Dividing a Whole Number by a Fraction
Dividing a whole number by a fraction answers the question: how many groups of the fraction fit into the whole number? For example, 6 ÷ 1/2 asks "how many halves are in 6?" The answer is 12, because each whole has 2 halves, and 6 wholes have 12 halves.
In Class 5, students learn the multiply by the reciprocal method: to divide by a fraction, multiply by its reciprocal (flip the fraction). This elegant rule makes division of fractions straightforward.
What is Dividing a Whole Number by a Fraction - Class 5 Maths (Fractions)?
To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction.
Reciprocal: The reciprocal of a/b is b/a (flip the numerator and denominator).
Rule:
Whole ÷ (a/b) = Whole × (b/a)
Why this works: Dividing by a fraction smaller than 1 gives a result larger than the original number. If you divide 6 by 1/2, you get 12 — there are 12 half-pieces in 6 wholes.
Dividing a Whole Number by a Fraction Formula
n ÷ (a/b) = n × (b/a) = (n × b) / a
Solved Examples
Example 1: Example 1: Whole number ÷ unit fraction
Problem: Calculate 6 ÷ 1/2.
Solution:
Step 1: Reciprocal of 1/2 = 2/1 = 2
Step 2: 6 × 2 = 12
Answer: 6 ÷ 1/2 = 12
Meaning: There are 12 halves in 6 wholes.
Example 2: Example 2: Whole number ÷ proper fraction
Problem: Calculate 8 ÷ 2/3.
Solution:
Step 1: Reciprocal of 2/3 = 3/2
Step 2: 8 × 3/2 = 24/2 = 12
Answer: 8 ÷ 2/3 = 12
Example 3: Example 3: Result is a mixed number
Problem: Calculate 5 ÷ 3/4.
Solution:
Step 1: Reciprocal of 3/4 = 4/3
Step 2: 5 × 4/3 = 20/3
Step 3: Convert: 20/3 = 6 2/3
Answer: 5 ÷ 3/4 = 6 2/3
Example 4: Example 4: Dividing by 1/4
Problem: Calculate 10 ÷ 1/4.
Solution:
Reciprocal of 1/4 = 4
10 × 4 = 40
Answer: 10 ÷ 1/4 = 40
Meaning: There are 40 quarter-pieces in 10 wholes.
Example 5: Example 5: Word problem — Ribbon cutting
Problem: Aditi has 12 metres of ribbon. She cuts pieces of 3/4 metre each. How many pieces can she cut?
Solution:
Number of pieces = 12 ÷ 3/4 = 12 × 4/3 = 48/3 = 16
Answer: Aditi can cut 16 pieces.
Example 6: Example 6: Word problem — Chapati making
Problem: Meera uses 2/5 kg of flour for each batch of chapatis. She has 4 kg of flour. How many batches can she make?
Solution:
Batches = 4 ÷ 2/5 = 4 × 5/2 = 20/2 = 10
Answer: Meera can make 10 batches.
Example 7: Example 7: Dividing by a fraction greater than 1/2
Problem: Calculate 9 ÷ 5/6.
Solution:
Reciprocal of 5/6 = 6/5
9 × 6/5 = 54/5 = 10 4/5
Answer: 9 ÷ 5/6 = 10 4/5
Example 8: Example 8: Verifying the answer
Problem: Calculate 7 ÷ 1/3 and verify.
Solution:
7 ÷ 1/3 = 7 × 3 = 21
Verify: 21 × 1/3 = 21/3 = 7 ✓
Answer: 21
Example 9: Example 9: Word problem — Painting
Problem: Dev can paint 1/6 of a wall in one hour. How many hours does he need to paint 3 walls?
Solution:
Hours needed = 3 ÷ 1/6 = 3 × 6 = 18
Answer: Dev needs 18 hours to paint 3 walls.
Key Points to Remember
- To divide by a fraction, multiply by its reciprocal (flip the fraction).
- The reciprocal of a/b is b/a. The reciprocal of a whole number n is 1/n.
- Dividing a whole number by a fraction less than 1 always gives a result larger than the original number.
- Dividing a whole number by a fraction equal to 1 gives the same number.
- To verify: multiply the answer by the original fraction and check if you get back the whole number.
- This concept answers: "How many groups of this fraction fit into the whole number?"
Practice Problems
- Calculate 4 ÷ 1/5.
- Calculate 15 ÷ 3/4.
- Calculate 7 ÷ 2/7.
- Ria has 6 litres of milk. She fills glasses with 3/4 litre each. How many glasses can she fill?
- Calculate 20 ÷ 5/8.
- A rope is 9 metres long. How many pieces of 3/8 m can be cut from it?
- Calculate 3 ÷ 4/5 and express as a mixed fraction.
- Arjun walks 1/3 km in one round. How many rounds does he need to walk 5 km?
Frequently Asked Questions
Q1. What does dividing a whole number by a fraction mean?
It asks: how many times does the fraction fit into the whole number? For example, 6 ÷ 1/3 means 'how many thirds are in 6?' Since each whole has 3 thirds, 6 wholes have 18 thirds.
Q2. Why does dividing by a fraction give a bigger answer?
Dividing by a number less than 1 always gives a result larger than the original. Think of it as: smaller pieces fit more times into the same amount. If each piece is 1/2, you need twice as many pieces.
Q3. What is a reciprocal?
The reciprocal of a fraction a/b is b/a — you flip the numerator and denominator. The reciprocal of 3/4 is 4/3. The reciprocal of 5 (or 5/1) is 1/5. A number times its reciprocal always equals 1.
Q4. How do I verify my answer?
Multiply your answer by the original fraction. The result should equal the whole number you started with. For example, if 8 ÷ 2/3 = 12, verify: 12 × 2/3 = 24/3 = 8. Correct.
Q5. Can I divide a whole number by a mixed number?
Yes. Convert the mixed number to an improper fraction first, then multiply by its reciprocal. Example: 6 ÷ 1 1/2 = 6 ÷ 3/2 = 6 × 2/3 = 12/3 = 4.
Q6. What is 0 divided by a fraction?
Zero divided by any fraction is always 0. There are zero pieces of any size in nothing. 0 ÷ 3/4 = 0 × 4/3 = 0.
Q7. Can a fraction be divided by a whole number?
Yes, but that is a different operation. To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number. For example, 3/4 ÷ 6 = 3/4 × 1/6 = 3/24 = 1/8.
Q8. Is this topic important for Class 5 CBSE exams?
Yes. Dividing whole numbers by fractions is part of the NCERT Class 5 fractions chapter and is commonly tested in school exams and olympiads.
Related Topics
- Dividing Fractions
- Reciprocal of a Fraction
- Fractions Revision (Grade 5)
- Adding Unlike Fractions
- Subtracting Unlike Fractions
- Adding Mixed Numbers
- Subtracting Mixed Numbers
- Multiplying Fractions
- Multiplying a Fraction by a Whole Number
- Fraction of a Number
- Fraction Word Problems (Grade 5)
- Proper, Improper and Mixed Fractions
