How to Do Long Division

The method, step by step

1. 01

   Set the calculation out

   12 ) 4728

   The divisor goes outside the bracket and the dividend inside it, because the answer is written above the dividend one digit at a time and needs the space. Keeping the columns lined up is not neatness for its own sake — the place value of every digit in the answer depends on which column it sits in.

2. 02

   Divide into the first digits that are large enough

   12 into 4 → 0, so use 47. 47 ÷ 12 = 3 r 11. Write 3 above the 7.

   Twelve does not fit into 4, so the first digit of the answer belongs above the 7 rather than above the 4. Placing that first digit correctly is where most wrong answers are decided, because everything after it inherits the same column.

3. 03

   Multiply back and subtract to find what is left

   3 × 12 = 36. 47 - 36 = 11.

   The subtraction tells you how much of the 47 the divisor could not account for. That leftover is not waste — it is carried into the next column as tens, which is why the remainder at each stage has to be smaller than the divisor. If it is not, the digit you chose was too small.

4. 04

   Bring down the next digit and repeat

   bring down 2 → 112. 112 ÷ 12 = 9 r 4. 9 × 12 = 108. 112 - 108 = 4.

   Bringing the digit down is what joins the leftover to the next place value: the 11 left over becomes 110, and the 2 brought down makes 112. The cycle is identical every time, which is the whole point of the method — one procedure repeated rather than a new decision at each stage.

5. 05

   Finish the last column

   bring down 8 → 48. 48 ÷ 12 = 4 r 0. 4 × 12 = 48. 48 - 48 = 0.

   There are no digits left to bring down, so the division is complete and the final subtraction gives the remainder. A zero here means the divisor goes in exactly; any other number is the remainder and has to be reported.

6. 06

   Check by multiplying back

   4728 ÷ 12 = 394. Check: 394 × 12 = 4728 ✓

   Multiplication undoes division, so multiplying the answer by the divisor should return the original number exactly. This is the one check that catches a misplaced digit, and it takes less time than re-reading the working. Where there is a remainder, the check becomes answer × divisor + remainder.

01

The cycle is the whole method

Long division looks long because it is written out, not because there is much to remember. There are four moves — divide, multiply, subtract, bring down — and they repeat in that order once for every digit of the dividend. A student who can say the cycle can do the method; a student who cannot is guessing at each stage.

The most common failure is not arithmetic at all. It is placing the first digit of the answer in the wrong column, which shifts every digit after it and produces an answer ten times too big or too small. Checking that the first digit sits above the last digit of the group you divided into prevents it.

• Divide — how many times does the divisor fit?
• Multiply — that digit by the divisor
• Subtract — to find what is left over
• Bring down — the next digit, and start again

02

What to do with a remainder

A remainder can be written three ways, and the context decides which is right. As a whole number remainder (394 remainder 2), as a fraction of the divisor (394 and 2/12, which simplifies to 394 and 1/6), or by continuing into decimal places (394.1666…). The national curriculum for England expects pupils to interpret remainders in all three forms.

The choice is usually made by the question rather than by preference. Dividing 50 pencils between 12 children leaves a whole-number remainder of 2, because half a pencil is not a useful answer. Dividing 50 metres of rope between 12 people does not.

03

When the divisor has two digits

Two-digit divisors are where long division stops being mechanical, because the number of times 12 goes into 112 is not a recalled fact. Estimating first is the technique: round the divisor to the nearest ten, see roughly how many times it fits, then adjust.

Writing out the first few multiples of the divisor before starting removes the guesswork entirely. For 12, listing 12, 24, 36, 48, 60, 72, 84, 96, 108, 120 down the side of the page turns every stage of the division into a lookup. It is not cheating; it is what the method is trying to make possible.

04

How we teach long division

We teach the estimate-then-adjust habit before the layout, because a student who cannot judge roughly how many times 12 goes into 112 will not be rescued by neat columns. Once the estimate is reliable, the written method becomes a way of recording work rather than a puzzle in itself.

Times table fluency is checked first. Long division is where weak tables surface, and diagnosing it as a division problem when it is a recall problem wastes weeks. A free demo class is usually enough to tell which of the two we are dealing with.