Multiplying a Fraction by a Whole Number
Multiplying a fraction by a whole number means finding a part of a whole number using repeated addition of fractions. For example, if you eat 1/4 of a chapati three times, you eat 3 × 1/4 = 3/4 of a chapati in total.
This concept builds directly on your understanding of fractions and multiplication from earlier classes. When you multiply 3 × 2/5, you are adding 2/5 three times: 2/5 + 2/5 + 2/5 = 6/5 = 1 1/5. But repeated addition takes too long for large numbers, so we use a simple rule instead.
In Class 5, you will learn the standard rule, practise simplifying results, handle mixed numbers, and solve real-life word problems involving fraction-whole number multiplication. This skill is needed for topics that come later: fraction of a number, percentage calculations, and ratio problems.
What is Multiplying a Fraction by a Whole Number - Class 5 Maths (Fractions)?
Multiplying a fraction by a whole number means adding that fraction to itself as many times as the whole number says.
For example, 4 × 2/5 means adding 2/5 four times:
2/5 + 2/5 + 2/5 + 2/5 = 8/5 = 1 3/5
Instead of adding repeatedly, we use this simple rule:
Whole Number × Numerator / Denominator
That is, a × (p/q) = (a × p) / q
The denominator stays the same. Only the numerator is multiplied by the whole number. This is because each copy of the fraction has the same-sized parts (same denominator), and you are simply counting how many parts you have in total.
Multiplying a Fraction by a Whole Number Formula
Whole Number × (Numerator / Denominator) = (Whole Number × Numerator) / Denominator
Steps to multiply a fraction by a whole number:
- Step 1: Write the whole number as it is and the fraction as numerator/denominator.
- Step 2: Multiply the whole number by the numerator of the fraction.
- Step 3: Keep the denominator the same.
- Step 4: Simplify the result if possible. Convert to a mixed number if the fraction is improper (numerator > denominator).
Shortcut (cancel before multiplying): If the whole number and the denominator share a common factor, divide both by that factor first. This keeps numbers small and avoids simplifying at the end.
Example of cancelling: 6 × 5/12. Here 6 and 12 share factor 6. Divide: 6 ÷ 6 = 1, 12 ÷ 6 = 2. So 6 × 5/12 = 1 × 5/2 = 5/2 = 2 1/2.
Types and Properties
Types of problems you will encounter:
- Proper fraction × whole number: The fraction is less than 1. Example: 5 × 3/7 = 15/7 = 2 1/7. The result is always greater than the original fraction but may be greater or less than the whole number.
- Unit fraction × whole number: The numerator of the fraction is 1. Example: 6 × 1/4 = 6/4 = 3/2 = 1 1/2. This is equivalent to dividing the whole number by the denominator.
- Mixed number × whole number: First convert the mixed number to an improper fraction, then multiply. Example: 3 × 2 1/5 = 3 × 11/5 = 33/5 = 6 3/5.
- Result is a whole number: When the whole number is a multiple of the denominator, the answer comes out to a whole number. Example: 15 × 2/5 = 30/5 = 6.
- Word problems: Finding a fraction of a quantity repeated several times, such as litres of water used daily over multiple days.
Solved Examples
Example 1: Example 1: Basic Multiplication
Problem: Calculate 3 × 2/5.
Solution:
Step 1: Multiply the whole number by the numerator: 3 × 2 = 6
Step 2: Keep the denominator the same: 5
Step 3: Result = 6/5. Since 6 > 5, this is an improper fraction. Convert: 6 ÷ 5 = 1 remainder 1, so 6/5 = 1 1/5.
Answer: 3 × 2/5 = 1 1/5
Example 2: Example 2: Unit Fraction
Problem: Calculate 8 × 1/4.
Solution:
Step 1: Multiply the numerator by the whole number: 8 × 1 = 8
Step 2: Denominator stays 4. Result = 8/4
Step 3: Simplify: 8 ÷ 4 = 2. The fraction reduces to a whole number.
Check: 8 × 1/4 means "one-quarter of 8", which is 2. Correct!
Answer: 8 × 1/4 = 2
Example 3: Example 3: Simplify Before Multiplying
Problem: Calculate 6 × 5/12.
Solution:
Step 1: Before multiplying, look for common factors. 6 and 12 share a factor of 6.
Step 2: Cancel: 6 ÷ 6 = 1, and 12 ÷ 6 = 2.
Step 3: Now multiply: 1 × 5/2 = 5/2
Step 4: Convert to mixed number: 5/2 = 2 1/2
Why cancel? Without cancelling, you get 6 × 5 = 30, then 30/12 = 5/2 = 2 1/2. Same answer, but larger numbers to work with.
