Division with whole-number remainders - Remainders by grouping
Theory
To divide by grouping, make as many equal groups as you can; whatever is left over is the remainder, always smaller than the group size. Check with total \(=\) group size \(\times\) groups \(+\) remainder.
To divide by grouping, build equal groups of the same size until you cannot make a full one. The number of groups is the answer.
The remainder is what is left over. It is always smaller than the group size.
Check with: total \(=\) group size \(\times\) number of groups \(+\) remainder.
Make full groups, then read off how many are left over.
| Total | Group size | Groups | Left over | Number sentence |
|---|---|---|---|---|
| \(23\) | \(4\) | \(5\) | \(3\) | \(23 = 4 \times 5 + 3\) |
| \(34\) | \(6\) | \(5\) | \(4\) | \(34 = 6 \times 5 + 4\) |
| \(45\) | \(8\) | \(5\) | \(5\) | \(45 = 8 \times 5 + 5\) |
How to divide by grouping
- Make equal groups. Build groups of the same size until you cannot make a full one.
- Count the leftover. Whatever is too little for another group is the remainder.
- Check by building back. Group size \(\times\) number of groups \(+\) remainder should give the total.
\(5\) groups, \(3\) left over.
| \(4 \times 5\) | \(=\) | \(20\) |
| \(23 - 20\) | \(=\) | \(3\) |
\(5\) groups, \(4\) left over.
| \(6 \times 5\) | \(=\) | \(30\) |
| \(34 - 30\) | \(=\) | \(4\) |
\(5\) bunches, \(5\) left over.
| \(8 \times 5\) | \(=\) | \(40\) |
| \(45\) | \(=\) | \(8 \times 5 + 5\) |
\(10\) still need a seat, so round up to \(3\) buses.
| \(20 \times 2\) | \(=\) | \(40\) |
| \(50 - 40\) | \(=\) | \(10\) |
Common pitfalls
Frequently asked questions
What is a remainder?
The amount left over after making all the full groups you can. It is always smaller than the group size.
How do you divide by grouping?
Build equal groups until you cannot make another full one. The number of full groups is the answer, and the leftover is the remainder.
How do you check a division with a remainder?
Build it back: group size \(\times\) number of groups \(+\) remainder should equal the total. For \(23 \div 4\), \(4 \times 5 + 3 = 23\).
Why is the remainder always smaller than the divisor?
Because if the leftover were as big as the group size, you could make one more full group.
When do you round up the answer?
When every item must fit, like seating people on buses. \(50\) students in buses of \(20\) need \(3\) buses, not \(2\), because \(10\) are still left.