Long Division

Long division is a step-by-step method for dividing large numbers that cannot be done mentally. It breaks a big division problem into a series of smaller, manageable steps. The process follows a repeating cycle: Divide, Multiply, Subtract, Bring down — sometimes remembered as “Does McDonald’s Sell Burgers?” or simply DMSB.

The Parts of a Division Problem

$\text{dividend} \div \text{divisor} = \text{quotient}$

• Dividend: the number being divided (the big number)
• Divisor: the number you are dividing by
• Quotient: the answer
• Remainder: whatever is left over after dividing

In long division format:

The DMSB Cycle

Every step of long division follows the same four-part cycle:

1. Divide: How many times does the divisor go into the current number?
2. Multiply: Multiply the divisor by that digit
3. Subtract: Subtract the result from the current number
4. Bring down: Bring down the next digit from the dividend

Repeat until there are no more digits to bring down.

Example 1: $846 \div 3$

Step 1 — Divide: How many times does 3 go into 8? 2 times (since $3 \times 2 = 6$ and $3 \times 3 = 9$ is too big).

Step 2 — Multiply: $3 \times 2 = 6$

Step 3 — Subtract: $8 - 6 = 2$

Step 4 — Bring down: Bring down the 4, making 24.

Repeat the cycle:

Divide: $3$ into $24$ = 8 (since $3 \times 8 = 24$).

Multiply: $3 \times 8 = 24$

Subtract: $24 - 24 = 0$

Bring down: Bring down the 6, making 6.

One more cycle:

Divide: $3$ into $6$ = 2

Multiply: $3 \times 2 = 6$

Subtract: $6 - 6 = 0$

No more digits to bring down.

$846 \div 3 = 282$

Check: $282 \times 3 = 846$ ✓

Example 2: Division with a Remainder — $529 \div 4$

Divide: $4$ into $5$ = 1 ($4 \times 1 = 4$)

Multiply: $4 \times 1 = 4$

Subtract: $5 - 4 = 1$

Bring down: Bring down the 2, making 12.

Divide: $4$ into $12$ = 3 ($4 \times 3 = 12$)

Multiply: $4 \times 3 = 12$

Subtract: $12 - 12 = 0$

Bring down: Bring down the 9, making 9.

Divide: $4$ into $9$ = 2 ($4 \times 2 = 8$)

Multiply: $4 \times 2 = 8$

Subtract: $9 - 8 = 1$

No more digits. The remainder is 1.

$529 \div 4 = 132 \text{ R } 1$

Check: $132 \times 4 + 1 = 528 + 1 = 529$ ✓

Example 3: Dividing by a Two-Digit Number — $1,547 \div 12$

When the divisor has two or more digits, the divide step requires estimation.

Divide: $12$ into $15$ = 1 ($12 \times 1 = 12$, $12 \times 2 = 24$ is too big)

Multiply: $12 \times 1 = 12$

Subtract: $15 - 12 = 3$

Bring down: Bring down the 4, making 34.

Divide: $12$ into $34$ = 2 ($12 \times 2 = 24$, $12 \times 3 = 36$ is too big)

Multiply: $12 \times 2 = 24$

Subtract: $34 - 24 = 10$

Bring down: Bring down the 7, making 107.

Divide: $12$ into $107$ = 8 ($12 \times 8 = 96$, $12 \times 9 = 108$ is too big)

Multiply: $12 \times 8 = 96$

Subtract: $107 - 96 = 11$

$1{,}547 \div 12 = 128 \text{ R } 11$

Check: $128 \times 12 + 11 = 1{,}536 + 11 = 1{,}547$ ✓

Extending to Decimal Answers

Instead of writing a remainder, you can continue dividing by adding a decimal point and zeros to the dividend.

Example 4: $7 \div 4$ as a decimal

$7 \div 4 = 1 \text{ R } 3$

Instead of stopping at the remainder, add a decimal point to the quotient and a zero to make the remainder 30:

Divide: $4$ into $30$ = 7 ($4 \times 7 = 28$)

Subtract: $30 - 28 = 2$

Bring down another 0 to make 20:

Divide: $4$ into $20$ = 5 ($4 \times 5 = 20$)

Subtract: $20 - 20 = 0$

$7 \div 4 = 1.75$

When to Use Remainders vs. Decimals

Estimation: Checking Your Work

Before doing long division, estimate the answer to catch major errors:

• $846 \div 3$: 900 ÷ 3 = 300, so the answer should be near 300. We got 282 ✓
• $1{,}547 \div 12$: 1,200 ÷ 12 = 100, and 1,800 ÷ 12 = 150, so the answer is between 100 and 150. We got 128 ✓

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Long division follows a repeating cycle: Divide, Multiply, Subtract, Bring down
• Always check your answer: quotient × divisor + remainder = dividend
• Estimate first to catch large errors
• Add a decimal point and zeros to continue past a remainder when you need a decimal answer
• For two-digit divisors, use estimation in the divide step — it is normal to adjust your guess

Return to Arithmetic for more foundational math topics.