In a division word problem, the arithmetic is the easy part. The real question is what the remainder means in the situation described, and there are five possible answers: keep it, round the quotient up, round the quotient down, turn it into a fraction, or turn it into a decimal. Dividing 187 students by 45 seats gives 4 remainder 7, but the answer is 5 buses, because the 7 extra students still need a ride.
The same numbers can produce three different final answers depending on what is being counted. This guide gives the rule for choosing, then works through 30 problems grouped by which interpretation applies.
Every worked problem below shows the question, the division, the quotient and remainder, the interpretation, and the final answer in words.
The Five Ways to Interpret a Remainder
| What the problem asks | What to do with the remainder | Typical wording |
|---|---|---|
| How many complete groups, and what is left? | Report both | ”how many are left over” |
| How many containers are needed? | Round the quotient up | ”how many … will be needed” |
| How many complete items can be made? | Round the quotient down | ”how many whole …”, “at most” |
| Share something divisible equally | Write the remainder as a fraction | ”shared equally”, cake, rope, time |
| Share money or a measurement | Write the remainder as a decimal | money, distance, litres |
The deciding question is always the same: can the leftover be split, and does the situation allow a partial one? A leftover student cannot be split and cannot be left behind, so the bus count goes up. A leftover pizza can be split, so it becomes a fraction. A leftover centimetre of ribbon too short to use is simply discarded, so the piece count goes down.
If the arithmetic itself is the sticking point rather than the interpretation, start with how to find the remainder and come back.
Group 1: Keep the Remainder (6 Problems)
These problems ask for both numbers. The quotient counts complete groups; the remainder counts what is left behind.
Problem 1
A shop packs 158 pencils into boxes of 12. How many full boxes are packed, and how many pencils are left over?
- Division: 158 ÷ 12
- Result: quotient 13, remainder 2
- Interpretation: 13 boxes are filled; 2 pencils cannot fill a fourteenth.
- Answer: 13 full boxes, with 2 pencils left over.
Problem 2
245 stickers are shared equally among 8 children, and any that cannot be shared are kept by the teacher. How many does each child get, and how many does the teacher keep?
- Division: 245 ÷ 8
- Result: quotient 30, remainder 5
- Interpretation: each child receives 30; the 5 that remain cannot be split evenly among 8.
- Answer: 30 stickers each, and the teacher keeps 5.
Problem 3
A printer has 500 sheets of paper and makes notebooks of 36 sheets each. How many complete notebooks are made, and how many sheets remain?
- Division: 500 ÷ 36
- Result: quotient 13, remainder 32
- Interpretation: 32 sheets is a large leftover but still short of the 36 needed.
- Answer: 13 notebooks, with 32 sheets left over.
Problem 4
A farm collects 97 eggs and packs them into cartons of 6. How many cartons are filled, and how many eggs are loose?
- Division: 97 ÷ 6
- Result: quotient 16, remainder 1
- Answer: 16 filled cartons, with 1 loose egg.
Problem 5
1,000 beads are threaded onto bracelets that take 24 beads each. How many bracelets are finished, and how many beads are spare?
- Division: 1,000 ÷ 24
- Result: quotient 41, remainder 16
- Answer: 41 bracelets, with 16 beads spare.
Problem 6
73 chairs are arranged in rows of 9. How many complete rows are formed, and how many chairs are left?
- Division: 73 ÷ 9
- Result: quotient 8, remainder 1
- Answer: 8 complete rows, with 1 chair left over.
Group 2: Round the Quotient Up (6 Problems)
Here the leftover still has to be dealt with, so one extra container, trip or page is required. The signal is wording like “how many will be needed”.
Problem 7
187 students are going on a trip. Each bus holds 45 students. How many buses are needed?
- Division: 187 ÷ 45
- Result: quotient 4, remainder 7
- Interpretation: 4 buses carry 180 students. The remaining 7 still need transport.
- Answer: 5 buses.
Problem 8
A library has 250 books to store. Each shelf holds 18 books. How many shelves are needed?
- Division: 250 ÷ 18
- Result: quotient 13, remainder 16
- Interpretation: 13 shelves hold 234 books; the last 16 need a fourteenth shelf.
- Answer: 14 shelves.
Problem 9
138 guests are attending a dinner and each table seats 8. How many tables must be set?
- Division: 138 ÷ 8
- Result: quotient 17, remainder 2
- Answer: 18 tables. The eighteenth seats only 2 people, but it is still required.
Problem 10
A truck can carry 90 kg per trip. How many trips are needed to move 415 kg of cement?
- Division: 415 ÷ 90
- Result: quotient 4, remainder 55
- Answer: 5 trips. The fifth trip carries only 55 kg.
