Reciprocal of a Fraction
The reciprocal of a fraction is obtained by swapping (interchanging) the numerator and the denominator. For example, the reciprocal of 3/4 is 4/3, and the reciprocal of 5/7 is 7/5.
The reciprocal is also called the multiplicative inverse because when a number is multiplied by its reciprocal, the result is always 1.
Understanding reciprocals is essential for dividing fractions. To divide by a fraction, we multiply by its reciprocal. This is the "invert and multiply" rule that you will use frequently.
In this chapter, you will learn how to find the reciprocal of different types of fractions — proper fractions, improper fractions, whole numbers, and mixed numbers.
What is Reciprocal of a Fraction?
Definition: The reciprocal (or multiplicative inverse) of a non-zero number a/b is b/a.
Key property:
a/b × b/a = 1
Rules:
- The reciprocal of a/b is b/a (flip the fraction).
- The reciprocal of a whole number n is 1/n (since n = n/1, its reciprocal is 1/n).
- The reciprocal of 1 is 1 (since 1 × 1 = 1).
- 0 has no reciprocal (since 0 × anything = 0, never 1).
- The reciprocal of a negative fraction is also negative: reciprocal of −a/b is −b/a.
- The reciprocal of the reciprocal gives back the original number.
Types of reciprocals:
- Reciprocal of a proper fraction (like 2/5) is an improper fraction (5/2).
- Reciprocal of an improper fraction (like 7/3) is a proper fraction (3/7).
- Reciprocal of a unit fraction (like 1/n) is the whole number n.
Reciprocal of a Fraction Formula
Reciprocal Formula:
Reciprocal of a/b = b/a (where a ≠ 0)
Division using reciprocal:
a/b ÷ c/d = a/b × d/c
Where:
- To divide by a fraction, multiply by its reciprocal.
- This is known as the "invert and multiply" rule.
Quick reference:
- Reciprocal of 3/4 = 4/3
- Reciprocal of 5 = 1/5
- Reciprocal of 1/8 = 8
- Reciprocal of −2/3 = −3/2
- Reciprocal of 1 = 1
- Reciprocal of 0 = does not exist
Derivation and Proof
Why does the "invert and multiply" rule work?
Consider the division: 2/3 ÷ 4/5
Step 1: Write as a complex fraction:
- 2/3 ÷ 4/5 = (2/3) / (4/5)
Step 2: Multiply both numerator and denominator by the reciprocal of the denominator (5/4):
- = (2/3 × 5/4) / (4/5 × 5/4)
- = (2/3 × 5/4) / 1
- = 2/3 × 5/4
- = 10/12 = 5/6
So dividing by 4/5 is the same as multiplying by 5/4 (the reciprocal of 4/5).
Why does a × (1/a) = 1?
By definition, a/b means a parts out of b equal parts. When you multiply a/b by b/a:
- (a/b) × (b/a) = (a × b) / (b × a) = ab/ab = 1
The numerator and denominator become the same, giving 1.
Types and Properties
Reciprocal problems can be classified as follows:
1. Finding the reciprocal of a proper fraction:
- Reciprocal of 2/7 = 7/2.
- A proper fraction's reciprocal is always an improper fraction.
2. Finding the reciprocal of an improper fraction:
- Reciprocal of 9/4 = 4/9.
- An improper fraction's reciprocal is always a proper fraction.
3. Finding the reciprocal of a whole number:
- Reciprocal of 6 = 1/6 (since 6 = 6/1).
4. Finding the reciprocal of a mixed number:
- First convert to improper fraction, then find the reciprocal.
- 2 1/3 = 7/3, so reciprocal = 3/7.
5. Finding the reciprocal of a negative number:
- Reciprocal of −5/8 = −8/5.
- The sign stays the same.
6. Using reciprocals for division:
- a/b ÷ c/d = a/b × d/c.
7. Verification problems:
- Multiply a number by its reciprocal to verify the answer is 1.
Solved Examples
Example 1: Example 1: Reciprocal of a proper fraction
Problem: Find the reciprocal of 3/8.
Solution:
Swap numerator and denominator:
- Reciprocal of 3/8 = 8/3
Verification: 3/8 × 8/3 = 24/24 = 1 ✓
Answer: The reciprocal of 3/8 is 8/3.
Example 2: Example 2: Reciprocal of a whole number
Problem: Find the reciprocal of 12.
Solution:
Step 1: Write 12 as a fraction: 12 = 12/1.
Step 2: Swap: Reciprocal = 1/12.
Verification: 12 × 1/12 = 12/12 = 1 ✓
Answer: The reciprocal of 12 is 1/12.
Example 3: Example 3: Reciprocal of a mixed number
Problem: Find the reciprocal of 3 2/5.
Solution:
Step 1: Convert to improper fraction: 3 2/5 = (3 × 5 + 2)/5 = 17/5.
Step 2: Swap: Reciprocal = 5/17.
Verification: 17/5 × 5/17 = 85/85 = 1 ✓
Answer: The reciprocal of 3 2/5 is 5/17.
Example 4: Example 4: Reciprocal of a negative fraction
Problem: Find the reciprocal of −4/9.
Solution:
Swap numerator and denominator (keeping the sign):
- Reciprocal of −4/9 = −9/4
Verification: (−4/9) × (−9/4) = 36/36 = 1 ✓
Answer: The reciprocal of −4/9 is −9/4.
Example 5: Example 5: Reciprocal of 1
Problem: Find the reciprocal of 1.
Solution:
1 = 1/1. Swap: Reciprocal = 1/1 = 1.
Verification: 1 × 1 = 1 ✓
Answer: The reciprocal of 1 is 1. (1 is the only positive number that is its own reciprocal.)
