How it works
Long division is a systematic method for dividing large numbers by hand. You break the problem into smaller steps, working left to right through the digits of the dividend. At each stage, you determine how many times the divisor fits into a portion of the dividend, subtract, bring down the next digit, and repeat until you've processed all digits. The result gives you three pieces of information: the quotient (whole number answer), the remainder (what's left over), and the decimal equivalent.
The formula
Dividend = (Divisor × Quotient) + Remainder
This relationship always holds true. Rearranged: Quotient = Dividend ÷ Divisor, with any leftover expressed as the remainder.
Worked example
Let's divide 456 by 12 step by step.
Step 1: Start with the leftmost digits of 456. Does 12 fit into 4? No. Does it fit into 45? Yes—12 goes into 45 three times (12 × 3 = 36).
Step 2: Write 3 above the 5 in 456. Subtract: 45 − 36 = 9. Bring down the next digit (6) to make 96.
Step 3: How many times does 12 fit into 96? Exactly 8 times (12 × 8 = 96). Write 8 above the 6.
Step 4: Subtract: 96 − 96 = 0. No remainder.
Result:
- Quotient: 38
- Remainder: 0
- Decimal result: 38.0
Verify using the formula: 456 = (12 × 38) + 0 = 456 ✓
Another example with a remainder: Divide 127 by 5.
- 5 goes into 12 twice (5 × 2 = 10); subtract to get 2.
- Bring down 7 to make 27.
- 5 goes into 27 five times (5 × 5 = 25); subtract to get 2.
- No more digits to bring down.
Result:
- Quotient: 25
- Remainder: 2
- Decimal result: 25.4
Verify: 127 = (5 × 25) + 2 = 125 + 2 = 127 ✓
Common mistakes to avoid
One frequent error is misaligning digits when bringing them down, which throws off all subsequent steps. Write each digit of the quotient directly above its corresponding dividend digit to stay organized. Another pitfall is forgetting to include a zero in the quotient when the divisor doesn't fit into a section—this shifts your answer and makes it completely wrong. If you're dividing 1024 by 32, for instance, and 32 doesn't fit into 10, you must write 0 in the tens place of your quotient before continuing. Finally, always check your work using the verification formula: multiply the divisor by the quotient, add the remainder, and confirm you get back the original dividend.