Long Division: Complete Guide to Division Symbols & Step-by-Step Examples

Long division is the step-by-step method for dividing numbers too large to solve mentally, using symbols like ÷, /, :, and the long division bracket, and following the DMSBR process: Divide, Multiply, Subtract, Bring down, Repeat. Struggling with it? You’re not alone—this is one of the most challenging topics for students in Montreal and across Quebec, from Anjou to Saint-Michel classrooms. This comprehensive guide covers everything from basic division to advanced techniques, with 20+ practice problems and real-world applications.

What is Long Division?

Long division is a mathematical method used to divide large numbers that are too difficult to solve mentally. Unlike simple division (like 20 ÷ 4 = 5), long division breaks the problem into smaller, manageable steps.

Why it’s called « long » division:

• The process takes multiple steps (not just one quick calculation)
• You write out each step vertically in a specific format
• It’s « longer » than basic mental math or short division
• The work is organized in columns to keep track of place values

Example: To solve 847 ÷ 7, you can’t easily do this in your head, so you use the long division algorithm to work through it step by step, arriving at the answer: 121.

What Are the Different Symbols for Division?

There are four main ways to represent division in mathematics. Understanding each symbol helps you read problems correctly and communicate your work clearly.

1. The Division Sign (Obelus): ÷

Symbol: ÷
Name: Obelus or division sign
How to read: « divided by »
Example: 24 ÷ 6 = 4 (read as « 24 divided by 6 equals 4 »)

When it’s used:

• Most common in elementary and middle school
• Found on calculators and keyboards
• Used in Quebec textbooks for primary students
• Standard in North America

2. The Fraction Slash (Solidus): /

Symbol: /
Name: Slash, solidus, or virgule
How to read: « divided by » or « over »
Example: 24/6 = 4 (read as « 24 divided by 6 » or « 24 over 6 »)

When it’s used:

• Common in computer programming and typing
• Used when writing fractions inline (in text)
• Standard in algebraic expressions
• Preferred for online calculators

Important note: 24/6 means the same thing as ²⁴⁄₆ (a fraction). Division and fractions are the same operation!

3. The Colon: :

Symbol: :
Name: Colon
How to read: « divided by » or « to » (in ratios)
Example: 24 : 6 = 4

When it’s used:

• Standard in Europe and some international textbooks
• Used to express ratios (3:1 means « 3 to 1 »)
• Common in proportions (a:b = c:d)
• Sometimes seen in Quebec French math materials

4. The Long Division Bracket: ) ̅

Symbol: A bracket with a bar extending to the right
Name: Long division bracket, division house, or division bar
How to read: « [divisor] into [dividend] »

Example format:

     4
   ___
6 | 24

Read as: « 6 into 24 equals 4 »

When it’s used:

• The primary symbol for long division problems
• Shows where to write the quotient (answer on top)
• Organizes work vertically with clear place values
• Essential format for Quebec examens ministériels

What Do Dividend, Divisor, Quotient, and Remainder Mean?

Before learning the long division algorithm, you need to know these essential terms:

Memory tip: « Divide the dividend by the divisor to get the quotient, with a possible remainder. »

What Is the DMSBR Method for Long Division?

The DMSBR method is the most popular way to remember the long division steps. Each letter represents an action:

💡 Memory Trick:
Dad, Mom, Sister, Brother, Really love pizza!
Or: Does McDonald’s Sell Burgers? Really!

Step-by-Step Example: 456 ÷ 4

Let’s work through a complete example using the DMSBR method.

Problem: 456 ÷ 4

Setup:

      ___
4 | 456

Step 1: Divide

Look at the first digit: 4
How many times does 4 go into 4? 1 time
Write 1 above the 4 in 456:

      1__
4 | 456

Step 2: Multiply

Multiply: 1 × 4 = 4
Write 4 below the first digit:

      1__
4 | 456
    4

Step 3: Subtract

Subtract: 4 – 4 = 0

      1__
4 | 456
    4
    ---
    0

Step 4: Bring Down

Bring down the next digit (5):

      1__
4 | 456
    4
    ---
    05

Step 5: Repeat – Divide Again

How many times does 4 go into 5? 1 time
Write 1 above the 5:

      11_
4 | 456
    4
    ---
    05

Multiply Again

1 × 4 = 4

      11_
4 | 456
    4
    ---
    05
    4

Subtract Again

5 – 4 = 1

      11_
4 | 456
    4
    ---
    05
    4
    ---
    1

Bring Down Again

Bring down the last digit (6):

      11_
4 | 456
    4
    ---
    05
    4
    ---
    16

Final Division

How many times does 4 go into 16? 4 times
Write 4 above the 6:

      114
4 | 456
    4
    ---
    05
    4
    ---
    16
    16
    ---
    0

✓ Final Answer: 456 ÷ 4 = 114
Check: 114 × 4 = 456 ✓

Example 2: Division with Remainder (157 ÷ 6)

Not all divisions result in whole numbers. Sometimes there’s a remainder—the amount left over.

