Dividing Fractions
Dividing fractions means finding how many times one fraction fits into another. For example, 1/2 ÷ 1/4 asks: "How many quarter-pieces fit inside a half?" The answer is 2, because two quarters make a half.
The method for dividing fractions is surprisingly simple: invert the second fraction (the divisor) and multiply. This is called the "Keep-Change-Flip" rule or the "invert and multiply" method. It works because dividing by a number is the same as multiplying by its reciprocal.
In Class 5, you will learn to divide a fraction by a fraction, a fraction by a whole number, a whole number by a fraction, and solve real-life word problems. This skill connects to ratio, proportion, and percentage calculations in higher classes.
What is Dividing Fractions - Class 5 Maths (Fractions)?
Dividing fractions means multiplying the first fraction (dividend) by the reciprocal of the second fraction (divisor).
a/b ÷ c/d = a/b × d/c
The reciprocal of c/d is d/c (flip the fraction). After flipping, you simply multiply the numerators together and the denominators together, then simplify.
Why does this work? Dividing by a number is the same as multiplying by its reciprocal. For example, 6 ÷ 2 = 6 × 1/2 = 3. The same logic applies to fractions.
Dividing Fractions Formula
a/b ÷ c/d = a/b × d/c = (a × d) / (b × c)
The Keep-Change-Flip (KCF) method:
- Keep the first fraction exactly as it is.
- Change the division sign (÷) to a multiplication sign (×).
- Flip the second fraction (find its reciprocal).
- Multiply the numerators together and the denominators together.
- Simplify the result. Convert to a mixed number if needed.
Memory aid: "Keep-Change-Flip" — three words, three steps. Some students also say "invert and multiply" (invert the divisor, then multiply).
For whole numbers: Write the whole number as a fraction over 1 before applying KCF. For example, 5 = 5/1.
Types and Properties
Types of fraction division problems:
- Fraction ÷ Fraction: The most common type. Example: 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2.
- Fraction ÷ Whole number: The fraction gets smaller. Example: 2/3 ÷ 4 = 2/3 × 1/4 = 2/12 = 1/6.
- Whole number ÷ Fraction: The result is larger than the whole number. Example: 5 ÷ 1/3 = 5 × 3 = 15.
- Mixed number ÷ Fraction: Convert the mixed number to an improper fraction first, then use KCF. Example: 2 1/2 ÷ 1/4 = 5/2 × 4/1 = 20/2 = 10.
- Fraction ÷ Fraction (same fractions): Any number divided by itself = 1. Example: 3/7 ÷ 3/7 = 1.
Solved Examples
Example 1: Example 1: Fraction ÷ Fraction (Basic)
Problem: Calculate 3/5 ÷ 2/7.
Solution:
Step 1 (Keep): Keep 3/5 as it is.
Step 2 (Change): Change ÷ to ×.
Step 3 (Flip): Flip 2/7 to get 7/2.
Step 4 (Multiply): 3/5 × 7/2 = (3 × 7) / (5 × 2) = 21/10
Step 5: Convert: 21/10 = 2 1/10
Answer: 3/5 ÷ 2/7 = 2 1/10
Example 2: Example 2: Fraction ÷ Fraction with Cancellation
Problem: Calculate 4/9 ÷ 2/3.
Solution:
Step 1: KCF: 4/9 × 3/2
Step 2: Before multiplying, cancel common factors. 4 and 2 share factor 2: 4 ÷ 2 = 2, 2 ÷ 2 = 1. Also 3 and 9 share factor 3: 3 ÷ 3 = 1, 9 ÷ 3 = 3.
Step 3: After cancelling: 2/3 × 1/1 = 2/3
Check: Without cancelling: 4/9 × 3/2 = 12/18 = 2/3. Same answer, but cancelling avoided larger numbers.
Answer: 4/9 ÷ 2/3 = 2/3
Example 3: Example 3: Whole Number ÷ Fraction
Problem: Calculate 6 ÷ 2/5.
Solution:
Step 1: Write 6 as 6/1. Apply KCF: 6/1 × 5/2
Step 2: Cancel: 6 and 2 share factor 2. So 6 ÷ 2 = 3 and 2 ÷ 2 = 1.
Step 3: 3/1 × 5/1 = 15
Meaning: There are 15 two-fifths in 6. Think of it as: if each piece is 2/5, you need 15 pieces to make 6 wholes.
