Long Division
Long division breaks a large division into small, repeated steps: divide, multiply, subtract, bring down. This NSW Year 7 maths subtopic consolidates primary-school skills, so there is no Stage 4 outcome of its own, and extends them to two-digit divisors.
You will estimate first, set out the working neatly, write the answer with a remainder, as a simplified mixed numeral or as a decimal, remember the zero in the quotient, and decide what a remainder means, such as how many tables are needed to seat 460 guests. It is part of Year 7 Whole Numbers.
Theory
Long division breaks a big division into small steps. You work along the number one digit at a time: divide, multiply, subtract, bring down. This Year 7 maths page shows the layout, zeros in the quotient, three ways to write a remainder and how to check.
In \(395 \div 17\), \(395\) is the dividend and \(17\) is the divisor. Long division is a method for dividing by a number with two or more digits; it is short division with every step written out.
Estimate first. Multiply the divisor by tens: \(17 \times 20 = 340\) and \(17 \times 30 = 510\), so the answer is between \(20\) and \(30\).
Start. \(17\) does not go into \(3\), so start with \(39\). Repeat four steps, one digit to the right each time: divide (how many times does \(17\) go in?), multiply, subtract, bring down the next digit.
Read the answer. The number on top is the quotient. What is left at the end is the remainder, and it must be smaller than the divisor: \(395 \div 17 = 23 \text{ remainder } 4\).
Check: quotient \(\times\) divisor \(+ \text{ remainder } =\) dividend: \(23 \times 17 + 4 = 391 + 4 = 395\). Make it divisible: take away the remainder: \(395 - 4 = 391 = 17 \times 23\).
Remainder as a fraction: put it over the divisor and simplify: \(23 \text{ remainder } 4 = 23\dfrac{4}{17}\). Remainder as a decimal: write a decimal point and zeros after the dividend, put the point in the quotient straight above, and keep dividing.
The four steps, again and again:
| Step | What you do |
|---|---|
| 1. Divide | How many times does the divisor go in? |
| 2. Multiply | Quotient digit × divisor |
| 3. Subtract | Take it away |
| 4. Bring down | The next digit |
How to do long division
- Estimate using multiples of the divisor (\(\times 10\), \(\times 20\), ...).
- Divide the first part of the dividend that is big enough.
- Multiply, subtract, bring down; repeat until no digits are left.
- Write the remainder (or keep going past the point for a decimal).
- Check: quotient \(\times\) divisor \(+\) remainder.
| \(67 \times 12\) | \(=\) | \(804\) |
\(804 \div 12 = 67\)
| \(24 \times 14 + 4\) | \(=\) | \(340\) |
\(24 \text{ remainder } 4\), so \(340 \div 14 = 24\dfrac{4}{14} = 24\dfrac{2}{7}\).
(i) with a remainder
(ii) as a mixed numeral
(iii) as a decimal
| \(23.6 \times 25\) | \(=\) | \(590\) |
(i) \(23 \text{ remainder } 15\)
(ii) \(23\dfrac{15}{25} = 23\dfrac{3}{5}\)
(iii) Bring down a \(0\) after the point: \(150 \div 25 = 6\), so \(23.6\)
| \(41 \times 18 + 4\) | \(=\) | \(742\) |
\(742 \div 18 = 41 \text{ remainder } 4\). The \(4\) people left still need a table: \(42\) tables.
| \(408 \times 21\) | \(=\) | \(8568\) |
\(21\) goes into \(16\) zero times, so write \(0\) and bring down the \(8\). Each row has \(408\) seedlings.
| \(19 \times 13 + 3\) | \(=\) | \(250\) |
| \(250 - 3\) | \(=\) | \(247\) |
\(250 \div 13 = 19 \text{ remainder } 3\), so subtract \(3\): \(247 = 13 \times 19\).
Common pitfalls
Frequently asked questions
What are the four steps of long division?
Divide, multiply, subtract, bring down.
When do you write a zero in the quotient?
When the divisor does not go into the number you have, after you bring down a digit.
How do you check a long division?
Quotient times divisor plus remainder equals the dividend.
How do you write the remainder as a fraction?
Put it over the divisor and simplify: \(24\) r \(4\) dividing by \(14\) is \(24\dfrac{2}{7}\).
How do you get a decimal answer?
Add a decimal point and zeros to the dividend and keep dividing.