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Long Division Calculator
The number being divided
The number to divide by
Understanding Long Division
Long division is a method for dividing large numbers that breaks the division process into a series of easier steps. It's one of the fundamental arithmetic operations taught in elementary school and remains useful for understanding division concepts, even in the age of calculators.
The Long Division Process
Step-by-Step Method (Divide, Multiply, Subtract, Bring Down)
- 1. Divide: Look at the first digit(s) of the dividend. How many times does the divisor fit into it?
- 2. Multiply: Multiply the divisor by the quotient digit you just found.
- 3. Subtract: Subtract the result from the current working number.
- 4. Bring Down: Bring down the next digit from the dividend.
- 5. Repeat: Continue the process until all digits have been used.
Example: 156 ÷ 12
Step 1: 15 ÷ 12 = 1 (12 fits into 15 once)
Write 1 above the 5
Step 2: 1 × 12 = 12
Write 12 below the 15
Step 3: 15 - 12 = 3
Subtract to get 3
Step 4: Bring down the 6
Working number is now 36
Step 5: 36 ÷ 12 = 3 (12 fits into 36 three times)
Write 3 next to the 1 in quotient
Step 6: 3 × 12 = 36
36 - 36 = 0
Final Answer: 156 ÷ 12 = 13 with remainder 0Key Terminology
Dividend
The number being divided. In 100 ÷ 7, the dividend is 100.
Example: In 45 ÷ 9, the dividend is 45
Divisor
The number you're dividing by. In 100 ÷ 7, the divisor is 7.
Example: In 45 ÷ 9, the divisor is 9
Quotient
The result of the division (whole number part). In 100 ÷ 7 = 14 R 2, the quotient is 14.
Example: 45 ÷ 9 = 5 (quotient is 5)
Remainder
What's left over after division. In 100 ÷ 7 = 14 R 2, the remainder is 2.
Example: 47 ÷ 9 = 5 R 2 (remainder is 2)
Visual Long Division Layout
Long division is typically written in a special format with the divisor outside, the dividend inside, and the quotient on top:
14 ← Quotient
-----
7 | 100 ← Dividend
7↓
---
30
28
---
2 ← Remainder
Divisor →The division symbol (⟌) acts as a bracket containing the dividend, with the divisor to the left and quotient above.
Types of Division Results
Even Division (No Remainder)
When the dividend is perfectly divisible by the divisor:
Examples: 100 ÷ 10 = 10 (remainder 0) 144 ÷ 12 = 12 (remainder 0) 81 ÷ 9 = 9 (remainder 0)
Division with Remainder
When there's a leftover amount after division:
Examples: 100 ÷ 7 = 14 R 2 50 ÷ 8 = 6 R 2 23 ÷ 5 = 4 R 3
The remainder is always less than the divisor.
Decimal Division
Expressing the result as a decimal number:
Examples: 100 ÷ 7 = 14.285714... 50 ÷ 8 = 6.25 23 ÷ 5 = 4.6
Mixed Number Form
Expressing the result as a whole number plus a fraction:
Examples: 100 ÷ 7 = 14 2/7 50 ÷ 8 = 6 2/8 = 6 1/4 23 ÷ 5 = 4 3/5
Checking Your Work
Verification Formula
To verify your long division is correct, use this formula:
(Divisor × Quotient) + Remainder = Dividend
Example with 100 ÷ 7 = 14 R 2:
(7 × 14) + 2 = 100 98 + 2 = 100 100 = 100 ✓ Correct!