Answer: 6 ÷ 2/5 = 15
Example 4: Example 4: Fraction ÷ Whole Number
Problem: Calculate 5/8 ÷ 5.
Solution:
Step 1: Write 5 as 5/1. Its reciprocal is 1/5.
Step 2: KCF: 5/8 × 1/5
Step 3: Cancel: 5 and 5 share factor 5. So 5 ÷ 5 = 1 and 5 ÷ 5 = 1.
Step 4: 1/8 × 1/1 = 1/8
Meaning: Dividing 5/8 by 5 gives each portion as 1/8 — makes sense since 5 × 1/8 = 5/8.
Answer: 5/8 ÷ 5 = 1/8
Example 5: Example 5: Unit Fraction Division
Problem: Calculate 1/2 ÷ 1/6.
Solution:
Step 1: KCF: 1/2 × 6/1
Step 2: = 6/2 = 3
Meaning: How many one-sixths fit in one-half? Since 1/6 + 1/6 + 1/6 = 3/6 = 1/2, exactly 3 sixths fit.
Visual check: Imagine a number line from 0 to 1/2. Mark every 1/6: 0, 1/6, 2/6 (=1/3), 3/6 (=1/2). You count 3 marks.
Answer: 1/2 ÷ 1/6 = 3
Example 6: Example 6: Mixed Number ÷ Fraction
Problem: Calculate 2 1/3 ÷ 1/2.
Solution:
Step 1: Convert 2 1/3 to improper fraction: 2 1/3 = (2 × 3 + 1)/3 = 7/3
Step 2: KCF: 7/3 × 2/1 = 14/3
Step 3: Convert: 14/3 = 4 2/3
Meaning: There are 4 2/3 halves in 2 1/3.
Answer: 2 1/3 ÷ 1/2 = 4 2/3
Example 7: Example 7: Word Problem (Ribbon)
Problem: Priya has 3/4 metre of ribbon. She cuts it into pieces of 1/8 metre each. How many pieces can she cut?
Solution:
Step 1: Number of pieces = 3/4 ÷ 1/8
Step 2: KCF: 3/4 × 8/1 = 24/4 = 6
Check: 6 pieces of 1/8 m each = 6/8 = 3/4 m. Correct!
Answer: Priya can cut 6 pieces.
Example 8: Example 8: Word Problem (Milk Distribution)
Problem: A milk booth has 4/5 litre of milk. Each glass holds 2/15 litre. How many glasses can be filled?
Solution:
Step 1: Number of glasses = 4/5 ÷ 2/15
Step 2: KCF: 4/5 × 15/2
Step 3: Cancel: 4 and 2 share factor 2 (gives 2/1). 15 and 5 share factor 5 (gives 3/1).
Step 4: 2/1 × 3/1 = 6
Answer: 6 glasses can be filled.
Example 9: Example 9: Word Problem (Rope Cutting)
Problem: Arjun has a rope 5 metres long. He cuts it into pieces, each 5/6 metre long. How many pieces does he get?
Solution:
Step 1: Number of pieces = 5 ÷ 5/6
Step 2: Write 5 as 5/1. KCF: 5/1 × 6/5
Step 3: Cancel: 5 and 5 cancel out. Result = 1/1 × 6/1 = 6
Check: 6 pieces × 5/6 m each = 30/6 = 5 m. Correct!
Answer: Arjun gets 6 pieces.
Example 10: Example 10: Word Problem (Ladoo Making)
Problem: Meera has 7/8 kg of ladoo mixture. She makes ladoos, each using 1/16 kg. How many ladoos can she make?
Solution:
Step 1: Number of ladoos = 7/8 ÷ 1/16
Step 2: KCF: 7/8 × 16/1
Step 3: Cancel: 8 and 16 share factor 8. 16 ÷ 8 = 2, 8 ÷ 8 = 1.
Step 4: 7/1 × 2/1 = 14
Check: 14 ladoos × 1/16 kg each = 14/16 = 7/8 kg. Correct!
Answer: Meera can make 14 ladoos.
Real-World Applications
Real-life uses of dividing fractions:
- Cooking: Dividing 3/4 cup of batter equally into 3 moulds: 3/4 ÷ 3 = 1/4 cup in each mould.
