Perform long division with detailed step-by-step solutions. Learn the division algorithm with clear explanations.
Basic: 84 ÷ 4
With remainder: 127 ÷ 5
Larger: 1234 ÷ 12
Three-digit: 456 ÷ 23
Perfect: 144 ÷ 12
Complex: 9876 ÷ 34
How many times does 12 go into 15? Once.
12 × 1 = 12. Write it below.
15 − 12 = 3. Must be less than 12.
Bring down the 6 to get 36. Repeat.
Long division breaks a big division problem into bite-sized steps. You work left to right through the dividend, one digit at a time, using a four-step cycle:
That's it. Four steps on a loop until you run out of digits. The number sitting on top of the bracket is your quotient, and whatever's left at the bottom is the remainder.
Division Terms:
Formula:
Divide resources equally among groups - splitting 150 books among 12 classrooms, distributing 240 candies to 15 children, or sharing costs.
Calculate unit prices (cost per item), fuel efficiency (miles per gallon), or speed (distance per hour) using division.
Divide total work hours into shifts, allocate project time across days, or determine how many items can be completed per hour.
Reduce recipe portions by dividing ingredient amounts, calculate servings per batch, or determine portions per person.
Calculate how many tiles fit in a room, how many boards can be cut from lumber, or partition materials across multiple projects.
Forgetting to bring down the next digit
After subtracting, always bring down the next digit from the dividend before continuing. Skipping this step leads to wrong answers.
Placing digits in the wrong position
Each quotient digit must align above the last digit of the portion being divided. Misalignment causes place value errors.
Incorrect subtraction
Double-check subtraction at each step. A small error early in the process propagates through all remaining steps.
Not checking the answer
Always verify: (divisor × quotient) + remainder should equal the original dividend. This catches mistakes immediately.
Best Practice
Follow the division algorithm systematically: Divide → Multiply → Subtract → Bring Down → Repeat. Check your work at the end.
| Method | Best For | Advantages | Limitations |
|---|---|---|---|
| Long Division | Any division problem, especially large numbers | Works for all cases; systematic; shows detailed steps | Time-consuming; requires careful tracking |
| Short Division | Single-digit divisors; mental math | Faster than long division; less writing required | Only practical for small divisors; harder to track steps |
| Chunking (Partial Quotients) | Learning division; building number sense | Intuitive; flexible; emphasizes understanding | Less efficient; more steps than long division |
| Repeated Subtraction | Very small numbers; understanding division concept | Shows division as repeated removal; conceptually clear | Extremely slow for large numbers; impractical |
The remainder represents what's "left over" after division when the dividend doesn't divide evenly by the divisor. Remainders can be expressed in multiple ways depending on the context.
Standard form showing quotient and remainder separately. Used when you need whole numbers only.
Remainder becomes the numerator over the divisor. Exact representation as a mixed number.
Continue division with decimal places. Most precise for measurements and calculations.
Choosing the Right Form: Use integer remainders for discrete items (people, objects), fractions for parts of wholes (recipes, measurements), and decimals for precision (money, science).
I watched my 9-year-old daughter cry over long division homework. She could do the steps — divide, multiply, subtract, bring down — like a robot. But when I asked "what are you actually doing?", she stared at me. No idea.
That's the problem with how long division gets taught. It's presented as an algorithm to memorize, not a process to understand. Divide, multiply, subtract, bring down. DMSB. Some teachers use "Does McDonald's Sell Burgers" as a mnemonic. Cute. But it doesn't explain why each step works.
Once I showed her what the algorithm is actually doing — distributing hundreds, then tens, then ones — the tears stopped. The math didn't change. Her understanding of it did.
You have 847 things and you need to split them equally among 3 groups. Think of 847 as 8 hundreds, 4 tens, and 7 ones — like having 8 hundred-dollar bills, 4 tens, and 7 singles.
Step 1: Divide the hundreds. How many times does 3 go into 8? Twice, with 2 left over. Each group gets 2 hundreds. You've distributed 600 of your 847. Write 2 above the 8.
Step 2: Handle the remainder. 8 - 6 = 2 hundreds left over. Convert those 2 hundreds into 20 tens. Add the 4 tens you already had: 24 tens total. "Bring down" is just combining the leftover with the next place value.
