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Last updated on July 4th, 2025

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Reciprocal of Fractions

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The reciprocal of a fraction means swapping numerator and denominator. It involves interchanging the numerator and the denominator of a fraction. For example, the reciprocal of 5/2 is 2/5. This process of reversing a fraction is also known as multiplicative inverse.

Reciprocal of Fractions for Australian Students
Professor Greenline from BrightChamps

What are Reciprocal Fractions?

The mathematical concept of reciprocal means the multiplicative inverse of a number. It is what you get when you reverse the positions of the numerator and the denominator. For example, the reciprocal of 2/7 is 7/2, 8/3 is 3/8, etc. 

 

When a fraction is multiplied by the reciprocal, it always results 1. Let us understand this by an example. Let’s say, 2, the reciprocal of 2 is ½ and then multiply 2 by its reciprocal, 2  1/2 = 1
 

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What are Fractions?

A fraction represents a portion of a whole or the ratio of two whole numbers. It can be represented in the form of p/q, where p and q are integers and q≠0 is called a fraction. It is a horizontal line that separates the two integers, called the fractional bar. The top number is called the numerator and the bottom number is called the denominator. For example, a ratio of 5:10 can be written as

 

        5/10 =>  p/q => NumeratorDenominator 

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Parts of Fractions

A fraction has two parts:

 

  • Numerator: Numerator means number above the fraction line. It tells us how many parts of the whole are being considered. It indicates the number of pieces being used. For instance, in a fraction 5/6, 5 is the numerator, meaning that you are working with 5 parts out of 6. 

     
  • Denominator: This is the number below the fraction line. It shows how many total numbers of equal parts the whole is divided into. It indicates how many pieces are there in the entire unit or set. For instance, in a fraction 5/6, 6 is the denominator, meaning the whole is divided into 6 equal parts. 
     
Professor Greenline from BrightChamps

Types of Fraction

There are three types of fractions. Let us understand more about fractions in detail.

 

  • Proper Fraction: In a proper fraction, the numerator is always smaller than the denominator. Proper fractions are always smaller than one. For example, 5/8 and 3/4 are proper fractions.

     
  • Improper Fractions: In an improper fraction, the numerator is equal to or greater than the denominator. Improper fractions are always greater than or equal to one when expressed as decimals. For example, 8/5 and 40/15 are improper fractions.

     
  • Mixed Fractions:  A mixed fraction is made of a whole number and a proper fraction. The whole number will be written first and then followed by a fraction. For example, 4 3/6 and 8 5/9 are mixed fractions.
Professor Greenline from BrightChamps

How to Find a Reciprocal Fraction?

To find the reciprocal of a fraction, follow these simple steps.

 

Step 1: Understand which one is the numerator and which is the denominator. 

 

Step 2: Reverse the numerator and denominator. Thus, the numerator becomes the new denominator and the denominator becomes the new numerator.

 

Step 3: Write the new fraction, this is the reciprocal of the original fraction.

 

Now, let’s look how to find the reciprocal of different types of fractions.
 

Professor Greenline from BrightChamps

Reciprocal of Mixed Fraction

Step 1: First, you have to convert the mixed number into an improper fraction. 

 

For example, 6 1/2  is a mixed fraction, it should be converted to improper fraction. For that, we follow these steps:
First, multiply the whole number by the denominator of the fraction.


Now, add the numerator to the result.


Write the sum as the new numerator, and denominator should be the same.


The answer is 13/2.

 

Step 2: Reverse the improper fraction now. 

 

        13/2 => 2/13

 

Step 3: The reciprocal of 6 1/2 is 2/13.
 

Professor Greenline from BrightChamps

Reciprocal of a Fraction with Exponents

Step 1: For example, 32/43 is a fraction with exponents. 

 

Step 2: Reverse the numerator and denominator, that is, 43/32. For reciprocal, powers of both numerator and denominator will also be swapped. 


Step 3: The reciprocal of 32/43 is 43/32.

Professor Greenline from BrightChamps

Reciprocal of a Negative Fraction

Step 1: Exchange the numerator for the denominator.  For example, – 3/2

 

            3/2 => 2/3

 

Step 2: The negative sign remains unchanged. So, the reciprocal of – 3/2 is – 2/3.
 

Professor Greenline from BrightChamps

How to Represent Reciprocal Fractions on Graph?

There are different types of reciprocal functions, one of which is expressed as f(x) = k/x, where k is a real number and x cannot be zero because division by zero is undefined. Now, let’s graph the function f(x) = 1/x by plotting x and y values.


In the reciprocal function f(x) = 1/x, x can never be 0 because division by zero is undefined. The graph reveals that it never touches the x-axis or y-axis. The y-axis is called the vertical asymptote and x-axis is the horizontal asymptote, as the hyperbola approaches it but never intersects it.
 

