Long Division

The idea in one sentence

Long division deals with one digit at a time, carrying whatever won’t divide evenly across to the next column.

The four steps, on repeat

To work out \(736 \div 4\):

Divide. How many 4s in 7? One, with 3 left over. Write 1 above the 7. Multiply and subtract. \(1 \times 4 = 4\), and \(7 - 4 = 3\). Bring down. Carry the 3 across to the next digit, making 33. Repeat. How many 4s in 33? Eight, with 1 left. Write 8. Carry the 1 to make 16. How many 4s in 16? Four exactly.

Divide, multiply, subtract, bring down — and again, until the digits run out. The rhythm is the method, and once it’s automatic the size of the number stops mattering.

What the carrying means

That “3 left over” isn’t an abstract leftover. When you divide 7 hundreds by 4, you get 1 hundred each with 3 hundreds undistributed. Those 3 hundreds become 30 tens, which join the 3 tens already there to make 33 tens.

So carrying is regrouping — the same trade as in subtraction, running the same direction: one of a bigger unit becomes ten of a smaller one. The shorthand hides it, which is why the method feels arbitrary until someone says what the little carried digit is.

When it doesn’t come out evenly

\(739 \div 4\) finishes with 3 left over. Three ways to report it, all correct in different situations:

• 184 remainder 3 — for whole things that can’t be split, like people or boxes.
• \(184\frac{3}{4}\) — the remainder over the divisor, as a fraction.
• 184.75 — carry on past the decimal point, bringing down zeros.

The context decides. Sharing 739 pence between 4 gives 184.75p; putting 739 eggs into boxes of 4 gives 184 full boxes and 3 spare eggs.

A zero in the answer counts

\(816 \div 4\): there are two 4s in 8, then zero 4s in 1, then four 4s in 16. The answer is 204, and that middle zero must be written. Skipping it gives 24, which is ten times too small — and an estimate catches it instantly, since 816 shared between 4 is clearly around 200.

Common mix-ups

• Dropping a zero from the middle of the answer.
• Forgetting to bring the next digit down, which ends the division early.
• Subtracting wrongly and carrying a number bigger than the divisor — if what’s left is 4 or more when dividing by 4, the previous digit was too small.
• Not estimating. Round first: \(736 \div 4\) is near \(800 \div 4 = 200\).

Check yourself

Work out \(952 \div 7\), and say what the first carried digit means.

\(952 \div 7 = 136\). The first step gives one 7 in 9 with 2 left — and those 2 are hundreds, which become 20 tens joining the 5 tens to make 25.