Common Long Division Patterns
Dividing by 10, 100, 1000
Simply move the decimal point left by the number of zeros:
4500 ÷ 10 = 450 4500 ÷ 100 = 45 4500 ÷ 1000 = 4.5
Dividing by 2 (Halving)
Split the number in half:
100 ÷ 2 = 50 88 ÷ 2 = 44 75 ÷ 2 = 37.5 or 37 R 1
Dividing by 5
Multiply by 2, then divide by 10:
85 ÷ 5 = (85 × 2) ÷ 10 = 170 ÷ 10 = 17
Real-World Applications
🍕 Sharing & Distribution
- • Dividing pizza slices among friends
- • Splitting bills or costs equally
- • Distributing items into groups
- • Allocating resources fairly
📊 Business & Finance
- • Calculating unit prices (total ÷ quantity)
- • Determining average costs
- • Computing rates and ratios
- • Budget allocation per category
📏 Measurements & Conversions
- • Converting units (inches to feet)
- • Calculating per-unit measurements
- • Finding dimensions (area ÷ width = length)
- • Recipe scaling and portions
⏱️ Time & Rate Calculations
- • Speed calculations (distance ÷ time)
- • Finding time per task
- • Determining frequency (events ÷ period)
- • Work rate problems
Division with Decimals
Dividing Decimals by Whole Numbers
Place the decimal point in the quotient directly above the decimal point in the dividend:
Example: 12.8 ÷ 4 3.2 ---- 4|12.8 12 --- 08 8 -- 0
Dividing by Decimals
Move the decimal point in the divisor to make it a whole number, then move the decimal point in the dividend the same number of places:
Example: 15.6 ÷ 1.2 Step 1: Convert 1.2 to 12 (move decimal 1 place) Step 2: Convert 15.6 to 156 (move decimal 1 place) Step 3: Calculate 156 ÷ 12 = 13 Answer: 15.6 ÷ 1.2 = 13
Tips for Success
✓ Do's
- • Line up digits carefully
- • Double-check each step
- • Verify with multiplication
- • Keep numbers aligned
- • Use estimation to check reasonableness
✗ Don'ts
- • Don't skip the multiplication step
- • Don't forget to bring down digits
- • Don't misalign place values
- • Don't divide by zero
- • Don't rush through subtraction
Common Mistakes to Avoid
❌ Forgetting to bring down the next digit
After subtracting, always bring down the next digit from the dividend before continuing.
❌ Misplacing the decimal point
When dividing decimals, carefully track where the decimal point should be in the quotient.
❌ Writing the wrong quotient digit
Make sure your quotient digit × divisor doesn't exceed the working number.
❌ Incorrect subtraction
Double-check subtraction steps—errors here cascade through the rest of the problem.
Beyond Basic Long Division
Polynomial Long Division
The same long division process applies to dividing polynomials in algebra. Instead of numbers, you work with algebraic terms, but the divide-multiply-subtract-bring down pattern remains the same.
Synthetic Division
A shortcut method for dividing polynomials when the divisor is in the form (x - c). It's faster but more abstract than long division.
Frequently Asked Questions
Why do we need long division when calculators exist?
Long division teaches fundamental understanding of how division works, develops mental math skills, helps verify calculator results, and is essential for algebraic operations like polynomial division. It's also useful when calculators aren't available.
What if the remainder is larger than the divisor?
The remainder should never be larger than the divisor. If it is, you haven't divided enough times. The remainder must always be less than the divisor, or the division isn't complete.
How do I know when to stop dividing?
For whole number division, stop when you've used all digits from the dividend. For decimal division, you can continue indefinitely, but typically stop after a certain number of decimal places (like 2 or 10 places) or when the pattern repeats.
What's the difference between quotient and result?
The quotient is the whole number part of the division result. The complete result may include a remainder (100 ÷ 7 = 14 R 2), where 14 is the quotient and 2 is the remainder. The decimal form (14.285...) is another way to express the complete result.
Can I divide a smaller number by a larger number?
Yes! When dividing a smaller number by a larger number, the quotient will be 0 with a remainder equal to the original number (e.g., 5 ÷ 10 = 0 R 5), or as a decimal less than 1 (e.g., 5 ÷ 10 = 0.5).
Why does my decimal division repeat?
Some divisions result in repeating decimals (like 1 ÷ 3 = 0.333...). This happens when the dividend can't be evenly divided by the divisor, creating a pattern that repeats infinitely. These are called repeating or recurring decimals.
How do I check if my long division is correct?
Use the formula: (Divisor × Quotient) + Remainder = Dividend. If this equation is true, your division is correct. For example, with 100 ÷ 7 = 14 R 2: (7 × 14) + 2 = 98 + 2 = 100 ✓