Long Division
What Is This?
Long division is a written method for dividing large numbers — numbers too big to divide mentally. It works by breaking the division into a series of smaller steps: divide, multiply, subtract, and bring down. You repeat these four steps until every digit of the dividend has been used.
Why Does It Matter?
Long division is one of the most important algorithms in arithmetic. It is used any time you need to split a quantity into equal groups, find how many times one number fits into another, calculate averages, or convert fractions to decimals. It also builds directly toward algebra and ratio work at higher grades.
Key Vocabulary
| Term | Meaning |
|---|---|
| Dividend | The number being divided (the one that goes inside the division bracket) |
| Divisor | The number you are dividing by (the one outside the bracket) |
| Quotient | The answer to a division problem |
| Remainder | The amount left over when a number does not divide evenly |
| Bring down | Moving the next digit of the dividend down to join the current remainder |
The Four-Step Cycle
Every long division problem follows the same four steps, repeated for each digit of the dividend:
- Divide — How many times does the divisor go into the current number?
- Multiply — Multiply the divisor by that quotient digit.
- Subtract — Subtract the product from the current number.
- Bring down — Bring down the next digit of the dividend.
Then repeat.
Method 1 — 2-Digit ÷ 1-Digit
Example: 84 ÷ 4
Set up:
_ _
4 ) 8 4
Step 1 — Divide: 8 ÷ 4 = 2. Write 2 above the 8. Step 2 — Multiply: 2 × 4 = 8. Write 8 below. Step 3 — Subtract: 8 − 8 = 0. Step 4 — Bring down: Bring down the 4.
Now: 4 ÷ 4 = 1. Write 1 above the 4. Multiply: 1 × 4 = 4. Subtract: 4 − 4 = 0. No remainder.
2 1
4 ) 8 4
8
--
0 4
4
---
0
Answer: 21 ✓
Method 2 — 3-Digit ÷ 1-Digit
Example: 738 ÷ 6
1 2 3
6 ) 7 3 8
6
---
1 3
1 2
---
1 8
1 8
---
0
- 7 ÷ 6 = 1 remainder 1 → write 1, multiply 1×6=6, subtract 7−6=1
- Bring down 3 → 13 ÷ 6 = 2 remainder 1 → write 2, multiply 2×6=12, subtract 13−12=1
- Bring down 8 → 18 ÷ 6 = 3 → write 3, multiply 3×6=18, subtract 18−18=0
Answer: 123 ✓
Method 3 — Division with a Remainder
Example: 95 ÷ 7
1 3
7 ) 9 5
7
---
2 5
2 1
---
4
- 9 ÷ 7 = 1 remainder 2 → write 1
- Bring down 5 → 25 ÷ 7 = 3 remainder 4 → write 3
- No more digits to bring down. Remainder = 4.
Answer: 13 remainder 4, written as 13 R4 ✓
Method 4 — 3-Digit ÷ 2-Digit
Example: 546 ÷ 13
4 2
13 ) 5 4 6
5 2
----
2 6
2 6
---
0
- 54 ÷ 13: 13 × 4 = 52 (13 × 3 = 39, 13 × 4 = 52, 13 × 5 = 65 — too big). Write 4.
- Subtract: 54 − 52 = 2. Bring down 6 → 26.
- 26 ÷ 13 = 2. Multiply: 2 × 13 = 26. Subtract: 26 − 26 = 0.
Answer: 42 ✓
What If the Divisor Doesn't Go In?
Sometimes the divisor is larger than the first digit of the dividend. In this case, look at the first two digits together.
Example: 312 ÷ 4
- 3 ÷ 4: 4 doesn't go into 3. Look at 31 instead.
- 31 ÷ 4 = 7 remainder 3. Write 7 above the 1 in 31.
- Bring down 2 → 32 ÷ 4 = 8.
0 7 8
4 ) 3 1 2
2 8
----
3 2
3 2
---
0
Answer: 78 ✓
Checking Your Answer
Multiply the quotient by the divisor, then add any remainder. The result should equal the dividend.
- Check: quotient × divisor + remainder = dividend
- 21 × 4 + 0 = 84 ✓
- 13 × 7 + 4 = 91 + 4 = 95 ✓
- 42 × 13 + 0 = 546 ✓
Estimating to Check
Before dividing, round the dividend to a compatible number.
Example: 738 ÷ 6
- Estimate: 720 ÷ 6 = 120
- Exact: 123 ✓ (close to 120)
Example: 546 ÷ 13
- Estimate: 520 ÷ 13 ≈ 40
- Exact: 42 ✓ (close to 40)
Worked Examples
Worked Example 1: 525 ÷ 5
1 0 5
5 ) 5 2 5
5
---
0 2
0 0
----
2 5
2 5
---
0
- 5 ÷ 5 = 1; bring down 2 → 2 ÷ 5 = 0 (write 0); bring down 5 → 25 ÷ 5 = 5.
Answer: 105 ✓
Worked Example 2: 847 ÷ 7
- 8 ÷ 7 = 1 R1; bring down 4 → 14 ÷ 7 = 2; bring down 7 → 7 ÷ 7 = 1.
Answer: 121 ✓
Worked Example 3 (Word Problem): 252 pencils are shared equally among 12 students. How many pencils does each student get?
252 ÷ 12:
- 25 ÷ 12 = 2 R1; bring down 2 → 12 ÷ 12 = 1.
Answer: 21 pencils each ✓
Common Mistakes
Mistake 1: Forgetting to write a zero in the quotient If the divisor doesn't go into the brought-down digit, you must write 0 in the quotient and bring down the next digit. Skipping the zero shifts all subsequent digits and gives the wrong answer.
Mistake 2: Subtracting incorrectly After multiplying, be careful with the subtraction step. A subtraction error early on will cause every subsequent step to be wrong.
Mistake 3: Not checking the answer Always multiply quotient × divisor + remainder to confirm you get the original dividend. This catches carrying and subtraction errors immediately.
Mistake 4: Stopping too early Every digit of the dividend must be used. Keep bringing down digits until there are none left.
Key Takeaways
- Long division follows four repeating steps: Divide → Multiply → Subtract → Bring down
- If the divisor is larger than the current digit, take two digits together
- Write a zero in the quotient whenever the divisor doesn't go into the current number
- A remainder is left over when the number does not divide evenly — write it as R followed by the remainder value
- Always check: quotient × divisor + remainder = dividend
- Estimate first to catch major errors
Practice and Resources
Ready to practise? Try our Grade 3-5 Long Division Worksheet with problems from 2-digit ÷ 1-digit up to 3-digit ÷ 2-digit, with and without remainders. Test yourself with the Grade 3-5 Math Practice Test.