Multiplying Fractions
Multiplying fractions is a fundamental operation introduced in Class 5. Unlike addition and subtraction of fractions, multiplication does not require a common denominator. The rule is straightforward: multiply the numerators together and multiply the denominators together.
This topic covers multiplication of a fraction by a fraction, a fraction by a whole number, and mixed numbers. It builds directly into concepts like "fraction of a number" and is essential for Class 6 and beyond.
Think of fraction multiplication as finding a part of a part. If you have half a pizza and you eat one-third of that half, you have eaten 1/2 × 1/3 = 1/6 of the whole pizza. The word "of" in maths almost always means multiplication when it appears with fractions.
Students should notice an important pattern: when you multiply a number by a proper fraction (less than 1), the result is smaller than the original number. This is different from multiplying by a whole number, where the result always gets larger. Understanding this helps students check if their answers make sense.
What is Multiplying Fractions - Class 5 Maths (Fractions)?
Multiplication of fractions means finding a part of a part. For example, 1/2 × 1/3 means "half of one-third" — which is 1/6.
a/b × c/d = (a × c) / (b × d)
Multiply numerators. Multiply denominators.
Key rules:
- No need for a common denominator.
- Always simplify the answer to lowest terms.
- You can cancel common factors before multiplying to make computation easier (cross-cancellation).
- When multiplying a fraction by a whole number, write the whole number as a fraction over 1.
Multiplying Fractions Formula
Fraction × Fraction: a/b × c/d = (a×c) / (b×d)
Fraction × Whole Number: a/b × n = (a×n) / b
Mixed × Mixed: Convert to improper fractions first, then multiply.
Types and Properties
Type 1: Fraction × Fraction
Multiply numerators, multiply denominators. Example: 2/3 × 4/5 = 8/15.
Type 2: Fraction × Whole Number
Write the whole number as a fraction with denominator 1. Example: 3/4 × 6 = 3/4 × 6/1 = 18/4 = 9/2 = 4 1/2.
Type 3: Mixed Number × Fraction
Convert the mixed number to an improper fraction, then multiply. Example: 2 1/3 × 3/5 = 7/3 × 3/5 = 21/15 = 7/5 = 1 2/5.
Type 4: Mixed Number × Mixed Number
Convert both to improper fractions, then multiply. Example: 1 1/2 × 2 1/3 = 3/2 × 7/3 = 21/6 = 7/2 = 3 1/2.
Cross-cancellation (shortcut):
Before multiplying, cancel any common factor between a numerator and the opposite denominator. This keeps numbers small.
Solved Examples
Example 1: Example 1: Fraction × Fraction
Problem: Calculate 2/5 × 3/7.
Solution:
Step 1: Multiply numerators: 2 × 3 = 6.
Step 2: Multiply denominators: 5 × 7 = 35.
Step 3: Result = 6/35. HCF(6, 35) = 1. Already in simplest form.
Answer: 2/5 × 3/7 = 6/35
Example 2: Example 2: Fraction × Whole Number
Problem: Calculate 3/4 × 8.
Solution:
Step 1: Write 8 as 8/1.
Step 2: Multiply: 3/4 × 8/1 = (3 × 8)/(4 × 1) = 24/4.
Step 3: Simplify: 24/4 = 6.
Answer: 3/4 × 8 = 6
Example 3: Example 3: With Cross-Cancellation
Problem: Calculate 4/9 × 3/8.
Solution:
Step 1: Look for common factors across the multiplication sign.
4 and 8 share factor 4: cancel to get 1 and 2.
3 and 9 share factor 3: cancel to get 1 and 3.
Step 2: Simplified multiplication: 1/3 × 1/2 = 1/6.
Answer: 4/9 × 3/8 = 1/6
Example 4: Example 4: Mixed Number × Fraction
Problem: Calculate 2 1/3 × 3/7.
Solution:
Step 1: Convert mixed number: 2 1/3 = (2 × 3 + 1)/3 = 7/3.
Step 2: Multiply: 7/3 × 3/7.
Step 3: Cross-cancel: 7 with 7 = 1, and 3 with 3 = 1.
Step 4: Result = 1/1 × 1/1 = 1.
Answer: 2 1/3 × 3/7 = 1
Example 5: Example 5: Mixed × Mixed
Problem: Calculate 1 1/2 × 2 2/5.
Solution:
Step 1: Convert: 1 1/2 = 3/2 and 2 2/5 = 12/5.
Step 2: Multiply: 3/2 × 12/5 = (3 × 12)/(2 × 5) = 36/10.
Step 3: Simplify: 36/10 = 18/5 = 3 3/5.
Answer: 1 1/2 × 2 2/5 = 3 3/5
Example 6: Example 6: Word Problem — Finding a Fraction of a Number
Problem: Ria has ₹360. She spends 2/3 of it. How much does she spend?
Solution:
Step 1: 2/3 of 360 = 2/3 × 360.
Step 2: 360 ÷ 3 = 120.
Step 3: 120 × 2 = 240.
Answer: Ria spends ₹240.
Example 7: Example 7: Word Problem — Area
Problem: A rectangular garden is 3/4 m long and 2/5 m wide. Find its area.
Solution:
Step 1: Area = length × breadth = 3/4 × 2/5.
Step 2: Multiply: (3 × 2)/(4 × 5) = 6/20.
Step 3: Simplify: 6/20 = 3/10.