Answer: 6 × 5/12 = 2 1/2
Example 4: Example 4: Larger Whole Number
Problem: Calculate 12 × 3/8.
Solution:
Step 1: Cancel common factors. GCD of 12 and 8 is 4. So 12 ÷ 4 = 3 and 8 ÷ 4 = 2.
Step 2: Multiply: 3 × 3/2 = 9/2
Step 3: Convert: 9 ÷ 2 = 4 remainder 1, so 9/2 = 4 1/2
Verification: Without cancelling: 12 × 3 = 36, then 36/8 = 9/2 = 4 1/2. Same answer.
Answer: 12 × 3/8 = 4 1/2
Example 5: Example 5: Result is a Whole Number
Problem: Calculate 15 × 2/5.
Solution:
Step 1: Cancel: 15 and 5 share factor 5. So 15 ÷ 5 = 3 and 5 ÷ 5 = 1.
Step 2: Multiply: 3 × 2/1 = 6/1 = 6
Why a whole number? Because 15 is a multiple of the denominator 5. Whenever the whole number is divisible by the denominator, the product will be a whole number.
Answer: 15 × 2/5 = 6
Example 6: Example 6: Mixed Number × Whole Number
Problem: Calculate 4 × 1 3/7.
Solution:
Step 1: Convert the mixed number to an improper fraction: 1 3/7 = (1 × 7 + 3)/7 = 10/7
Step 2: Multiply: 4 × 10/7 = 40/7
Step 3: Convert to mixed number: 40 ÷ 7 = 5 remainder 5, so 40/7 = 5 5/7
Alternative method: 4 × 1 3/7 = 4 × 1 + 4 × 3/7 = 4 + 12/7 = 4 + 1 5/7 = 5 5/7. This works too!
Answer: 4 × 1 3/7 = 5 5/7
Example 7: Example 7: Word Problem (Chapatis)
Problem: Aman eats 3/4 of a chapati at each meal. He eats 3 meals a day. How many chapatis does he eat in a day?
Solution:
Step 1: Identify: We need to find 3 × 3/4 (3 meals, each with 3/4 chapati).
Step 2: Multiply the numerator: 3 × 3 = 9. Denominator = 4. Result = 9/4.
Step 3: Convert to mixed number: 9 ÷ 4 = 2 remainder 1, so 9/4 = 2 1/4.
Interpretation: Aman eats 2 full chapatis and a quarter of another chapati each day.
Answer: Aman eats 2 1/4 chapatis in a day.
Example 8: Example 8: Word Problem (Ribbon)
Problem: Priya needs 5/6 metre of ribbon for one gift box. She has to wrap 6 gift boxes. How much ribbon does she need in total?
Solution:
Step 1: Total ribbon = 6 × 5/6
Step 2: Cancel: 6 and 6 share factor 6. So 6 ÷ 6 = 1 and 6 ÷ 6 = 1.
Step 3: Multiply: 1 × 5/1 = 5
Check: If each box needs 5/6 m and there are 6 boxes, that is exactly 5 metres. Makes sense!
Answer: Priya needs 5 metres of ribbon.
Example 9: Example 9: Word Problem (Pocket Money)
Problem: Dev saves 2/5 of his pocket money every week. His weekly pocket money is Rs.200. How much does he save in 4 weeks?
Solution:
Step 1: Savings per week = 2/5 of 200 = (2 × 200)/5 = 400/5 = Rs.80
Step 2: Savings in 4 weeks = 4 × 80 = Rs.320
Alternative one-step method: Total savings = 4 × (2/5 × 200) = 4 × 80 = 320. Or: (4 × 2/5) × 200 = 8/5 × 200 = 1600/5 = 320.
Answer: Dev saves Rs.320 in 4 weeks.
Example 10: Example 10: Word Problem (Distance)
Problem: Neha walks 7/10 km to reach school. She walks this distance 5 days a week (to school only, not back). What is the total distance she walks to school in a week?
Solution:
Step 1: Total distance = 5 × 7/10
Step 2: Multiply numerator: 5 × 7 = 35. Denominator = 10. Result = 35/10.
Step 3: Simplify: GCD of 35 and 10 is 5. So 35/10 = 7/2 = 3 1/2.
Verification: 5 × 0.7 = 3.5 km = 3 1/2 km. Correct!
Answer: Neha walks 3 1/2 km to school in a week.
