Reciprocal of a Fraction
The reciprocal of a fraction is obtained by flipping the fraction upside down — swapping the numerator and the denominator. For example, the reciprocal of 3/7 is 7/3.
Reciprocals are essential for dividing fractions. Instead of dividing by a fraction, we multiply by its reciprocal. This makes fraction division as easy as fraction multiplication. The reciprocal is also called the multiplicative inverse because a number multiplied by its reciprocal always gives 1.
In Class 5, you will learn how to find reciprocals of proper fractions, improper fractions, whole numbers, and mixed numbers. You will also understand why zero has no reciprocal and practise verifying reciprocals by multiplication.
What is Reciprocal of a Fraction - Class 5 Maths (Fractions)?
The reciprocal (or multiplicative inverse) of a fraction p/q is q/p, where both p and q are not zero.
Reciprocal of p/q = q/p
where p ≠ 0 and q ≠ 0
Key property: A fraction multiplied by its reciprocal always equals 1.
p/q × q/p = (p × q) / (q × p) = 1
This property is what makes reciprocals so useful. When we need to "undo" multiplication by a fraction, we multiply by its reciprocal.
Reciprocal of a Fraction Formula
Reciprocal of a/b = b/a
a/b × b/a = 1
How to find the reciprocal step by step:
- If the number is a fraction p/q: swap numerator and denominator. Reciprocal = q/p.
- If the number is a whole number n: write it as n/1, then swap. Reciprocal = 1/n.
- If the number is a unit fraction 1/n: swap to get n/1 = n (a whole number).
- If the number is a mixed number: convert to an improper fraction first, then swap numerator and denominator.
Important: The number 0 has no reciprocal because 1/0 is undefined (division by zero is not allowed).
Types and Properties
Reciprocals of different types of numbers:
| Type | Number | Reciprocal | Verification (Product = 1?) |
|---|---|---|---|
| Proper fraction | 2/5 | 5/2 | 2/5 × 5/2 = 10/10 = 1 ✓ |
| Improper fraction | 7/3 | 3/7 | 7/3 × 3/7 = 21/21 = 1 ✓ |
| Whole number | 6 (= 6/1) | 1/6 | 6 × 1/6 = 6/6 = 1 ✓ |
| Unit fraction | 1/9 | 9 (= 9/1) | 1/9 × 9 = 9/9 = 1 ✓ |
| Mixed number | 2 1/3 (= 7/3) | 3/7 | 7/3 × 3/7 = 21/21 = 1 ✓ |
Special cases:
- The number 1 is its own reciprocal: 1/1 = 1, and 1 × 1 = 1.
- The number 0 has no reciprocal: 1/0 is undefined. Division by zero is not allowed in mathematics.
- The reciprocal of a reciprocal gives back the original number. Reciprocal of reciprocal of 3/5 = reciprocal of 5/3 = 3/5.
Solved Examples
Example 1: Example 1: Reciprocal of a Proper Fraction
Problem: Find the reciprocal of 3/8.
Solution:
Step 1: The fraction is 3/8. Swap the numerator (3) and denominator (8).
Step 2: Reciprocal of 3/8 = 8/3
Verification: 3/8 × 8/3 = (3 × 8)/(8 × 3) = 24/24 = 1 ✓
Answer: The reciprocal of 3/8 is 8/3 (or 2 2/3 as a mixed number).
Example 2: Example 2: Reciprocal of a Whole Number
Problem: Find the reciprocal of 5.
Solution:
Step 1: Write 5 as a fraction: 5 = 5/1
Step 2: Swap numerator and denominator: Reciprocal = 1/5
Verification: 5 × 1/5 = 5/5 = 1 ✓
Meaning: The reciprocal of 5 is one-fifth. When you multiply any number by its reciprocal, you always get 1.
Answer: The reciprocal of 5 is 1/5.
Example 3: Example 3: Reciprocal of a Unit Fraction
Problem: Find the reciprocal of 1/12.
Solution:
Step 1: The fraction is 1/12. Swap: numerator becomes 12, denominator becomes 1.
