Dividing Numbers
Every question with a fully worked solution.
- Dividing Numbers - Video - Parts of a Division Problem: Dividend, Divisor, Quotient, & Remainder Watch
Theory
Division shares a number into equal groups. Short division works from the left, one digit at a time, and carries each remainder to the next digit. Check by multiplying: quotient \(\times\) divisor \(+\) remainder \(=\) dividend. This Year 7 maths page shows the words, short division and what to do with a remainder.
The words. In \(22 \div 7 = 3 \text{ r } 1\), \(22\) is the dividend, \(7\) is the divisor, \(3\) is the quotient and \(1\) is the remainder (what is left over). As a mixed numeral, \(22 \div 7 = 3\dfrac{1}{7}\).
Short division. Estimate first. Divide each digit, starting from the left, and write the answer digit above it. Carry each remainder to the next digit as a small number. If the first digit is smaller than the divisor, use the first two digits.
Check by multiplying. Quotient \(\times\) divisor \(+\) remainder \(=\) dividend: \(144 \times 4 + 3 = 579\). With no remainder, dividend \(=\) divisor \(\times\) quotient.
A remainder three ways. Write the remainder over the divisor to make a mixed numeral, then change it to a decimal: \(19 \div 4 = 4 \text{ r } 3 = 4\dfrac{3}{4} = 4.75\).
A 2-digit divisor. Write out some useful multiples of the divisor first, then divide in the same way (or use long division).
Make it divisible. Add the divisor take away the remainder: \(150 \div 7 = 21 \text{ r } 3\), so add \(7 - 3 = 4\) to get \(154 = 7 \times 22\).
What does the remainder mean? \(30 \div 4 = 7 \text{ r } 2\):
| Problem | The remainder | Answer |
|---|---|---|
| \(30\) people sit at tables of \(4\) | the \(2\) extra still need a table | round up: \(8\) tables |
| \(30\) muffins in boxes of \(4\) | \(2\) muffins don’t fill a box | round down: \(7\) full boxes |
| \(\$30\) shared by \(4\) people | money splits into cents | \(7\dfrac{1}{2}\), so \(\$7.50\) each |
How to use short division
- Estimate the answer.
- Divide the first digit (or first two digits) and write the answer above.
- Carry the remainder to the next digit and repeat.
- Check: quotient \(\times\) divisor \(+\) remainder \(=\) dividend.
| \(8 \times 72\) | \(=\) | \(576\) |
Dividend \(=\) divisor \(\times\) quotient \(= 576\).
| \(26 \times 6 + 2\) | \(=\) | \(158\) |
\(1\) is smaller than \(6\), so start with \(15\): \(2 \text{ r } 3\); then \(38 \div 6 = 6 \text{ r } 2\). \(158 \div 6 = 26 \text{ r } 2\).
(i) with a remainder
(ii) as a mixed numeral
(iii) as a decimal
| \(26 \times 4 + 3\) | \(=\) | \(107\) |
(i) \(26 \text{ r } 3\)
(ii) \(26\dfrac{3}{4}\)
(iii) \(26.75\)
(ii) The dividend is \(675\) and the divisor is \(12\). Find the remainder.
| \(15 \times 23\) | \(=\) | \(345\) |
| \(12 \times 56\) | \(=\) | \(672\) |
| \(675 - 672\) | \(=\) | \(3\) |
(i) \(34 \div 15 = 2 \text{ r } 4\), \(45 \div 15 = 3\): quotient \(23\)
(ii) \(67 \div 12 = 5 \text{ r } 7\), \(75 \div 12 = 6 \text{ r } 3\): remainder \(3\)
(i) How many boxes are completely full?
(ii) How many boxes are needed to pack every cupcake?
| \(15 \times 12 + 10\) | \(=\) | \(190\) |
\(190 \div 12 = 15 \text{ r } 10\).
(i) \(15\) full boxes (round down)
(ii) \(16\) boxes (the \(10\) extra still need a box)
| \(216 \times 17\) | \(=\) | \(3672\) |
Multiples of \(17\): \(34\), \(102\). \(36 \div 17 = 2 \text{ r } 2\); \(27 \div 17 = 1 \text{ r } 10\); \(102 \div 17 = 6\). Each school gets \(216\) books.
(i) What is the number?
(ii) What is the smallest number you must add to \(274\) to make it divisible by \(8\)?
| \(45 \times 6 + 4\) | \(=\) | \(274\) |
| \(34 \times 8\) | \(=\) | \(272\) |
| \(8 - 2\) | \(=\) | \(6\) |
(i) \(274\)
(ii) \(274 \div 8 = 34 \text{ r } 2\), so add \(8 - 2 = 6\) (to get \(280 = 8 \times 35\)).
Common pitfalls
Frequently asked questions
What are the dividend, divisor and quotient?
In \(22 \div 7 = 3 \text{ r } 1\): dividend \(22\), divisor \(7\), quotient \(3\), remainder \(1\).
How do you check a division?
Quotient times divisor plus remainder should give the dividend.
How do you write a remainder as a fraction?
Put it over the divisor: \(19 \div 4 = 4\dfrac{3}{4}\).
When do you round up a division answer?
When the leftover items still need a group, such as an extra table or bus.
How do you make a number divisible by 7?
Add \(7\) minus the remainder.