Answer: Area = 3/10 sq m
Example 8: Example 8: Word Problem — Fraction of a Fraction
Problem: Meera had 3/4 of a cake. She gave 1/3 of what she had to her friend. What fraction of the whole cake did her friend get?
Solution:
Step 1: 1/3 of 3/4 = 1/3 × 3/4.
Step 2: Cross-cancel: 3 and 3 cancel to 1.
Step 3: Result = 1/1 × 1/4 = 1/4.
Answer: Her friend got 1/4 of the whole cake.
Example 9: Example 9: Multiplying Unit Fractions
Problem: Calculate 1/4 × 1/5 × 1/2.
Solution:
Step 1: Multiply all numerators: 1 × 1 × 1 = 1.
Step 2: Multiply all denominators: 4 × 5 × 2 = 40.
Step 3: Result = 1/40.
Answer: 1/4 × 1/5 × 1/2 = 1/40
Example 10: Example 10: Word Problem — Weight
Problem: A bag of rice weighs 5 1/4 kg. Dev buys 2/3 of a bag. What is the weight Dev bought?
Solution:
Step 1: Convert: 5 1/4 = 21/4.
Step 2: Multiply: 2/3 × 21/4.
Step 3: Cross-cancel: 2 and 4 share factor 2: 2/4 = 1/2. Also 3 and 21 share factor 3: 21/3 = 7.
Step 4: Result = 1/1 × 7/2 = 7/2 = 3 1/2.
Answer: Dev bought 3 1/2 kg of rice.
Real-World Applications
Where multiplying fractions is used:
- Finding a part of a quantity: "2/3 of 60 students" = 2/3 × 60 = 40 students
- Area calculations: Length × breadth when dimensions are fractions
- Cooking: Scaling recipes — "make 3/4 of the recipe" means multiply each ingredient by 3/4
- Speed and time: Distance = speed × time when given in fractional hours
- Discounts: "1/5 off" means multiply price by 1/5 to find the discount
Key Points to Remember
- Multiply numerators. Multiply denominators. No common denominator needed.
- To multiply a fraction by a whole number, write the whole number as a fraction over 1.
- For mixed numbers, always convert to improper fractions first.
- Use cross-cancellation to simplify before multiplying — it keeps numbers small.
- Multiplying a fraction by a proper fraction gives a smaller result (a part of a part is smaller).
- Multiplying a fraction by a whole number greater than 1 gives a larger result.
- Always simplify the final answer to lowest terms.
- "a/b of n" means a/b × n — the word "of" means multiply.
Practice Problems
- Calculate 3/5 × 5/9.
- Find the product: 4/7 × 14.
- Calculate 2 1/4 × 2/3.
- Arjun had ₹480. He saved 5/8 of it. How much did he save?
- A plot of land is 3 1/2 m long and 2/5 m wide. Find its area.
- Multiply 1 2/3 × 2 1/5. Express the answer as a mixed number.
- Aditi has 4/5 of a ribbon. She uses 1/2 of what she has. What fraction of the full ribbon did she use?
- Calculate 5/6 × 3/10 × 4. Simplify fully.
Frequently Asked Questions
Q1. Do I need a common denominator to multiply fractions?
No. Unlike addition and subtraction, multiplication does not require a common denominator. Simply multiply numerators together and denominators together.
Q2. What does 'of' mean in fraction problems?
The word 'of' means multiply. For example, '2/3 of 45' means 2/3 × 45 = 30.
Q3. What is cross-cancellation?
Before multiplying, if a numerator and the opposite denominator share a common factor, divide both by that factor. This simplifies the multiplication and avoids large numbers. For example, in 4/9 × 3/8, cancel 4 with 8 (both divide by 4) and 3 with 9 (both divide by 3) to get 1/3 × 1/2 = 1/6.
Q4. Why does multiplying by a proper fraction make the number smaller?
A proper fraction is less than 1. Taking a part of something (like 2/3 of 12) always gives a result smaller than the original. 2/3 × 12 = 8, which is less than 12.
Q5. How do I multiply mixed numbers?
Convert each mixed number to an improper fraction first, then multiply. For example, 2 1/3 × 1 1/2 = 7/3 × 3/2 = 21/6 = 7/2 = 3 1/2.
Q6. Can I multiply more than two fractions at once?
Yes. Multiply all the numerators together and all the denominators together. Use cross-cancellation to simplify first. For example, 1/2 × 2/3 × 3/4 = 6/24 = 1/4.
Q7. What is the product of a number and its reciprocal?
Always 1. The reciprocal of a/b is b/a, and a/b × b/a = ab/ab = 1. For example, 3/5 × 5/3 = 15/15 = 1.
Q8. Is the product of two fractions always a fraction?
Yes, but it might simplify to a whole number. For example, 2/3 × 3/2 = 6/6 = 1, which is a whole number.
Q9. Is multiplying fractions covered in NCERT Class 5?
Yes. Multiplying fractions — including fraction by fraction, fraction by whole number, and mixed number multiplication — is a key topic in the Fractions chapter of NCERT Class 5 Mathematics.
Related Topics
- Multiplying a Fraction by a Whole Number
- Fraction of a Number
- Fractions Revision (Grade 5)
- Adding Unlike Fractions
- Subtracting Unlike Fractions
- Adding Mixed Numbers
- Subtracting Mixed Numbers
- Reciprocal of a Fraction
- Dividing Fractions
- Fraction Word Problems (Grade 5)
- Proper, Improper and Mixed Fractions
- Comparing and Ordering Fractions (Grade 5)