Example 6: Example 6: Why 0 has no reciprocal
Problem: Does 0 have a reciprocal?
Solution:
If the reciprocal of 0 existed, say x, then:
- 0 × x = 1
- But 0 × (any number) = 0, never 1.
Therefore, no number x can satisfy 0 × x = 1.
Answer: No, 0 does not have a reciprocal.
Example 7: Example 7: Division using reciprocals
Problem: Divide 5/6 by 2/3.
Solution:
Step 1: Find the reciprocal of 2/3 → 3/2.
Step 2: Multiply: 5/6 × 3/2 = 15/12.
Step 3: Simplify: 15/12 = 5/4 = 1 1/4.
Answer: 5/6 ÷ 2/3 = 5/4 (or 1 1/4).
Example 8: Example 8: Reciprocal of a unit fraction
Problem: Find the reciprocal of 1/15.
Solution:
Swap: Reciprocal of 1/15 = 15/1 = 15.
Verification: 1/15 × 15 = 15/15 = 1 ✓
Answer: The reciprocal of 1/15 is 15.
Example 9: Example 9: Reciprocal in word problem
Problem: A piece of ribbon is 7/8 m long. It is to be cut into pieces of length 1/4 m each. How many pieces can be cut?
Solution:
Number of pieces = Total length ÷ Length of each piece:
- = 7/8 ÷ 1/4
- = 7/8 × 4/1 (multiply by reciprocal of 1/4)
- = 28/8 = 7/2 = 3 1/2
Answer: 3 complete pieces can be cut (with 1/2 piece remaining).
Example 10: Example 10: Reciprocal of a reciprocal
Problem: Find the reciprocal of the reciprocal of 7/11.
Solution:
Step 1: Reciprocal of 7/11 = 11/7.
Step 2: Reciprocal of 11/7 = 7/11.
The reciprocal of the reciprocal gives back the original number.
Answer: The reciprocal of the reciprocal of 7/11 is 7/11.
Real-World Applications
- The most important application of reciprocals is in dividing fractions. To divide by a fraction, multiply by its reciprocal: a/b ÷ c/d = a/b × d/c. This is the foundation of fraction division.
Unit Rate Problems:
- If 3/4 of a task is done in 1 hour, the rate per unit is found by taking the reciprocal: the whole task takes 4/3 hours.
Speed, Time, and Distance:
- If speed = distance/time, then time = distance × (1/speed) = distance × reciprocal of speed. Reciprocals help switch between speed and time calculations.
Scaling Recipes:
- If a recipe serves 3/4 of a batch and you need a full batch, multiply all quantities by the reciprocal (4/3) to scale up.
Ratios and Proportions:
- Reciprocals are used in inverse proportion problems. If speed and time are inversely proportional, doubling speed halves the time (multiply by 1/2, the reciprocal of 2).
Key Points to Remember
- The reciprocal of a/b is b/a (flip the fraction).
- A number multiplied by its reciprocal always equals 1.
- The reciprocal is also called the multiplicative inverse.
- Reciprocal of a whole number n is 1/n.
- Reciprocal of a unit fraction 1/n is n.
- 0 has no reciprocal (no number multiplied by 0 gives 1).
- The reciprocal of 1 is 1, and the reciprocal of −1 is −1.
- A negative number's reciprocal is also negative.
- The reciprocal of the reciprocal is the original number.
- To divide by a fraction: multiply by its reciprocal.
Practice Problems
- Find the reciprocal of 7/9.
- Find the reciprocal of 15.
- Find the reciprocal of 4 3/7 (mixed number).
- Find the reciprocal of −5/11.
- Verify that 8/3 and 3/8 are reciprocals of each other.
- Divide: 3/5 ÷ 9/10 using reciprocals.
- A car travels 5/6 km in 1/3 hour. Find its speed (distance ÷ time).
- Find a number whose reciprocal is equal to itself.
Frequently Asked Questions
Q1. What is the reciprocal of a fraction?
The reciprocal of a fraction a/b is b/a — you swap the numerator and denominator. For example, the reciprocal of 3/5 is 5/3.
Q2. What is another name for reciprocal?
The reciprocal is also called the multiplicative inverse, because a number times its reciprocal equals 1 (the multiplicative identity).
Q3. Does 0 have a reciprocal?
No. The reciprocal of 0 would need to satisfy 0 × x = 1, but 0 times anything is 0, so no such x exists. Zero has no reciprocal.
Q4. What is the reciprocal of a whole number?
The reciprocal of a whole number n is 1/n. For example, the reciprocal of 8 is 1/8, and the reciprocal of 1 is 1.
Q5. How do you find the reciprocal of a mixed number?
First convert the mixed number to an improper fraction, then flip it. Example: 2 3/4 = 11/4, so its reciprocal is 4/11.
Q6. Is the reciprocal of a negative number also negative?
Yes. The reciprocal of −a/b is −b/a. The sign does not change. For example, the reciprocal of −2/3 is −3/2.
Q7. Why is the reciprocal used for dividing fractions?
Dividing by a fraction a/b is the same as multiplying by its reciprocal b/a. This is because (a/b) × (b/a) = 1, and dividing is the inverse of multiplying.
Q8. What number is its own reciprocal?
Only 1 and −1 are their own reciprocals. 1 × 1 = 1, and (−1) × (−1) = 1.
Q9. Is the reciprocal of a proper fraction always an improper fraction?
Yes. If a/b is proper (a < b), then its reciprocal b/a has numerator > denominator, making it improper. For example, reciprocal of 2/5 is 5/2 (improper).
Q10. What is the reciprocal of the reciprocal of a number?
The reciprocal of the reciprocal gives back the original number. If you flip a fraction twice, you return to the original. Example: 3/7 → 7/3 → 3/7.