Problem: 157 ÷ 6

      26 R 1
6 | 157
    12↓
    ---
    37
    36
    ---
    1

Steps:

1. 6 into 15 goes 2 times (6 × 2 = 12)
2. 15 – 12 = 3
3. Bring down 7 → now we have 37
4. 6 into 37 goes 6 times (6 × 6 = 36)
5. 37 – 36 = 1
6. Nothing left to bring down, so 1 is the remainder

Answer: 157 ÷ 6 = 26 R 1 (or 26 remainder 1)

How to write the answer:

• With remainder: 26 R 1
• As a mixed number: 26 ¹⁄₆
• As a decimal: 26.166… (repeating)

Check: (26 × 6) + 1 = 156 + 1 = 157 ✓

Example 3: Dividing by Two-Digit Numbers (432 ÷ 12)

When dividing by two-digit numbers, estimation skills become important.

Problem: 432 ÷ 12

       36
12 | 432
     36↓
     ---
     72
     72
     ---
     0

Steps:

1. 12 into 43 goes 3 times (12 × 3 = 36)
2. 43 – 36 = 7
3. Bring down 2 → now we have 72
4. 12 into 72 goes 6 times (12 × 6 = 72)
5. 72 – 72 = 0 (no remainder!)

Answer: 432 ÷ 12 = 36

Estimation tip: When dividing by 12, think « how many times does 10 go into this? » to get close, then adjust.

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Common Mistakes Students Make

❌ Mistake #1: Forgetting to bring down the next digit

WRONG: After subtracting, starting a new division without bringing down
CORRECT: Always bring down the next digit before dividing again

❌ Mistake #2: Writing the quotient in the wrong place

WRONG: Writing quotient digits randomly above the bracket
CORRECT: Write each quotient digit directly above the corresponding dividend digit

❌ Mistake #3: Subtracting incorrectly

WRONG: 15 – 12 = 7 (calculation error)
CORRECT: Double-check each subtraction carefully

❌ Mistake #4: Dividing by the dividend instead of the divisor

WRONG: For 24 ÷ 6, dividing 6 by 24
CORRECT: The divisor (6) goes INTO the dividend (24)

❌ Mistake #5: Not checking the answer

WRONG: Assuming your answer is correct
CORRECT: Always multiply quotient × divisor (+ remainder) to verify

Practice Problems

Easy Problems (1-digit divisor)

Problem 1: 84 ÷ 4
Solution: 21
Check: 21 × 4 = 84 ✓

Problem 2: 96 ÷ 8
Solution: 12
Check: 12 × 8 = 96 ✓

Problem 3: 135 ÷ 5
Solution: 27
Check: 27 × 5 = 135 ✓

Problem 4: 248 ÷ 8
Solution: 31
Check: 31 × 8 = 248 ✓

Problem 5: 567 ÷ 9
Solution: 63
Check: 63 × 9 = 567 ✓

Medium Problems (with remainders)

Problem 6: 125 ÷ 4
Solution: 31 R 1
Check: (31 × 4) + 1 = 124 + 1 = 125 ✓

Problem 7: 287 ÷ 7
Solution: 41
Check: 41 × 7 = 287 ✓

Problem 8: 543 ÷ 6
Solution: 90 R 3
Check: (90 × 6) + 3 = 540 + 3 = 543 ✓

Problem 9: 729 ÷ 9
Solution: 81
Check: 81 × 9 = 729 ✓

Problem 10: 856 ÷ 8
Solution: 107
Check: 107 × 8 = 856 ✓

Hard Problems (2-digit divisors)

Problem 11: 624 ÷ 13
Solution: 48
Check: 48 × 13 = 624 ✓

Problem 12: 945 ÷ 15
Solution: 63
Check: 63 × 15 = 945 ✓

Problem 13: 1,147 ÷ 11
Solution: 104 R 3
Check: (104 × 11) + 3 = 1,144 + 3 = 1,147 ✓