Step 2: Reciprocal of 1/12 = 12/1 = 12
Verification: 1/12 × 12 = 12/12 = 1 ✓
Pattern: The reciprocal of any unit fraction 1/n is always the whole number n.
Answer: The reciprocal of 1/12 is 12.
Example 4: Example 4: Reciprocal of an Improper Fraction
Problem: Find the reciprocal of 9/4.
Solution:
Step 1: Swap numerator and denominator: Reciprocal of 9/4 = 4/9
Verification: 9/4 × 4/9 = 36/36 = 1 ✓
Note: The reciprocal of an improper fraction (greater than 1) is always a proper fraction (less than 1), and vice versa.
Answer: The reciprocal of 9/4 is 4/9.
Example 5: Example 5: Reciprocal of a Mixed Number
Problem: Find the reciprocal of 3 1/5.
Solution:
Step 1: Convert the mixed number to an improper fraction: 3 1/5 = (3 × 5 + 1)/5 = 16/5
Step 2: Swap numerator and denominator: Reciprocal = 5/16
Verification: 16/5 × 5/16 = 80/80 = 1 ✓
Important: Never try to find the reciprocal of a mixed number directly (do NOT write 5/3 1). Always convert to an improper fraction first.
Answer: The reciprocal of 3 1/5 is 5/16.
Example 6: Example 6: Verification Problem
Problem: Verify that 4/7 and 7/4 are reciprocals of each other.
Solution:
Step 1: Multiply them together: 4/7 × 7/4
Step 2: = (4 × 7) / (7 × 4) = 28/28 = 1
Since the product is exactly 1, they are confirmed reciprocals. ✓
Rule: Two numbers are reciprocals of each other if and only if their product equals 1.
Answer: Yes, 4/7 and 7/4 are reciprocals of each other.
Example 7: Example 7: Which Number Has No Reciprocal?
Problem: Which of the following has no reciprocal: 1, 0, 1/3, 5?
Solution:
Step 1: Reciprocal of 1 = 1/1 = 1 ✓ (exists)
Step 2: Reciprocal of 0 = 1/0 → undefined (division by zero) ✗
Step 3: Reciprocal of 1/3 = 3 ✓ (exists)
Step 4: Reciprocal of 5 = 1/5 ✓ (exists)
Explanation: Zero has no reciprocal because there is no number that gives 1 when multiplied by 0. Any number multiplied by 0 is always 0, never 1.
Answer: 0 has no reciprocal.
Example 8: Example 8: Finding a Missing Number Using Reciprocals
Problem: Fill in the blank: 5/9 × ___ = 1
Solution:
Step 1: A number multiplied by its reciprocal equals 1.
Step 2: The missing number is the reciprocal of 5/9 = 9/5.
Verification: 5/9 × 9/5 = 45/45 = 1 ✓
General rule: If a × b = 1, then b is the reciprocal of a, and a is the reciprocal of b.
Answer: The missing number is 9/5 (or 1 4/5).
Example 9: Example 9: Application in Division
Problem: Aman has 3/4 kg of sugar. He packs it into small bags of 1/8 kg each. How many bags can he fill?
Solution:
Step 1: Number of bags = 3/4 ÷ 1/8
Step 2: To divide by a fraction, multiply by its reciprocal. Reciprocal of 1/8 is 8.
Step 3: 3/4 × 8 = 3/4 × 8/1 = 24/4 = 6
Interpretation: There are 6 one-eighth portions in three-quarters. Aman fills 6 bags.
Answer: Aman can fill 6 bags.
Example 10: Example 10: Reciprocal of the Reciprocal
Problem: The reciprocal of a number is 4/11. What is the original number?
Solution:
Step 1: If the reciprocal of the number is 4/11, then the number is the reciprocal of 4/11.
Step 2: Reciprocal of 4/11 = 11/4
Verification: 11/4 × 4/11 = 44/44 = 1 ✓
Rule: The reciprocal of the reciprocal always gives back the original number. This is like flipping a coin twice — you end up back where you started.
Answer: The number is 11/4 (or 2 3/4).
Real-World Applications
Why are reciprocals useful in real life?
