The words decide the answer
Before this page: be able to divide by a one-digit number and record a remainder using short division.
A remainder is the amount left after making as many whole groups as possible. The calculation tells you the quotient and remainder; the question tells you what to do with them.
29 ÷ 4= 7 remainder 1
29 = 4 × 7 + 1
Here, 7 is the number of complete groups and 1 is left. Because the remainder is part of the original 29, it cannot simply be dropped. It must be reported, shared, or allowed for.
One division, four possible answers
Each situation below begins with 29 ÷ 4 = 7 remainder 1. Scroll through the example to see why the same arithmetic does not always produce the same final answer.
Example: interpret 29 ÷ 4
29 ÷ 4 = 7 r 1
Start with the division
29 ÷ 4 gives 7 remainder 1. The context will decide what that result means.
Read the question before choosing
State the remainder
Write the quotient and remainder when that form is requested. In a context, answer the exact question and name the leftover amount.
29 counters shared between 4 pupils: 7 each, 1 counter left.
Give the exact value
If the quantity can be split, write the remainder as a fraction of the divisor or continue to a decimal.
29 litres shared equally between 4 tanks: 7¼ litres = 7.25 litres each.
Round down
Count only full groups. The leftover items do not make another complete one.
29 flowers, 4 in each bouquet: 7 complete bouquets, with 1 flower left.
Round up
Add another group when every item or person must be accommodated.
29 passengers, 4 in each car: 8 cars. The eighth car is not full.
Fractions and decimals describe the same share
When the leftover can be divided, place it over the divisor. This turns the remainder into the fractional part of the quotient.
7 r 1→7 + ¼→7¼=7.25
The denominator is 4, not 29: the leftover 1 is being shared between the same four groups. Use a fraction if the question requests a fraction or an exact non-terminating value. Use a decimal if the question requests one and round only when an accuracy is specified.
Compare the possible answers
Change the numbers to see every interpretation of the same whole-number division. Then use the wording of a problem to select the answer that fits.
53 ÷ 8 = 6 remainder 5
The remainder is non-zero, so the complete-groups and groups-needed answers differ by one.
Three questions, one calculation
A theatre group has 94 costumes. Each rail holds 8 costumes. The calculation is 94 ÷ 8 = 11 remainder 6.
How many full rails can be displayed?
Only complete rails count, so the answer is 11 full rails.
How many costumes remain after filling them?
The remainder answers the question, so 6 costumes remain.
How many rails are needed for every costume?
The 6 remaining costumes need another rail, so 12 rails are needed.
Notice that each answer includes a unit. “11 remainder 6” is correct arithmetic, but it does not by itself answer any of the three questions.
Common mistakes
- Ignoring the remainderA leftover cannot disappear. Explain whether it is left over, shared, or placed in another group.
- Always rounding to the nearest whole numberComplete groups force the answer down; fitting everything in forces it up.
- Giving a fraction of an indivisible objectDecimal cars, people and boxes are not usable answers. Decide whether another whole one is needed.
- Omitting unitsWrite what the number counts or measures: cars, bouquets, counters, litres or another stated unit.
Summary
- Divide and check that the remainder is smaller than the divisor.
- Read the whole question, especially the noun and what is being asked.
- State the leftover, give an exact share, count complete groups or add another group as the context requires.
- Do not use ordinary rounding when the situation forces the answer up or down.
- Write the final answer with its unit and a short sentence when the interpretation matters.