Problem 14: 2,496 ÷ 24
Solution: 104
Check: 104 × 24 = 2,496 ✓

Problem 15: 3,675 ÷ 35
Solution: 105
Check: 105 × 35 = 3,675 ✓

Challenge Problems

Problem 16: 4,823 ÷ 17
Solution: 283 R 12
Check: (283 × 17) + 12 = 4,811 + 12 = 4,823 ✓

Problem 17: 7,854 ÷ 42
Solution: 187
Check: 187 × 42 = 7,854 ✓

Problem 18: 9,216 ÷ 64
Solution: 144
Check: 144 × 64 = 9,216 ✓

Problem 19: 12,675 ÷ 75
Solution: 169
Check: 169 × 75 = 12,675 ✓

Problem 20: 15,488 ÷ 88
Solution: 176
Check: 176 × 88 = 15,488 ✓

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Division with Decimals

Long division also works with decimals in the quotient (answer).

Example: 25 ÷ 4

       6.25
4 | 25.00
    24
    ---
    10
     8
    ---
    20
    20
    ---
     0

Steps:

1. 4 into 25 goes 6 times (4 × 6 = 24)
2. 25 – 24 = 1
3. Add a decimal point and zero: 1.0 becomes 10
4. 4 into 10 goes 2 times (place 2 after decimal)
5. 10 – 8 = 2
6. Bring down another zero: 20
7. 4 into 20 goes 5 times exactly

Answer: 25 ÷ 4 = 6.25

Where Is Long Division Used in Real Life?

1. Sharing Food Equally

Problem: You have 156 cookies to share equally among 12 friends. How many cookies does each person get?
Solution: 156 ÷ 12 = 13 cookies each

2. Calculating Speed

Problem: A car travels 480 km in 6 hours. What’s the average speed?
Solution: 480 ÷ 6 = 80 km/h

3. Budget Planning

Problem: You save $2,400 in a year. How much per month?
Solution: 2,400 ÷ 12 = $200 per month

4. Class Grouping

Problem: A teacher has 135 students to divide into 9 equal groups. How many students per group?
Solution: 135 ÷ 9 = 15 students per group

5. Recipe Scaling

Problem: A recipe for 8 people uses 960 ml of milk. How much per person?
Solution: 960 ÷ 8 = 120 ml per person

6. Construction Materials

Problem: A builder needs 2,856 bricks for 42 identical sections. How many bricks per section?
Solution: 2,856 ÷ 42 = 68 bricks per section

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Why Does Long Division Matter?

Mastering long division is essential because:

• Foundation for higher math: Needed for algebra, fractions, and calculus
• Quebec curriculum requirement: Tested in Primary 3-6 and Secondary 1-2
• Examens ministériels: Appears on provincial standardized tests
• Real-world problem solving: Used in budgeting, cooking, construction
• Mental math development: Improves number sense and estimation
• SAT/ACT preparation: Required for standardized tests
• Career applications: Engineering, science, finance, trades

Tips for Success

1. Master multiplication tables first

• Know your times tables 1-12 by heart
• This makes estimating quotients much faster

2. Keep work organized

• Line up place values carefully
• Write neatly so you don’t confuse digits
• Use graph paper if needed

3. Always check your answer

• Multiply quotient × divisor
• Add remainder if there is one
• Should equal the original dividend

4. Estimate before solving

• Round numbers to estimate the answer
• This helps you catch major errors
• Example: 487 ÷ 9 ≈ 500 ÷ 10 = 50

5. Practice regularly

• Do 5-10 problems daily
• Start easy, gradually increase difficulty
• Time yourself to build speed

Connection to Other Math Topics

Now that you understand long division, explore these related topics:

• Fractions – division is the same as fractions!
• Multiplication – the inverse operation of division
• Decimals – division often results in decimal answers
• Ratios & Proportions – uses division concepts

Summary

Key Takeaways:

• Long division breaks large division problems into manageable steps
• Four symbols: ÷, /, :, and the long division bracket ) ̅
• DMSBR method: Divide, Multiply, Subtract, Bring down, Repeat
• Always check your work by multiplying quotient × divisor (+ remainder)
• Master your multiplication tables for faster division
• Essential for Quebec math curriculum and real-world applications

Frequently Asked Questions (FAQ)