- Sewing and craft: How many 1/3 m pieces can be cut from 2 m of cloth? 2 ÷ 1/3 = 6 pieces.
- Sharing food: Dividing half a pizza among 4 friends: 1/2 ÷ 4 = 1/8 of the pizza each.
- Packing and packaging: Filling 3/5 kg bags from a 6 kg sack: 6 ÷ 3/5 = 10 bags.
- Time and scheduling: If a task takes 1/4 hour, how many tasks in 3 hours? 3 ÷ 1/4 = 12 tasks.
- Construction: How many tiles of 1/6 m width fit in a 4 m wall? 4 ÷ 1/6 = 24 tiles.
Key Points to Remember
- To divide fractions: Keep-Change-Flip (Keep the first fraction, Change ÷ to ×, Flip the second fraction).
- Formula: a/b ÷ c/d = a/b × d/c = (a × d)/(b × c).
- For whole numbers: write the whole number as a fraction over 1 before applying KCF.
- For mixed numbers: convert to an improper fraction first, then apply KCF.
- Cancel common factors before multiplying to keep numbers small.
- Always simplify the final answer and convert improper fractions to mixed numbers.
- Dividing by a proper fraction (less than 1) gives a result larger than the original.
- Dividing by a number greater than 1 gives a result smaller than the original.
- Any fraction divided by itself equals 1.
Practice Problems
- Calculate 5/6 ÷ 1/3.
- Calculate 7/10 ÷ 7/5. Simplify your answer.
- Calculate 8 ÷ 2/3.
- Calculate 3/7 ÷ 6.
- Ria has 2/3 litre of juice. She pours it into cups of 1/9 litre each. How many cups can she fill?
- Calculate 1 3/4 ÷ 1/2. (Convert the mixed number first.)
- Dev has a 5/6 m wooden plank. He needs pieces of 5/12 m each. How many pieces can he cut?
- Neha divides 4/5 kg of flour equally into 8 containers. How much flour is in each container?
Frequently Asked Questions
Q1. What is the rule for dividing fractions?
Multiply the first fraction by the reciprocal (flip) of the second fraction. This is called Keep-Change-Flip. For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8.
Q2. Why do we flip the second fraction when dividing?
Division asks 'how many times does the divisor fit into the dividend?' Flipping and multiplying gives the same answer mathematically. Dividing by a fraction is equivalent to multiplying by its reciprocal — this is a proven mathematical fact.
Q3. What happens when you divide a fraction by itself?
The answer is always 1. For example, 3/7 ÷ 3/7 = 3/7 × 7/3 = 21/21 = 1. Any number divided by itself equals 1 (except 0).
Q4. Can the answer be greater than both fractions?
Yes. When you divide by a proper fraction (less than 1), the answer is larger than the dividend. For example, 1/2 ÷ 1/4 = 2, which is greater than both 1/2 and 1/4.
Q5. How do I divide a whole number by a fraction?
Write the whole number over 1, then flip the fraction and multiply. For example, 6 ÷ 3/4 = 6/1 × 4/3 = 24/3 = 8.
Q6. How do I divide a fraction by a whole number?
Write the whole number as a fraction (over 1), flip it to get 1 over the number, then multiply. For example, 2/3 ÷ 4 = 2/3 × 1/4 = 2/12 = 1/6.
Q7. What is the difference between dividing and multiplying fractions?
In multiplication, you multiply numerators and denominators directly. In division, you first flip the divisor, then multiply. Division is multiplication by the reciprocal.
Q8. Can I divide by zero in fractions?
No. Division by zero is undefined in mathematics. 0/0 and any fraction divided by 0 have no answer. Zero has no reciprocal.
Q9. Is this topic in the NCERT Class 5 syllabus?
Yes. Dividing fractions is part of the NCERT/CBSE Class 5 Maths curriculum under the Fractions chapter. Students learn the invert-and-multiply method and apply it to word problems.
Related Topics
- Reciprocal of a Fraction
- Multiplying Fractions
- Fractions Revision (Grade 5)
- Adding Unlike Fractions
- Subtracting Unlike Fractions
- Adding Mixed Numbers
- Subtracting Mixed Numbers
- Multiplying a Fraction by a Whole Number
- Fraction of a Number
- Fraction Word Problems (Grade 5)
- Proper, Improper and Mixed Fractions
- Comparing and Ordering Fractions (Grade 5)