- Dividing fractions: Division by a fraction is done by multiplying by the reciprocal. This is the most important and most common use of reciprocals in Class 5 and beyond.
- Speed and rate problems: If a painter takes 3/4 hours per wall, the reciprocal 4/3 tells you how many walls per hour — useful for calculating work done.
- Sharing food equally: Dividing 1/2 a pizza among 4 friends means 1/2 ÷ 4, which uses the reciprocal of 4 (i.e., 1/4): 1/2 × 1/4 = 1/8 per person.
- Unit conversion: If 1 inch = 2.54 cm, then 1 cm = 1/2.54 inches. Reciprocals help reverse conversion factors.
- Science: In physics, if speed = distance/time, then time = distance × reciprocal of speed. Reciprocals appear in formulas everywhere.
Key Points to Remember
- The reciprocal of p/q is q/p (flip the fraction).
- A number multiplied by its reciprocal always equals 1.
- The reciprocal of a whole number n is 1/n.
- The reciprocal of a unit fraction 1/n is the whole number n.
- For mixed numbers, always convert to an improper fraction first, then find the reciprocal.
- Zero (0) has no reciprocal because 1/0 is undefined. No number multiplied by 0 gives 1.
- The number 1 is its own reciprocal (1/1 = 1).
- The reciprocal of an improper fraction is always a proper fraction, and vice versa.
- Reciprocals are essential for the "invert and multiply" method of dividing fractions.
Practice Problems
- Find the reciprocal of 7/11.
- What is the reciprocal of 8?
- Find the reciprocal of 1/15.
- Find the reciprocal of 4 2/3. (Convert to an improper fraction first.)
- Fill in the blank: 6/13 × ___ = 1.
- Is 3/5 the reciprocal of 5/3? Verify by multiplying them together.
- Priya says the reciprocal of 0 is 0. Is she correct? Explain why or why not.
- The reciprocal of a number is 2/9. What is the number? Express it as a mixed number.
Frequently Asked Questions
Q1. What is the reciprocal of a fraction?
The reciprocal of a fraction p/q is q/p, obtained by swapping the numerator and denominator. For example, the reciprocal of 2/7 is 7/2. The key test: their product must equal 1.
Q2. What is another name for reciprocal?
The reciprocal is also called the multiplicative inverse. When a number is multiplied by its multiplicative inverse, the product is always 1. For example, 3/4 and 4/3 are multiplicative inverses of each other.
Q3. Does zero have a reciprocal?
No. The reciprocal of 0 would be 1/0, which is undefined because division by zero is not allowed. No number exists that gives 1 when multiplied by 0.
Q4. What is the reciprocal of 1?
The reciprocal of 1 is 1 itself, because 1/1 = 1 and 1 × 1 = 1. This makes 1 the only positive whole number that is its own reciprocal.
Q5. How do I find the reciprocal of a mixed number?
First convert the mixed number to an improper fraction. Then flip it. For example, 2 3/4 = 11/4, so its reciprocal is 4/11. Never try to flip a mixed number directly.
Q6. Why do we use reciprocals in division of fractions?
Dividing by a fraction is the same as multiplying by its reciprocal. For instance, 3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10. This converts a division problem into a multiplication problem, which is easier to solve.
Q7. Is the reciprocal of a proper fraction always improper?
Yes. If the original fraction is less than 1 (proper), flipping it gives a fraction greater than 1 (improper). For example, reciprocal of 2/5 is 5/2. And vice versa: reciprocal of an improper fraction is always proper.
Q8. What happens if I take the reciprocal of a reciprocal?
You get back the original number. The reciprocal of the reciprocal of p/q is p/q itself. Flipping twice brings you back to where you started. Example: reciprocal of 3/4 is 4/3; reciprocal of 4/3 is 3/4.
Q9. Is this topic in the NCERT Class 5 syllabus?
Yes. Reciprocals are introduced in Class 5 as part of the Fractions chapter, especially as a prerequisite for division of fractions. It is part of the NCERT/CBSE curriculum and appears in school exams.
Related Topics
- Dividing Fractions
- 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)