Step 3: Divide the tens. How many times does 3 go into 24? Eight times, exactly. Each group gets 8 tens. No remainder. Write 8 above the 4.
Step 4: Divide the ones. Bring down the 7. How many times does 3 go into 7? Twice, with 1 left over. Each group gets 2 ones, and there's 1 left that can't be evenly split. Write 2 above the 7. Remainder 1.
Each group gets 282, and one item is left over. That's all long division does — it distributes place values one at a time, from biggest to smallest, carrying leftovers forward.
847 ÷ 3 — Step by Step
"Bring down" is the most mechanical-sounding step, and it's the one kids understand least. Here's what it actually means: when you have 2 hundreds left over after distributing hundreds, you're converting those 2 hundreds into 20 tens. Then you combine them with the tens you haven't distributed yet (4 tens). That gives you 24 tens to work with.
It's the same thing you do with money. If you have 2 hundred-dollar bills and need to make change, you swap them for 20 tens. "Bring down" is just making change between place values.
This connects directly to how polynomial long division works in algebra — same algorithm, different objects. Instead of distributing hundreds and tens, you're distributing x² terms and x terms. The "bring down" step is identical.
Multiply the quotient by the divisor and add the remainder. If you get back to the original number, you're right.
This works every time. It's the fastest way to verify long division, and it reinforces that division is just multiplication in reverse. Your GPA calculation is really just a division problem too — total quality points divided by total credit hours.
The 847 ÷ 3 walkthrough above covers the basic loop. These four add the situations that actually stop people: a two-digit divisor, a division that comes out exact, a decimal divisor, and numbers too big to want to write out.
24 doesn't go into 3, and it doesn't go into 33 more than once. Start there:
Answer: 137 remainder 19, or 137.7916… as a decimal (add a decimal point and keep bringing down zeros).
Check: 137 × 24 + 19 = 3288 + 19 = 3307 ✓
17 into 60 → 3 (51), remainder 9. Bring down 5 → 95. 17 into 95 → 5 (85), remainder 10. Bring down 2 → 102. 17 into 102 → 6 exactly, remainder 0.
Answer: 356, no remainder.
A remainder of 0 is the signal to stop — don't keep going into decimals when the problem divides cleanly. Check: 356 × 17 = 6052 ✓
You can't divide by a decimal directly, so shift both numbers by the same amount until the divisor is whole. Move one place right: 1.4 → 14, and 33.32 → 333.2.
Now it's ordinary long division: 14 into 33 → 2 (28), remainder 5. Bring down 3 → 53. 14 into 53 → 3 (42), remainder 11. Bring down 2 → 112. 14 into 112 → 8 exactly.
Answer: 23.8
Shifting both numbers is safe because it multiplies top and bottom by 10 — the ratio doesn't change. Moving only one of them is the classic error.
Both numbers end in three zeros, so cancel them before writing anything down: 3,000,000 ÷ 175,000 becomes 3000 ÷ 175. That's the same division with far less to write.
175 into 300 → 1 (175), remainder 125. Bring down 0 → 1250. 175 into 1250 → 7 (1225), remainder 25. Continue into decimals: the answer repeats.
Answer: 17.142857… (17 remainder 25,000 in the original numbers)
Cancelling trailing zeros in pairs is always allowed — you're dividing both sides by the same power of ten.
Divide, multiply, subtract, bring down — repeated for each digit. Divide the divisor into the current number, multiply the result by the divisor, subtract to find the remainder, then bring down the next digit. Continue until you've processed all digits. Any final remainder is written as "R" followed by the number.
When the last subtraction step leaves a number smaller than the divisor, that's your remainder. For 847 ÷ 3, the final subtraction gives 1, which is less than 3, so the answer is 282 R1. You can also express the remainder as a fraction (282⅓) or continue dividing with decimal places (282.333...).
Multiply the quotient by the divisor, then add the remainder. The result should equal the original dividend. For 847 ÷ 3 = 282 R1: 282 × 3 + 1 = 847. If it doesn't match, recheck your subtraction steps — that's where most errors hide.
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