Professor Greenline from BrightChamps

Operations on Reciprocal Fractions

Now, let’s understand how to do operations on reciprocal fractions. 

 

Addition of Reciprocal Fractions

 

Step 1: Start by finding the reciprocal of each fraction. The reciprocal of a fraction a/b is b/a. For example, 3/4 and 5/6, its reciprocal will be 4/3 and 6/5.

 

Step 2: After the reciprocals, the fractions will be added. 
 

        4/3 + 6/5  

    Here, the fractions are with different denominator. So, we have to make fractions with similar denominators. Therefore, LCM of 3 and 5 is 15.

        4 x 5 / 3 x 5 = 20 /15 

6 x 3 / 5 x 3 = 18 /15

    Now, add them together
        
20 / 15 + 18 / 15 = 48 / 30

 

Step 3: Finally, simplify the resulting fraction if possible.

        48 / 30 => 8 / 5
 

Professor Greenline from BrightChamps

Real-Life Applications of Reciprocal of Fractions

  • Cooking and Recipe: If a recipe calls for 2/3 cup of flour per serving, but you want to find how many servings you can make with 1 cup, you take the reciprocal (3/2) and multiply. By multiplying 1 cup by the reciprocal (3/2), you determine that you can make 1 1/2 servings.           1 x 3 / 2 = 3 / 2 = 1.5 servings

     
  • Speed and Time: If a car travels at 4/5 of a mile per minute, its time per mile is the reciprocal (5/4 minutes per mile).

     
  • Construction and Scaling: If a worker takes 2/3 of an hour to complete a task, the reciprocal (3/2) helps determine how many tasks they can complete per hour (1.5 tasks per hour). 

     
  • Exchange Rates and Currency Conversion: If 1 dollar = 3/4 euros, then the reciprocal (4/3) tells you that 1 euro = 4/3 dollars. 
Max Pointing Out Common Math Mistakes

Common Mistakes of Reciprocal of Fractions and How to Avoid Them

Understanding reciprocals is essential in math, especially while working with fractions and equations. However, many students make mistakes while finding reciprocals. Here are five common mistakes and how to avoid them.
 

Mistake 1

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Forgetting to reverse the whole numbers.
 

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Whole numbers are often mistakenly left unchanged when finding reciprocals. Always convert whole numbers into fraction by writing 1 over it (e.g., 5 = 5/1) and then find the reciprocal by flipping them (1/5). 

Mistake 2

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Reversing the fraction only for some fractions.

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While solving any equation that require its reciprocals, some students only reverse one part of the fraction. Always remember to reverse the numerator and denominator for every fraction in the equation before any mathematical operations.

Mistake 3

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Not reciprocating mixed numbers properly.

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Some students flip only the fractional part of a mixed number and ignore the whole number. Convert mixed numbers into improper fractions first. For example, 2 1/3 becomes 7/3, then the reciprocal is 3/7. 

Mistake 4

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Incorrectly simplifying after reversing.
 

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After finding the reciprocal, some students forget to simplify or mistakenly simplify before flipping. Always reverse the fraction first, then simplify if needed. For example, 8/12 has a reciprocal of 12/8, which simplifies to 3/2. 

Mistake 5

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Reciprocal of zero
 

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Some mistakenly think 0 has a reciprocal, but 1/0 is undefined. Remember that zero has no reciprocal because division by zero is impossible.
 

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Solved Examples for Reciprocal of Fractions

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Problem 1

What is the reciprocal of 5/7?

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 7/5

Explanation

To find the reciprocal, swap the numerator and denominator. The fraction 5/7 becomes 7/5.

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Max, the Girl Character from BrightChamps

Problem 2

Find the reciprocal of 9.

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1/9
 

Explanation

A whole number 9 can be written as 9/1. Reversing it gives 1/9. 
 

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Max, the Girl Character from BrightChamps

Problem 3

What is the reciprocal of 1 2/5?

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5/7
 

Explanation

First, convert the mixed number 1 2/5 to an improper fraction:
 

            1 2/5 = 7/5
 

        Then, flip it to get 5/7.

 

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Max, the Girl Character from BrightChamps

Problem 4

What is the reciprocal of 12/16 in the simplest form?

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4/3
 

Explanation

Flip 12/16 to get 16/12. Then simplify:

 

            16/12 = 4/3

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Problem 5

Find the reciprocal of 0.

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 Undefined
 

Explanation

The reciprocal of 0 is 1/0, which is undefined because division by zero is impossible. 

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

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Fun Fact

: She loves to read number jokes and games.

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