Division with whole-number remainders - Remainders by grouping
Theory
When objects are put into equal groups, some may be left over. The amount left over is the remainder. This Year 5 maths page shows how to make groups and find the remainder, write a number sentence such as \(27 = 5 \times 5 + 2\), and decide whether a problem needs the answer rounded up (for example, how many buses) or down (how many full boxes).
Dividing \(27\) pencils into groups of \(5\) makes \(5\) complete groups with \(2\) pencils left over. We write \(27 \div 5 = 5 \text{ r } 2\), where r means remainder.
The same fact as a number sentence: \(27 = 5 \times 5 + 2\). Groups times group size, plus the remainder, gives back the total.
The remainder is always smaller than the group size. If it were not, you could make another group.
In a problem, think about what the remainder means. “How many full boxes?” ignores it (round down). “How many boxes are needed for all of them?” needs one more box (round up).
A division with a remainder can be checked with multiplication:
How to divide by grouping
- Make groups of the given size (or skip count by the group size).
- Count the complete groups.
- Count what is left over: this is the remainder.
- Check: groups \(\times\) size \(+\) remainder \(=\) total.
Deciding what the answer means
- Read the question again.
- Complete groups only? Use the quotient.
- Must every item fit? Add one to the quotient.
| \(5 \times 5\) | \(=\) | \(25\) |
| \(27 - 25\) | \(=\) | \(2\) |
\(5\) groups with \(2\) left over.
(i) How many cartons are full?
(ii) How many eggs are left over?
| \(6 \times 6\) | \(=\) | \(36\) |
| \(40 - 36\) | \(=\) | \(4\) |
(i) \(6\) full cartons.
(ii) \(4\) eggs.
| \(22 \div 4\) | \(=\) | \(5 \text{ r } 2\) |
The \(2\) children left over still need a table, so \(6\) tables are needed.
| \(8 \times 4 + 5\) | \(=\) | \(37\) |
\(37 = 8 \times 4 + 5\).
| \(4 \times 7\) | \(=\) | \(28\) |
| \(31 - 28\) | \(=\) | \(3\) |
\(7\) each, with \(3\) left over.
| \(6 \times 5\) | \(=\) | \(30\) |
| \(35 - 30\) | \(=\) | \(5\) |
\(5\) pieces, with \(5\) m left.
| \(58 \div 6\) | \(=\) | \(9 \text{ r } 4\) |
\(9\) boats leave \(4\) people behind. Everyone must go, so \(10\) boats are needed.
Common pitfalls
Frequently asked questions
What is a remainder?
The amount left over when a number cannot be divided into equal groups exactly. Dividing \(27\) into fives gives \(5\) groups with \(2\) left over.
How do you write a division with a remainder?
Write the quotient, then r and the remainder: \(27 \div 5 = 5 \text{ r } 2\). As a number sentence, \(27 = 5 \times 5 + 2\).
Can a remainder be bigger than the number you divide by?
No. If it were, you could make one more group, so the remainder is always smaller.
When do you round the answer up?
When every item must fit, such as buses or tables for everyone. \(22\) children at tables of \(4\) need \(6\) tables.
When do you ignore the remainder?
When only complete groups count, such as full cartons or whole pieces of ribbon.