# Reciprocal Formula – Definition, Examples, Practice Problems, FAQs

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## What Is the Reciprocal Formula?

The reciprocal formula calculates the reciprocal of a given number. The reciprocal of a non-zero number “a” is “1 divided by a.” It is expressed as $\frac{1}{a}$ and represents the multiplicative inverse of the number. In other words, when a number and its reciprocal are multiplied together, the result is 1.

Reciprocal is defined as the multiplicative inverse of a number. If ‘n’ is any non-zero number, the reciprocal will be $\frac{1}{n}$. This means we have to convert the original number to the upside-down form. If the number is multiplied by its reciprocal, the product is equal to one.

## Reciprocal Formula

Reciprocal of a non-zero real number $= \frac{1}{Number}$

## How to Find Reciprocal

To find the reciprocal of a number, just flip its fraction form.

Example 1: Find the reciprocal of 7.

The given number is 7. In fraction form, we can write it as $\frac{7}{1}$.

Flipping $\frac{7}{1}$ (interchanging the numerator and denominator), we get $\frac{1}{7}$.

Thus, the reciprocal of 7 will be $\frac{1}{7}$.

Similarly, the reciprocal of a fraction can be written by interchanging the values of the numerator and the denominator.

Example 2: Find the reciprocal of $\frac{5}{3}$.

Reciprocal formula: $x = \frac{1}{x}$

Here, $x = \frac{5}{3}$

Reciprocal of $\frac{5}{3} = \frac{1}{\frac{5}{3}} = \frac{3}{5}$.

Thus, the reciprocal of the fraction $\frac{5}{3} = \frac{3}{5}$.

## Facts about the Reciprocal Formula

• Except zero, all the numbers have reciprocal.
• On multiplying a number and its reciprocal, the product is always one.
• In mathematics, the term “reciprocal” is often used to describe the multiplicative inverse of a number.
• The reciprocal of -1 is -1.
• The reciprocal of 1 is 1.
• Reciprocal of a proper fraction is an improper fraction, and vice-versa.

## Conclusion

In this article, we explored the concept of reciprocals and how they are obtained by flipping the fraction form of a number. Understanding reciprocals is essential in various mathematical applications. To reinforce our understanding, let’s work through some examples and test our knowledge with practice MCQs.

## Solved Examples on the Reciprocal Formula

Example 1: What is the reciprocal of 5 and 8?

Solution:

The given numbers are 5 and 8. To find the solution, we have to use reciprocal formula:

Reciprocal formula: Reciprocal of $x = \frac{1}{x}$.

Reciprocal of $5 = \frac{1}{5}$

Reciprocal of $8 = \frac{1}{8}$

Thus, the reciprocal of 5 is, $\frac{1}{5}$ and the reciprocal of 8 is $\frac{1}{8}$.

Example 2: What is the reciprocal of $\frac{7}{2}$?

Solution:

The given fraction is $\frac{7}{2}$ . To find the solution, we have to use the reciprocal formula:

Reciprocal formula: Reciprocal of $x = \frac{1}{x}$

Here, $x = \frac{7}{2}$

Reciprocal of $\frac{7}{2} = \frac{1}{\frac{7}{2}} = \frac{2}{7}$.

Thus, the reciprocal of the fraction $\frac{7}{2} = \frac{2}{7}$.

Example 3: Find the reciprocal of $-\;\frac{1}{3}$.

Solution:

The given fraction is $-\;\frac{1}{3}$ . To find the solution, we have to use the reciprocal formula:

Reciprocal formula: Reciprocal of $x = \frac{1}{x}$.

Here, $x = -\;\frac{1}{3}$

Reciprocal of $\frac{-\;1}{3} = \frac{1}{-\;\frac{1}{3}} = -\;\frac{3}{1}$

Thus, the reciprocal of the fraction $-\;\frac{1}{3}$ is  $-\;3$.

## Practice Problems on the Reciprocal Formula

1

### What is the reciprocal of 6 and $\frac{1}{6}$?

$6, \frac{1}{6}$
$\frac{1}{6} ,\; 6$
$\frac{1}{6} ,\; \frac{-\;1}{6}$
$6,\; -\;6$
CorrectIncorrect
Correct answer is: $\frac{1}{6} ,\; 6$
Reciprocal of $6 = \frac{1}{6}$
Reciprocal of $\frac{1}{6} = 6$
2

### What is the reciprocal of $\frac{5}{2}$?

$\frac{5}{2}$
$-\;\frac{5}{2}$
$\frac{2}{5}$
$-\;\frac{2}{5}$
CorrectIncorrect
Correct answer is: $\frac{2}{5}$
The given fraction is $\frac{5}{2}$ . To find the solution, we have to use the reciprocal formula:
Reciprocal of $x = \frac{1}{x}$.
Here, $x = \frac{5}{2}$
Reciprocal of $\frac{5}{2} = \frac{1}{\frac{5}{2}} = \frac{2}{5}$
3

### Find the reciprocal of $-\;\frac{1}{9}$.

$\frac{1}{9}$
$-\;\frac{1}{9}$
$-\;9$
$9$
CorrectIncorrect
Correct answer is: $-\;9$
Reciprocal formula: Reciprocal of $x = \frac{1}{x}$.
Here, $x = -\;\frac{1}{9}$
Reciprocal of $-\;\frac{1}{9} = \frac{1}{-\;\frac{1}{9}} = -\;\frac{9}{1} = -\;9$

To find the reciprocal of a mixed fraction, convert it into an improper fraction. Then take the reciprocal of the improper fraction. For example: $4\frac{3}{2}$ is a mixed fraction. Converting it into an improper fraction will be  $4\frac{3}{2} = \frac{11}{2}$. Now, find the reciprocal by interchanging the denominator and numerator $\frac{2}{11}$ . The reciprocal of $4\frac{3}{2}$ is $\frac{2}{11}$ .
If the reciprocal of the number is multiplied by itself, the value will result in unity which is one. For example, $2 \times \frac{1}{2} = 1$.
The reciprocal of any negative number -x will be the inverse of the number along with the negative sign $-\frac{1}{x}$ . For example, the reciprocal of -5 will be $-\frac{1}{5}$ .