Reciprocal of Fractions
2026-02-28 11:56 Diff
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Last updated on December 9, 2025

The reciprocal of a fraction means swapping numerator and denominator. It involves interchanging the numerator and 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.

What are Reciprocal Fractions?

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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 \( \frac{2}{7} \) is \( \frac{7}{2} \), \( \frac{8}{3} \) is \( \frac{3}{8} \), etc. 

When a fraction is multiplied by the reciprocal, it always results in 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 \( \frac{1}{2} \) = 1
 

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 \(\frac{p}{q} \), where p and q are integers, and q≠0. Here, \(\frac{p}{q} \) 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
 

        \(\frac{5}{10} \) ⇒ \(\frac{p}{q} \) ⇒ \(\frac{\text{Numerator}}{\text{Denominator}} \)

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. 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. 
     

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Types of Fraction

  • Proper fraction: In a proper fraction, the numerator is always smaller than the denominator. Proper fractions are always smaller than one. For example, \( \frac{5}{8} \) and \( \frac{3}{4} \) 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,\( \frac{8}{5} \)and \( \frac{40}{15} \) 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,\( 4 \frac{3}{6} \) and \( 8 \frac{5}{9} \) are mixed fractions.
  • Zero fraction: A zero fraction is a fraction whose numerator is 0 and denominator is not 0. The value of a zero fraction is always 0. For example, \(\frac{0}{3}\), \(\frac{0}{7}\).

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.
 

Reciprocal of Mixed Fraction

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

For example, \( 6 \frac{1}{2} \) 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 \( \frac{13}{2} \)

Step 2: Reverse the improper fraction now. 

      

\( \frac{13}{2} \Rightarrow \frac{2}{13} \)

Step 3: The reciprocal of 6 \( \frac{1}{2} \) is \( \frac{2}{13} \).
 

Reciprocal of a Fraction with Exponents

For fractions with exponents, reverse the numerator and denominator; the exponents remain with their respective numbers.

Step 1: For example, \( \frac{3^2}{4^3} \) is a fraction with exponents. 

Step 2: Reverse the numerator and denominator, that is, \( \frac{4^3}{3^2} \) For reciprocal, powers of both numerator and denominator will also be swapped. 

Step 3: The reciprocal of \( \frac{3^2}{4^3} \) is \( \frac{4^3}{3^2} \)

Reciprocal of a Negative Fraction

Step 1: Exchange the numerator for the denominator.  For example, \( -\frac{3}{2} \)

\( -\frac{3}{2} \Rightarrow -\frac{2}{3} \)

Step 2: The negative sign remains unchanged. So, the reciprocal of \( -\frac{3}{2} \text{ is } -\frac{2}{3} \).
 

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 \(f(x) = 1/x \) by plotting corresponding x and y values.

In the reciprocal function \(f(x) = 1/x\),

  • x ≠ 0 because division by zero is undefined.
     
  • The graph 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.

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,\( \frac{3}{4} \) and \( \frac{5}{6} \), its reciprocal will be \( \frac{4}{3} \) and \( \frac{6}{5} \).

Step 2: Once the reciprocals are found, the fractions can be added.

\( \frac{4}{3} + \frac{6}{5} \)

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

\( \frac{4 \times 5}{3 \times 5} = \frac{20}{15} \)
 

\( \frac{6 \times 3}{5 \times 3} = \frac{18}{15} \)
 

 Now, add them together

         \( \frac{20}{15} + \frac{18}{15} = \frac{38}{15} \)

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

Reciprocal of Fraction with Exponents

Fractions with exponents are written in the form \((a/b)^p\), where a and b are whole numbers (b ≠ 0), and p is any rational number. When dealing with a reciprocal fraction, especially one with a negative exponent, it becomes its reciprocal raised to the same positive exponent.

For example, consider the fraction \((4/5)^{-3}\). When you see a negative exponent, it simply tells you to take the reciprocal of the fraction. Instead of keeping the negative sign, you flip the fraction and make the exponent positive: \((4/5) ^{- 3}\) = \((5/4)^{ 3}\). Now, if you want to find the reciprocal of this new expression, you take its reciprocal, and you end up with \((4/5) ^{3}\). In general, to understand what the reciprocal of a fraction with exponents is, we first separate the numerator and denominator. \((a/b)^5=a^5/b^5\).

To find the reciprocal of a fraction, we interchange the numerator and denominator along with their respective exponents. Therefore, the reciprocal of \(a^5/b^5\) becomes \(b^5/a^5\).
This method helps when you want to find the reciprocal of a fraction or check answers. Understanding the reciprocals of fractions makes it easy to solve exponent problems in algebra and arithmetic.

Tips and Tricks to Master Reciprocal of Fractions

Understanding reciprocals helps simplify division, check calculations, and solve fraction problems quickly and accurately.
 

  • To find the reciprocal, keep the numerator and denominator.
  • Rewrite whole numbers as fractions over one before finding the reciprocal.
  • Multiplying any fraction by its reciprocal, you will always get 1.
  • When the fraction is negative, the reciprocal should stay negative.
  • Dividing by a fraction is the same as multiplying by its reciprocal.
  • Parents can use visuals to draw fraction bars to help children see numerators and denominators before flipping.
  • Teachers can bring reciprocals to life with hands-on tools like cards or manipulatives, letting students physically turn fractions over to understand the concept.
  • Children can use a shortcut when dividing by a fraction: flip it and multiply instead of dividing.

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.
 

Real-Life Applications of Reciprocal of Fractions

  • If a recipe needs \( \frac{2}{3} \) a cup of flour per serving and you have 1 cup, take the reciprocal \( \left(\frac{3}{2}\right) \) and multiply. This shows that 1 cup of flour makes \( 1\tfrac{1}{2} \) 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. 
  • If a recipe uses 1/4 cup per portion, taking the reciprocal (4/1) shows you can make 4 portions with 1 cup.

Download Worksheets

Problem 1

What is the reciprocal of 5/7?

Okay, lets begin

 \( \frac{7}{5} \)

Explanation

To find the reciprocal, swap the numerator and denominator. The fraction \( \frac{5}{7} \) becomes \( \frac{7}{5} \).

Well explained 👍

Problem 2

Find the reciprocal of 9.

Okay, lets begin

\( \frac{1}{9} \)
 

Explanation

A whole number 9 can be written as \( \frac{9}{1} \). Reversing it gives \( \frac{1}{9} \). 
 

Well explained 👍

Problem 3

What is the reciprocal of 1 2/5?

Okay, lets begin

\( \frac{5}{7} \)
 

Explanation

First, convert the mixed number\( 1 \frac{2}{5} \) to an improper fraction:
 

          \( 1 \frac{2}{5} \) = \( \frac{7}{5} \)
 

        Then, flip it to get \( \frac{5}{7} \).

Well explained 👍

Problem 4

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

Okay, lets begin

\( \frac{4}{3} \)
 

Explanation

Flip \( \frac{12}{16} \) to get \( \frac{16}{17} \). Then simplify:

           \( \frac{16}{17} \) = \( \frac{4}{3} \)

Well explained 👍

Problem 5

Find the reciprocal of 0.

Okay, lets begin

 Undefined
 

Explanation

The reciprocal of 0 is\( \frac{1}{0} \), which is undefined because division by zero is impossible. 

Well explained 👍

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.

Fun Fact

: She loves to read number jokes and games.