Division with remainders
In Year 3 (NSW Stage 2, outcome MA2-MR-01), students learn division with remainders. They share or group a number into equal parts, count the number of full groups, and name the amount that is left over as the remainder, writing it as, for example, 16 divided by 5 equals 3 remainder 1.
This topic also covers the quotient, divisor and remainder, checking an answer with multiplication and addition, and knowing the remainder is always smaller than the divisor. By Year 4 these skills help students decide when to round up or down in real-world problems such as packing boxes or seating people on buses.
Theory
Division with remainders is a Year 3 skill: sharing or grouping a number into equal parts and naming what is left over. This page explains the quotient, the divisor and the remainder, how to check a division with multiplication, and when to round up or down in word problems.
Some numbers will not split into equal groups exactly. When that happens, the amount left over is called the remainder. For example, \(16\) counters put into groups of \(5\) make \(3\) full groups with \(1\) counter left over, so \(16 \div 5 = 3\) remainder \(1\).
Every division has a name for each part. The dividend is the number being shared (\(16\)). The divisor is the size of each group (\(5\)). The quotient is how many full groups you make (\(3\)). The remainder is what is left over (\(1\)).
Sharing works the same way. When you share, the divisor is the number of groups. \(10\) shared equally among \(3\) children gives \(3\) each, with \(1\) left over, so \(10 \div 3 = 3\) remainder \(1\).
The remainder is always smaller than the divisor. If the leftover is as big as the divisor, you can make another full group. So when you group by \(5\), the remainder can only be \(1\), \(2\), \(3\) or \(4\).
Division with remainders has two facts worth remembering. The first names every part; the second is how you check an answer.
How to divide with a remainder
- Make full groups. Count in the divisor (\(5, 10, 15, \dots\)) and stop before you go past the dividend. That count is the quotient.
- Find the leftover. Take the last multiple away from the dividend. What is left is the remainder, and it must be less than the divisor.
- Check it. Multiply the quotient by the divisor and add the remainder. If you get the dividend back, the answer is right.
| \(3\text{ groups of }4\) | \(=\) | \(12\) |
| \(15 - 12\) | \(=\) | \(3\text{ left over}\) |
| \(15 \div 4\) | \(=\) | \(3\text{ remainder }3\) |
Three full groups, with \(3\) left over.
| \(8 \times 3 + 2\) | \(=\) | \(24 + 2\) |
| \(24 + 2\) | \(=\) | \(26\) |
It gives back \(26\), so the answer is correct.
| \(40 \div 9\) | \(=\) | \(4\text{ remainder }4\) |
| \(4\text{ buses}\) | \(\to\) | \(36\text{ seats}\) |
| \(4 + 1\) | \(=\) | \(5\text{ buses}\) |
The \(4\) leftover students still need a seat, so \(5\) buses are needed.
| \(4 \times 7 + 3\) | \(=\) | \(28 + 3\) |
| \(28 + 3\) | \(=\) | \(31\) |
The number was \(31\).
Common pitfalls
Frequently asked questions
What is a remainder in division?
A remainder is the amount left over when a number cannot be shared into equal groups exactly. For \(16 \div 5\) you make \(3\) full groups of \(5\), which is \(15\), and \(1\) is left over, so the remainder is \(1\).
How do you write a division with a remainder?
Write the quotient first, then the word remainder, then the leftover. For example \(16 \div 5 = 3\) remainder \(1\). The \(3\) is the number of full groups and the \(1\) is what is left over.
How do you check a division with a remainder?
Multiply the quotient by the divisor and then add the remainder. You should get back the number you started with. For \(16 \div 5 = 3\) remainder \(1\), check \(3 \times 5 + 1 = 16\).
Can a remainder be bigger than the divisor?
No. The remainder is always less than the divisor. If the leftover is the same size as the divisor or larger, you can make another full group, so the quotient should be one bigger.
What is the difference between the quotient and the remainder?
The quotient is how many full equal groups you can make. The remainder is how many are left over after those groups are made. In \(11 \div 3 = 3\) remainder \(2\), the quotient is \(3\) and the remainder is \(2\).
When do you round up in a division word problem?
Round up when the leftovers still need a place, such as buses, boxes or cars. If \(40\) students share minibuses that hold \(9\), then \(40 \div 9 = 4\) remainder \(4\), but \(5\) buses are needed so everyone has a seat.