Division with whole-number remainders - Remainders using division facts
Theory
Instead of drawing groups, use your multiplication facts. To work out \(43 \div 6\), find the largest multiple of \(6\) that is not more than \(43\) (\(42 = 6 \times 7\)); the quotient is \(7\) and the remainder is \(43 - 42 = 1\). This Year 5 maths page shows the method, how to check the answer and what the largest remainder can be.
In \(43 \div 6 = 7 \text{ r } 1\), \(43\) is the dividend, \(6\) is the divisor, \(7\) is the quotient and \(1\) is the remainder.
The quotient comes from the largest multiple of the divisor that is not greater than the dividend: the multiples of \(6\) are \(\ldots,\ 36,\ 42,\ 48\), and \(42\) is the largest that fits in \(43\).
Check: quotient \(\times\) divisor \(+\) remainder \(=\) dividend. \(7 \times 6 + 1 = 43\).
The remainder is always less than the divisor. When dividing by \(7\), the largest possible remainder is \(6\).
Division with a remainder and its check:
How to divide using facts
- List or recall multiples of the divisor.
- Find the largest that is not greater than the number.
- The quotient is how many times the divisor fits; the remainder is the difference.
- Check: quotient \(\times\) divisor \(+\) remainder.
| \(9 \times 6\) | \(=\) | \(54\) |
| \(58 - 54\) | \(=\) | \(4\) |
\(54\), so \(58 \div 9 = 6 \text{ r } 4\).
| \(8 \times 9\) | \(=\) | \(72\) |
| \(75 - 72\) | \(=\) | \(3\) |
\(75 \div 8 = 9 \text{ r } 3\).
| \(9 \times 8 + 3\) | \(=\) | \(75\) |
The check gives back \(75\), so the answer is right.
(i) \(43 \div 6\)
(ii) \(26 \div 3\)
(iii) \(61 \div 8\)
| \(43 - 42\) | \(=\) | \(1\) |
| \(26 - 24\) | \(=\) | \(2\) |
| \(61 - 56\) | \(=\) | \(5\) |
(i) \(1\).
(ii) \(2\).
(iii) \(5\).
| \(7 - 1\) | \(=\) | \(6\) |
\(6\). A remainder of \(7\) would make another group of \(7\).
(i) How many cards does each player get?
(ii) How many are left over?
(iii) How many more cards are needed to give everyone one more?
| \(8 \times 8\) | \(=\) | \(64\) |
| \(65 - 64\) | \(=\) | \(1\) |
| \(8 - 1\) | \(=\) | \(7\) |
(i) \(8\) each.
(ii) \(1\).
(iii) \(7\) more cards.
| \(7 \times 7\) | \(=\) | \(49\) |
| \(52 - 49\) | \(=\) | \(3\) |
A remainder of \(10\) is bigger than \(7\), so one more group fits. The answer is \(7 \text{ r } 3\).
Common pitfalls
Frequently asked questions
How do you divide using times tables?
Find the largest multiple of the divisor that fits in the number. For \(43 \div 6\), \(6 \times 7 = 42\), so the answer is \(7\) with \(1\) left over.
What is the difference between quotient and remainder?
The quotient is the number of whole groups; the remainder is what is left. In \(75 \div 8 = 9 \text{ r } 3\), the quotient is \(9\) and the remainder is \(3\).
How do you check a division with a remainder?
Multiply the quotient by the divisor and add the remainder: \(9 \times 8 + 3 = 75\).
What is the largest possible remainder?
One less than the divisor. Dividing by \(7\), the largest remainder is \(6\).
Must the remainder be smaller than the divisor?
Yes. Otherwise another whole group could be made.