes for a school project. If each baking tray holds 12 cupcakes, how many trays will be needed to bake all the cupcakes? Answer: 90/12 = 7 R6 which means that at least 8 trays will be needed to bake the 90 cupcakes (or use the same tray 8 times).
- 99 kids go to lunch at 11:10 am in the cafeteria. If one table can hold 10 kids, how many tables are needed so that each kid has a place to sit? Answer: 99/10 = 9 R9 which means that at least 10 tables are needed so that all the kids will have a place to sit.
- Marsha is planning a party and is going to order pizzas for lunch. If there are 15 guests that each will eat 2 slices of pizza, how many pizzas are needed if each pizza has 8 slices? Answer: 15X2 = 30 slices, 30/8 = 3 R6 which means that at least 4 pizzas are needed to make sure that all 15 guests can have at least 2 slices.
- One huge box can hold 144 balls. If Macy and Mindy have 1500 toy balls, how many boxes are needed to be able to store all of the balls? Answer: 1500/144 = 10 R60 which means that at least 11 huge boxes will be needed to ensure that all the balls can be stored.
- One file folder can hold 5 small reports. If Mark has to file 66 small reports, how many file folders will be needed to make sure that all the reports get filed? Answer: 66/5 = 13 R1 which means that at least 14 file folders will be needed to file all the reports.
© 2019 Christopher Wanamaker