Division

What is Division?  

Division is a basic arithmetic operation that means splitting or sharing something into equal parts or groups. It shows how many times one number fits into another or how much each part gets when something is divided equally. It is the opposite of multiplication.  

Definition:  

Division tells us how many groups we can make or how many items go into each group when we split something evenly.  

Example:  

12 ÷ 3 = 4  

This means:  

If we divide 12 objects into 3 equal groups, each group will have 4 objects.  

You can also read it as:  

“How many times does 3 go into 12?” → Answer: 4 times  

Key Terms:  

Dividend: The number to be divided.  

Divisor: The number that divides the dividend.  

Quotient: The result of division.  

Remainder: What is left over (if any).  

 

Table of Content

• Properties of Division
• Methods of Division
• Word Problems on Division
• Division of Fractions and Decimals
• Division Tricks and Mental Math
• Practice & Assessment
• Conclusion
• FAQs On Division

 

Properties of Division

Division by 1  

When any number is divided by 1, the result is the number itself.  

Example:  

7 ÷ 1 = 7  

100 ÷ 1 = 100  

 

Division by the Number Itself  

If you divide any number by itself, the answer is always 1.  

Example:  

9 ÷ 9 = 1  

45 ÷ 45 = 1  

Exception: 0 ÷ 0 is undefined and has no clear answer.  

 

Zero in Division  

Zero divided by a number (0 ÷ a): The result is 0.  

Example:  

0 ÷ 5 = 0  

0 ÷ 100 = 0  

 

A number divided by zero (a ÷ 0): 

This is undefined and not allowed in mathematics.  

Example:  

5 ÷ 0 = undefined  

100 ÷ 0 = undefined  

 

Division is Not Commutative  

In division, changing the order of numbers gives a different answer.  

Example:  

12 ÷ 4 = 3  

4 ÷ 12 = 0.33  

So, a ÷ b ≠ b ÷ a  

 

Division is Not Associative  

Changing the grouping of numbers also changes the result.  

Example:  

(16 ÷ 4) ÷ 2 = 4 ÷ 2 = 2  

16 ÷ (4 ÷ 2) = 16 ÷ 2 = 8  

So, (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)  

 

Methods of Division

Objective: Practice different techniques for division.  

Short Division (mental math method):  

Best for smaller numbers. Example: 36 ÷ 6.  

 

Long Division (standard algorithm):  

Useful for larger numbers. Involves step-by-step breakdown:  

Divide, Multiply, Subtract, Bring down, Repeat.  

 

Division Using Multiplication Facts:  

Helps reinforce times tables and speeds up division.  

 

Repeated Subtraction Method:  

Subtract the divisor repeatedly from the dividend to find the quotient.  

 

Division with Remainders  

Objective: Learn how to handle non-exact division.  

 

Exact Division:  

When remainder = 0.  

Example: 20 ÷ 4 = 5.  

 

When the number cannot be divided exactly:

Example: 17 ÷ 3 = 5 remainder 2.  

 

Write as:  

Dividend = Divisor × Quotient + Remainder.  

 

Word Problems on Division

1. Sharing Chocolates  

Problem: Riya has 24 chocolates. She wants to share them equally among 6 friends. How many chocolates will each friend get?  

Solution:  

Total chocolates = 24  

Number of friends = 6  

Chocolates per friend = 24 ÷ 6 = 4  

Answer: Each friend will get 4 chocolates.  

 

2. Distributing Apples  

Problem: A fruit vendor has 96 apples. He wants to pack them equally into 12 boxes. How many apples will go in each box?  

Solution:  

Total apples = 96  

Number of boxes = 12  

Apples per box = 96 ÷ 12 = 8  

Answer: Each box will have 8 apples.  

 

3. Books on Shelves  

Problem: A library has 180 books. If 15 books are placed on each shelf, how many shelves are needed?  

Solution:  

Total books = 180  

Books per shelf = 15  

Number of shelves = 180 ÷ 15 = 12  

Answer: 12 shelves are needed.  

 

Division of Fractions and Decimals

Objective: Transition from whole numbers to fractional division.  

Reciprocal Rule:  

ab ÷ cd = ab × dc.  

Division of Decimals:  

Adjust the decimal point in the dividend and divisor.  

Practice with currency, weight, and length units.  

 

Division Tricks and Mental Math

Objective: Improve speed and accuracy.  

Divisibility rules for 2, 3, 4, 5, 6, 9, and 10.  

(e.g., a number is divisible by 3 if the sum of its digits is divisible by 3.)  

Divide by 5 using a shortcut: Multiply by 2 and move the decimal point one place left.  

(e.g., 260 ÷ 5 = (260 × 2) ÷ 10 = 520 ÷ 10 = 52).  

 

Practice & Assessment

A. Short Division Problems  

• 12 ÷ 3 =  

• 20 ÷ 5 =  

• 45 ÷ 9 =  

• 18 ÷ 6 =  

• 35 ÷ 7 =  

• 64 ÷ 8 =  

• 90 ÷ 10 =  

• 81 ÷ 9 =  

• 56 ÷ 7 =  

• 72 ÷ 8 =  

 

B. Long Division Word Problems  

• A class of 36 students needs to be divided into equal groups of 6. How many groups will there be?  

• 120 pencils are to be packed equally into 10 boxes. How many pencils go in each box?  

• A book has 240 pages. If you read 30 pages each day, how many days will it take to finish?  

• A farmer has 96 apples and wants to place them equally in 12 baskets. How many apples per basket?  

• A shopkeeper has ₹1,350 and wants to distribute it equally among 9 workers. How much will each get?  

• A school has 180 chairs and places 15 chairs in each row. How many rows are formed?  

• 250 stickers are shared equally among 5 friends. How many stickers per friend?  

• A library has 432 books and places 36 books on each shelf. How many shelves are needed?  

• A car travels 540 km in 6 hours. How many kilometers does it travel each hour?  

• There are 640 students divided into 8 houses. How many students per house?  

 

Conclusion

By the end of this self-study syllabus, learners will have a strong understanding of division and its real-life applications. They will be able to perform division operations with and without remainders, confidently solve word problems involving division, and use the long division method for larger numbers. Additionally, students will handle division involving decimals and fractions while developing mental math strategies to divide efficiently and accurately in various contexts.  

 

Related Links

Square Root Long Division Method: Master the square root long division method with step-by-step examples and practice questions.

 

Frequently Asked Questions On Division

1. What is division?

Answer: Division is a basic arithmetic operation used to split a number into equal parts. It shows how many times one number is contained in another. For example, 12 ÷ 3 = 4 means 12 is divided into 3 equal parts, each part being 4.

 

2. What is division in maths?

Answer: In mathematics, division is the process of finding out how many times one number fits into another. It is the opposite of multiplication. For example, in 20 ÷ 5 = 4, 20 is the dividend, 5 is the divisor, and 4 is the quotient.

 

3. What is division in district?

Answer: In administrative terms, a division is a region that consists of a group of districts. It is used for governance and is usually overseen by a Divisional Commissioner. For example, a state may be divided into several divisions, each managing multiple districts.

 

4. What is the division of 4?

Answer: This depends on the divisor.
Examples:
4 ÷ 1 = 4
4 ÷ 2 = 2
4 ÷ 4 = 1
4 ÷ 0 = undefined (division by zero is not allowed)

 

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