Long Division

You've got a bag of 738 sweets and 6 friends to share them with (yourself included). Everyone must get exactly the same amount — no arguing. How many sweets does each person get?

That's 738 \div 6, and it's far too big to just see the answer in your head. This is exactly what long division is for: a reliable, step-by-step method that will divide any number by another, chewing through the digits one at a time from left to right until the whole thing is done. Once you know the rhythm, no division is too big to beat.

The rhythm: divide, multiply, subtract, bring down

Long division repeats the same four steps for every digit of the number you're dividing (the dividend). Say them like a little chant:

We write it in "bus stop" notation: the dividend sits inside the bus shelter, the divisor stands outside on the left, and the answer (the quotient) grows along the roof on top — each answer digit lined up directly above the digit it came from.

Worked example 1 — the sweets, shared exactly

Let's settle the sweets: 738 \div 6. Read top to bottom; each block is one round of the chant. 

        1 2 3       ← quotient (the answer)
     ┌────────
   6 │ 7 3 8
       6           DIVIDE: 6 into 7 goes 1 time.  MULTIPLY: 6 × 1 = 6
       ─
       1 3         SUBTRACT: 7 − 6 = 1.  BRING DOWN the 3 → 13
       1 2         DIVIDE: 6 into 13 goes 2 times.  MULTIPLY: 6 × 2 = 12
       ───
         1 8       SUBTRACT: 13 − 12 = 1.  BRING DOWN the 8 → 18
         1 8       DIVIDE: 6 into 18 goes 3 times.  MULTIPLY: 6 × 3 = 18
         ───
           0       SUBTRACT: 18 − 18 = 0 — nothing left, so it divides exactly

So 738 \div 6 = 123. Each friend gets 123 sweets, with none left over. Crunch through the digits one at a time and even a scary-looking sum falls apart into easy little steps.

Worked example 2 — when there's a bit left over

Real life rarely divides so neatly. Suppose you have 227 stickers to pack into boxes of 4. How many full boxes can you make, and how many stickers are left rattling around? 

        5 6         ← quotient
     ┌──────
   4 │ 2 2 7
       2 0         4 into 22 goes 5 times; 4 × 5 = 20
       ──
         2 7       22 − 20 = 2, bring down the 7 → 27
         2 4       4 into 27 goes 6 times; 4 × 6 = 24
         ──
           3       27 − 24 = 3 — and 4 won't go into 3, so we STOP

The 3 that's left over is the remainder:

227 \div 4 = 56 \text{ r } 3.

Now read the remainder in context. You can fill 56 full boxes, and 3 stickers are left over (not enough for a 57th box). The remainder is a real thing you can point at — the leftovers.

Worked example 3 — carrying on into the decimals

Sometimes you don't want a leftover — you want to keep going and split the remainder up too. Just imagine the dividend has a decimal point and a string of zeros after it: 3 = 3.000. Then keep the chant running past the point.

Let's do 3 \div 8:

        0 . 3 7 5     ← quotient
     ┌────────────
   8 │ 3 . 0 0 0
       0             8 into 3 goes 0 times, so write 0 · then bring down
       3 0           8 into 30 goes 3 times; 8 × 3 = 24
       2 4
       ──
         6 0         30 − 24 = 6, bring down a 0 → 60; 8 × 7 = 56
         5 6
         ──
           4 0       60 − 56 = 4, bring down a 0 → 40; 8 × 5 = 40
           4 0
           ──
             0       40 − 40 = 0 — done, it divides exactly

So 3 \div 8 = 0.375. The decimal point in the answer lines up straight above the point in the dividend. This is the secret machine that turns any fraction into a decimal — because \tfrac{3}{8} is just "3 divided by 8".

Whether it divides exactly, stops at a remainder, or runs on into decimals, the working is always the same four steps repeating. Here's the classic two-digit divisor case too, so you can see the rhythm doesn't change when the divisor gets bigger — 432 \div 16 = 27: 

        2  7        ← quotient (answer)
     ┌────────
  16 │ 4 3 2
       3 2          16 × 2 = 32   (DIVIDE: 43 ÷ 16 = 2, MULTIPLY)
       ───
       1 1 2        43 − 32 = 11  (SUBTRACT), then BRING DOWN the 2 → 112
       1 1 2        16 × 7 = 112  (DIVIDE: 112 ÷ 16 = 7, MULTIPLY)
       ─────
           0        112 − 112 = 0 (SUBTRACT) — nothing left, so it divides exactly

The divisor 16 goes into 43 twice (the 4 alone is too small), then into 112 exactly seven times — giving 432 \div 16 = 27.

That last line is a brilliant safety net: multiply your answer back by the divisor, add the remainder, and you should land exactly on the number you started with. For example 2: 56 \times 4 + 3 = 224 + 3 = 227. Tick!

These two traps swallow more marks than anything else — guard against both:

Long division has a fearsome reputation as the trickiest thing in primary maths. But peek under the hood and it's gentle: it's just repeated subtraction, sped up. Sharing 738 among 6 could be done by taking away 6 again and again and counting how many times — but that would take 123 subtractions! Long division cleverly takes away big chunks (like 600, then 120, then 18) all at once, place value by place value. Same idea, hugely faster.

And it hides a beautiful surprise. Because dividing top-by-bottom turns every fraction into a decimal, some fractions never stop. Try 1 \div 3: you get 0.3333\ldots and the same remainder keeps coming back forever, so the 3s go on and on. That's your first glimpse that not every number is "tidy" — a repeating decimal that runs to infinity — which you'll explore in terminating and recurring decimals.

Before diving into the steps, make a rough estimate — it catches silly errors. For 738 \div 6, round to friendly numbers: 6 goes into 720 about 120 times, so the answer should be "a bit over 120". Our exact answer, 123, fits perfectly.

If your long division had spat out 1230 or 13, the estimate would instantly wave a red flag — usually a missing or misplaced zero in the quotient. Estimate first, work carefully, then check with \text{quotient} \times \text{divisor} + \text{remainder}. Three layers of safety, and you'll almost never get a division wrong.

See it explained