Problem 11
92 photographs are placed in an album whose pages hold 6 photos each. How many pages are used?
- Division: 92 ÷ 6
- Result: quotient 15, remainder 2
- Answer: 16 pages, the last one holding just 2 photos.
Problem 12
A post office sorts 1,450 letters into mailbags that hold 200 letters each. How many bags are needed?
- Division: 1,450 ÷ 200
- Result: quotient 7, remainder 50
- Answer: 8 bags.
Group 3: Round the Quotient Down (6 Problems)
Now the leftover is unusable. It is too small to make another complete item, so it is discarded and only the quotient is reported.
Problem 13
A 314 cm ribbon is cut into pieces 40 cm long. How many full pieces can be cut?
- Division: 314 ÷ 40
- Result: quotient 7, remainder 34
- Interpretation: the leftover 34 cm is shorter than 40 cm, so it cannot make another piece.
- Answer: 7 pieces, with 34 cm of scrap.
Problem 14
Tickets cost $14 each. With $95, how many tickets can be bought?
- Division: 95 ÷ 14
- Result: quotient 6, remainder 11
- Interpretation: $11 is not enough for a seventh ticket.
- Answer: 6 tickets, with $11 unspent.
Problem 15
A recipe uses 65 g of flour per batch. How many complete batches can be made from 500 g?
- Division: 500 ÷ 65
- Result: quotient 7, remainder 45
- Answer: 7 batches, with 45 g of flour left.
Problem 16
268 marbles are put into bags of 25. How many bags can be filled completely?
- Division: 268 ÷ 25
- Result: quotient 10, remainder 18
- Answer: 10 full bags. The 18 spare marbles do not fill an eleventh.
Problem 17
A film festival has a 720 minute slot. Each film runs 95 minutes. How many complete films fit?
- Division: 720 ÷ 95
- Result: quotient 7, remainder 55
- Answer: 7 films, with 55 minutes of the slot unused.
Problem 18
Fence panels are 5 m wide. How many whole panels fit along an 84 m boundary?
- Division: 84 ÷ 5
- Result: quotient 16, remainder 4
- Answer: 16 whole panels, leaving a 4 m gap.
Compare problems 13 and 7 carefully. Both have a leftover, and the answers move in opposite directions. Ribbon scrap is thrown away, so the count goes down. Leftover students must travel, so the count goes up.
Group 4: Write the Remainder as a Fraction (4 Problems)
When the object being shared can genuinely be cut, the leftover is divided too. The fraction is always remainder over divisor.
Problem 19
7 pizzas are shared equally among 4 friends. How much does each friend get?
- Division: 7 ÷ 4
- Result: quotient 1, remainder 3
- Interpretation: each friend gets 1 whole pizza, and the 3 remaining pizzas are each cut into 4, giving 3/4 more.
- Answer: 1 3/4 pizzas each.
Problem 20
A 26 m rope is cut into 8 equal parts. How long is each part?
- Division: 26 ÷ 8
- Result: quotient 3, remainder 2
- Fraction: 2/8, which simplifies to 1/4.
- Answer: 3 1/4 m each.
Problem 21
45 kg of rice is divided equally into 6 sacks. How much goes in each sack?
- Division: 45 ÷ 6
- Result: quotient 7, remainder 3
- Fraction: 3/6, which simplifies to 1/2.
- Answer: 7 1/2 kg per sack.
Problem 22
100 cakes are shared equally between 8 stalls. How many does each stall receive?
- Division: 100 ÷ 8
- Result: quotient 12, remainder 4
- Fraction: 4/8, which simplifies to 1/2.
- Answer: 12 1/2 cakes each.
The full method for building and simplifying these mixed numbers is in remainders as fractions.
Group 5: Write the Remainder as a Decimal (4 Problems)
Money and measurements are normally written as decimals rather than fractions, so the division continues past the decimal point.
Problem 23
$58 is shared equally among 5 people. How much does each receive?
- Division: 58 ÷ 5
- Result: quotient 11, remainder 3
- Continue: 3 ÷ 5 = 0.6
- Answer: $11.60 each.
Problem 24
A 147 mile journey is split equally over 4 days. How far is driven each day?
- Division: 147 ÷ 4
- Result: quotient 36, remainder 3
- Continue: 3 ÷ 4 = 0.75
- Answer: 36.75 miles per day.
Problem 25
90 litres of juice is poured equally into 16 containers. How much is in each?
- Division: 90 ÷ 16
- Result: quotient 5, remainder 10
- Continue: 10 ÷ 16 = 0.625
- Answer: 5.625 litres per container.
Problem 26
$206 in wages is split equally among 8 workers. How much does each get?
