Word problems with a remainder
When a division doesn't come out evenly, you get a remainder. In a pure maths problem ("32 ÷ 5") writing "6 r 2" is enough. In a word problem you must decide what the remainder means for the story. Sometimes you round up, sometimes you round down, sometimes the remainder is the answer.
Three ways to handle a remainder
Round up — when you can't leave anyone behind
The remainder forces you to use one more of whatever you're counting.
32 children are going on a trip in 5-seat cars. How many cars are needed?
32 ÷ 5 = 6 r 2. Six cars fit only 30 children. The remaining 2 still need a car. So 7 cars are needed.
The clue: everybody must be served / fit / carried.
Round down — when only complete groups count
The remainder doesn't fit, so we ignore it.
A baker has 50 eggs. Each cake needs 6 eggs. How many cakes can the baker make?
50 ÷ 6 = 8 r 2. The 2 leftover eggs are not enough for another cake. So 8 cakes.
The clue: only finished/full groups count.
The remainder is the answer
Sometimes the question is "what's left over" — so the remainder is the answer.
Mum bakes 23 cookies and shares them equally among 4 children. How many cookies are left after the share?
23 ÷ 4 = 5 r 3. Each child gets 5; 3 cookies are left.
The clue: how many are left, missed, surplus.
Reading the question carefully
The same arithmetic (50 ÷ 6 = 8 r 2) can give three different answers depending on the question:
50 children form teams of 6. How many teams? 8 teams (round down).
50 children form teams of 6. How many teams are needed to seat everyone? 9 teams (round up).
50 children form teams of 6. How many children are not in a team? 2 children (the remainder).
Always re-read the question after the arithmetic, and write the answer in a sentence.
A worked example
A school orders 100 packets of crisps for a sports day. Each box holds 12 packets. How many full boxes are there, and how many extra packets?
100 ÷ 12 = 8 r 4.
- Full boxes: 8 boxes.
- Extra packets that don't fill a box: 4 packets.
Same story, different question: "How many boxes does the school need to order to have 100 packets?"
8 full boxes have only 96 packets. To have at least 100, you need 9 boxes (you'll have 8 extra packets, but that's fine).
Common mistakes
- Writing "r 3" as the final answer to a real-life question. A word problem needs a complete sentence, not the bare remainder.
- Always rounding up. Round up only when everyone or everything must be served. For "how many full boxes", round down.
- Forgetting the remainder when it matters. "How many cookies are left?" — the remainder is the answer.
- Confusing how-many-groups and how-many-per-group. 24 ÷ 4 could mean "4 groups of 6" or "6 groups of 4". The story tells you which.