Long Division
Long division is a written method for dividing large numbers by systematically breaking down the calculation into manageable steps. The process involves repeatedly estimating how many times the divisor fits into portions of the dividend, then multiplying, subtracting, and bringing down the next digit. This algorithm produces exact quotients and remainders for any division problem, no matter the size of the numbers involved.
Why it matters
Long division appears throughout practical situations requiring precise calculations with larger numbers. A baker dividing 384 cupcakes among 16 party boxes needs exactly 24 per box. Financial calculations often require long division: splitting a £2,847 inheritance among 3 beneficiaries yields £949 each. In Year 6 SATs and GCSE Foundation papers, long division questions test computational fluency with multi-digit numbers. The method underpins polynomial division in A-level mathematics and forms the foundation for understanding decimal expansions. Construction workers calculating materials, shop managers distributing stock, and anyone working with precise measurements rely on long division's systematic approach to handle calculations that mental arithmetic cannot manage efficiently.
How to solve long division
Long division — how to
- See how many times the divisor fits into the first digits of the dividend.
- Multiply, subtract, bring down the next digit.
- Repeat until nothing is left. Express remainder as a decimal.
Example: 728 ÷ 10: 72 r 8 → 72.8.
Worked examples
Share 12 sweets equally among 3 friends. How many does each get?
Answer: 4
- Understand what division means → 12 ÷ 3 — Division means sharing equally. Imagine splitting 12 sweets among 3 friends so everyone gets the same amount.
- How many times does 3 fit into 12? → 3 × 4 = 12 — We ask: '3 times what equals 12?' The answer is 4, because 3 × 4 = 12.
- Check: no leftovers → 12 - 12 = 0 — There is nothing left over. 12 divides evenly by 3.
- Write the answer → 12 ÷ 3 = 4 — Each friend gets 4. That is our answer!
- Verify by multiplying back → 4 × 3 = 12 ✓ — Multiply the answer by the divisor: 4 × 3 = 12. Correct!
How many groups of 4 can you make from 52?
Answer: 13
- Understand what division means → 52 ÷ 4 — Division means sharing equally. Imagine splitting 52 sweets among 4 friends so everyone gets the same amount.
- How many times does 4 fit into 52? → 4 × 13 = 52 — We ask: '4 times what equals 52?' The answer is 13, because 4 × 13 = 52.
- Check: no leftovers → 52 - 52 = 0 — There is nothing left over. 52 divides evenly by 4.
- Write the answer → 52 ÷ 4 = 13 — Each friend gets 13. That is our answer!
- Verify by multiplying back → 13 × 4 = 52 ✓ — Multiply the answer by the divisor: 13 × 4 = 52. Correct!
317 ÷ 7 = _______
Answer: 45.2857
- Understand the division → 317 ÷ 7 — We want to share 317 equally among 7 groups. Sometimes it does not divide perfectly, and we get leftovers.
- How many whole times does 7 go into 317? → 7 × 45 = 315 — 7 fits into 317 a total of 45 whole times. That accounts for 315 out of 317.
- Find the remainder (leftovers) → 317 - 315 = 2 — Subtract what we used: 317 - 315 = 2. There are 2 left that could not be shared evenly.
- Turn the remainder into a decimal → 2 ÷ 7 = 0.2857 — Divide the leftover 2 by 7 to get the decimal part: 0.2857. Think of it as cutting the remaining pieces into smaller equal slices.
- Combine whole part and decimal → 45 + 0.2857 = 45.2857 — The whole part is 45 and the decimal part is 0.2857, giving 45.2857.
- Verify by multiplying back → 45.2857 × 7 ≈ 317 ✓ — Multiply the answer by the divisor: 45.2857 × 7 should be close to 317.
Common mistakes
- Writing 84 ÷ 12 = 6 instead of 7 by incorrectly estimating how many times 12 fits into 84, often caused by rushing the multiplication check
- Calculating 145 ÷ 8 = 18.1 instead of 18.125 by stopping the decimal expansion too early or making errors in the decimal place values
- Finding 273 ÷ 15 = 17.2 instead of 18.2 by misplacing digits when bringing down numbers or making subtraction errors in the working