- Division: 206 ÷ 8
- Result: quotient 25, remainder 6
- Continue: 6 ÷ 8 = 0.75
- Answer: $25.75 each.
The technique for continuing a division past the remainder is covered in remainders as decimals.
Group 6: Two-Step and Unit Problems (4 Problems)
The last group needs a calculation before the division, or converts one unit into another.
Problem 27
Three crates each hold 48 apples. The apples are repacked into bags of 10. How many bags are filled, and how many apples are left?
- First step: 3 × 48 = 144 apples
- Division: 144 ÷ 10
- Result: quotient 14, remainder 4
- Answer: 14 bags, with 4 apples left over.
Problem 28
A task takes 500 minutes. How long is that in hours and minutes?
- Division: 500 ÷ 60
- Result: quotient 8, remainder 20
- Interpretation: the quotient counts whole hours and the remainder gives the leftover minutes.
- Answer: 8 hours 20 minutes.
Problem 29
A timer runs for 1,000 seconds. Express that in minutes and seconds.
- Division: 1,000 ÷ 60
- Result: quotient 16, remainder 40
- Answer: 16 minutes 40 seconds.
Problem 30
Today is Monday. What day of the week will it be in 100 days?
- Division: 100 ÷ 7
- Result: quotient 14, remainder 2
- Interpretation: 14 complete weeks land back on a Monday, and the remainder 2 moves 2 days forward.
- Answer: Wednesday.
Problems 28 to 30 show why remainders matter beyond the classroom. Any repeating cycle, whether hours, weeks or positions in a rotation, is read from the remainder, which is the idea that modular arithmetic is built on.
How to Decide Which Interpretation to Use
Work through three questions in order:
- Does the problem ask for the leftover? Words like “left over”, “remaining” or “how many are spare” mean report the remainder as it stands.
- Must the leftover still be accommodated? Buses, tables, boxes, trips and pages all mean round up.
- Is the leftover unusable, or must the item be complete? Cutting, buying and making all mean round down.
If none of these apply, the object is divisible and the answer is a fraction or a decimal. Choose a fraction for food, rope and shares. Choose a decimal for money, distance and volume.
Common Mistakes in Remainder Word Problems
- Writing the decimal from a calculator as the answer. 187 ÷ 45 = 4.155… is not “4 buses” and is not “4.16 buses”. Convert to quotient and remainder first, then interpret.
- Rounding to the nearest whole number out of habit. 250 ÷ 18 = 13.9 rounds to 14 correctly, but 314 ÷ 40 = 7.85 must round down to 7, not up to 8. The context decides, not the decimal.
- Attaching the remainder to the wrong unit. In 500 ÷ 36, the answer is 13 notebooks and 32 sheets, not 32 notebooks.
- Writing the remainder over the wrong number. The fraction is remainder ÷ divisor. For 26 ÷ 8 remainder 2, that is 2/8, not 2/26 and not 2/3.
- Forgetting to simplify. 3/6 should be written as 1/2.
- Skipping the first step in a two-step problem. Problem 27 divides 144, not 48.
Word Problem Practice
Ready to try some unaided? A graded set with a complete answer key is in division with remainders practice problems, and the Remainder Calculator checks any quotient and remainder instantly.
Remainder Word Problem FAQ
How do you know whether to round up or down in a division word problem?
Ask whether the leftover still needs to be handled. If it does, as with passengers who still need a seat, round up. If it is unusable, as with ribbon too short to cut, round down. The arithmetic is identical; only the situation changes the answer.
What does the remainder mean in a word problem?
It is the part of the total that the divisor could not cover with whole groups. Its meaning comes from the units of the dividend, so dividing 97 eggs by 6 leaves a remainder of 1 egg, not 1 carton.
When should a remainder be written as a fraction instead of left as a whole number?
Write it as a fraction when the object can be divided and the problem shares everything out, such as pizza, rope, flour or time. Leave it whole when the object cannot be split, such as people, cars or eggs.
Why does a calculator not show the remainder?
A calculator divides on the number line and reports one decimal value, so 158 ÷ 12 comes out as 13.1666… rather than 13 remainder 2. To recover the remainder, multiply the whole-number part by the divisor and subtract, or use a tool that reports both. The full method is in how to find the remainder on a calculator.
Can the remainder be bigger than the divisor?
No. If it is, the quotient was too small and at least one more group could have been formed. In 500 ÷ 36 the remainder 32 is close to 36 but still below it, which is what makes 13 the correct quotient.
What if the division comes out exactly?
Then the remainder is 0, every group is complete, and no interpretation is needed. Dividing 144 by 12 gives 12 with nothing left, which means 12 is a factor of 144. That case is covered in what a remainder of zero means.