Real-World Applications
Real-life uses of multiplying fractions by whole numbers:
- Cooking: If a recipe needs 3/4 cup of flour and you want to make 3 batches, you calculate 3 × 3/4 = 2 1/4 cups. This tells you exactly how much flour to measure.
- Shopping: Buying 5 packets of dal, each weighing 3/4 kg, gives 5 × 3/4 = 3 3/4 kg total. You can check if your bag can hold that much.
- Distance and travel: If an auto-rickshaw covers 7/8 km per trip and makes 4 trips, total = 4 × 7/8 = 3 1/2 km. This helps in planning fuel needs.
- Time management: If each class period is 3/4 hour and there are 6 periods, total = 6 × 3/4 = 4 1/2 hours of classes per day.
- Construction: If each tile needs 2/5 kg of adhesive and you have 20 tiles, total adhesive = 20 × 2/5 = 8 kg.
Key Points to Remember
- To multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator unchanged.
- Formula: a × (p/q) = (a × p) / q.
- Always simplify the answer. Convert improper fractions to mixed numbers.
- You can cancel common factors between the whole number and the denominator before multiplying to make calculation easier.
- For mixed numbers, first convert to an improper fraction, then multiply.
- Multiplying a fraction by a whole number is the same as repeated addition of that fraction.
- If the whole number is a multiple of the denominator, the answer will be a whole number.
- Always estimate to check your answer: 4 × 2/5 is roughly 4 × 0.4 = 1.6, which matches 1 3/5.
Practice Problems
- Calculate 7 × 3/8. Express your answer as a mixed number.
- Find the value of 9 × 2/3. Is the answer a whole number?
- Ria buys 4 packets of sugar, each weighing 5/6 kg. What is the total weight of sugar?
- Simplify: 10 × 7/15. Cancel common factors before multiplying.
- Arjun runs 3/5 km every morning. How far does he run in 6 days?
- Calculate 5 × 2 1/4. (Hint: Convert the mixed number to an improper fraction first.)
- A painter uses 2/7 litre of paint for one wall. How much paint does he need for 14 walls?
- Meera drinks 3/8 litre of milk every day. How much milk does she drink in a week (7 days)? Express as a mixed number.
Frequently Asked Questions
Q1. What is the rule for multiplying a fraction by a whole number?
Multiply the numerator of the fraction by the whole number, and keep the denominator the same. Then simplify the result. For example, 4 × 3/5 = 12/5 = 2 2/5.
Q2. Do we multiply both the numerator and denominator by the whole number?
No. Only the numerator is multiplied by the whole number. The denominator stays unchanged. Multiplying both numerator and denominator by the same number would give an equivalent fraction with the same value, not the product.
Q3. Can the answer be a whole number?
Yes. When the whole number is a multiple of the denominator, the result is a whole number. For example, 6 × 1/3 = 6/3 = 2. Another example: 10 × 3/5 = 30/5 = 6.
Q4. How do I multiply a mixed number by a whole number?
First convert the mixed number to an improper fraction. Then multiply the numerator by the whole number, keeping the denominator the same. For example, 3 × 1 2/5 = 3 × 7/5 = 21/5 = 4 1/5.
Q5. What does 'cancel before multiplying' mean?
It means dividing the whole number and the denominator by their greatest common factor before multiplying. This makes the numbers smaller and easier to work with. For example, in 6 × 5/12, divide 6 and 12 by 6 to get 1 × 5/2 = 5/2.
Q6. Is 3 × 2/5 the same as 2/5 + 2/5 + 2/5?
Yes. Multiplying a fraction by a whole number is the same as repeated addition. 3 × 2/5 = 2/5 + 2/5 + 2/5 = 6/5 = 1 1/5. The multiplication rule is just a faster way to get the same answer.
Q7. What is the difference between 'fraction of a number' and 'fraction times a whole number'?
They are the same operation. Finding 2/5 of 15 means calculating 2/5 × 15 = 6. The word 'of' in fractions means multiplication. So 'fraction of a number' and 'fraction times whole number' give identical results.
Q8. Why should I convert the answer to a mixed number?
An improper fraction like 11/4 is harder to visualise. Converting to 2 3/4 makes it easier to understand the quantity in real life. In exams, always express the answer in simplest form, usually as a mixed number.
Q9. Is this topic covered in the NCERT Class 5 syllabus?
Yes. Multiplying fractions by whole numbers is part of the NCERT/CBSE Class 5 Maths curriculum under the chapter on Fractions. It is a foundational skill tested in school exams and competitive assessments.
Related Topics
- Multiplying Fractions
- 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)
