Before this page: be able to divide by a one-digit number and record a remainder using short division.
Making sense of a remainder
A remainder is the amount left after making as many whole groups as possible. The calculation gives the quotient and remainder; the question says what to do with them.
29 ÷ 4 = 7 remainder 1
29 = 4 × 7 + 1
The 1 is one of the 29, so the answer has to say what happens to it.
One calculation, different answers
Example: interpret 29 ÷ 4
29 ÷ 4 = 7 r 1
Start with the division
What the 29 counts decides what the 7 and the 1 mean.
Full groups or enough groups
What is left
7 remainder 1, or in a context, 1 counter left.
Exact share
If the quantity can be split, the 1 is shared too: 7¼ or 7.25 each.
Complete groups
The 1 left over does not fill another group, so only the 7 full ones count.
Groups needed
When every one of the 29 must have a place, the 1 left over needs an 8th group.
Sharing the remainder
When the remainder can be divided, place it over the divisor:
7 r 1→7 + ¼→7¼=7.25
The denominator is 4, not 29: the remainder 1 is being shared between the same four groups.
A fraction can express an exact share even when its decimal is not terminating. For example, 29 ÷ 3 = 9⅔ = 9.666…, with the sixes continuing forever. Round only if a question asks for a particular accuracy.
Explore the different answers
53 ÷ 8 = 6 remainder 5
5 left over: 6 complete groups leave them out, and 7 groups fit them in.
Putting the context first
A theatre group has 94 costumes. Each rail holds 8 costumes. The calculation is 94 ÷ 8 = 11 remainder 6.
How many rails can be filled?
Only complete rails count, so the answer is 11 full rails.
How many costumes remain after filling them?
11 rails hold 88 costumes, and 94 − 88 = 6 costumes remain.
How many rails are needed for every costume?
The 6 remaining costumes need another rail, so 12 rails are needed.
Each answer includes a unit. “11 remainder 6” records the arithmetic; the nouns turn its two numbers into costumes and rails.
Key points
- A remainder is smaller than the divisor.
- Complete groups exclude an unfinished group; enough groups for everything include it.
- An exact share divides the remainder too, giving a fraction or decimal.