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## Finding the Reciprocal

#### Introduction

The reciprocal of an integer is that number divided by 1. So, the reciprocal of 3 is $rac{1}{3}$. Fractions also have reciprocals, which just involves swapping the numerator and the denominator. So, the reciprocal of $rac{3}{4}$ is $rac{4}{3}$, or 1 $rac{1}{3}$.

#### Terms

Reciprocal - A number's reciprocal is 1 divided by that number.
Denominator - The bottom number of a fraction, it tells you how many parts of a whole a fraction is made up of (e.g. $$rac{1}{3}$$ has a denominator of 3, meaning you split a whole up into three parts.)
Numerator - The top number of a fraction, it tells you how many of the denominator segments you have. (e.g. $$rac{2}{3}$$ has a numerator of 2 and a denominator of 3, meaning you split a whole up into three parts, and then keep two of those parts.)
Mixed Fraction - A fraction with a whole number and a fraction (e.g. 6 $$rac{1}{3}$$ )
Improper Fraction - A top-heavy fraction, where the numerator is bigger than the denominator (e.g. $$rac{19}{3}$$)

## Lesson

To find the reciprocal of an integer, just divide 1 by that number.

The reciprocal of 5 is $rac{1}{5}$

The reciprocal of 7 is $rac{1}{7}$

The reciprocal of 86932 is $rac{1}{86932}$

### Fractions

The reciprocal of a fraction is the fraction flipped upside down. Just swap the denominator and the numerator.

The reciprocal of $rac{2}{7}$ is $rac{7}{2}$

The reciprocal of $rac{6}{19}$ is $rac{19}{6}$

The reciprocal of $rac{487}{6198}$ is $rac{6198}{487}$

If the numerator is 1, then the reciprocal will be a whole number:

The reciprocal of $rac{1}{7}$ is $rac{7}{1}$, or 7.

### Mixed Fractions

To find the reciprocal of a mixed fraction, you need to turn it into an improper (or top-heavy) fraction.

So, to find the reciprocal of 6 $rac{1}{3}$, we create the improper fraction $rac{19}{3}$. The reciprocal is then the numerator and denominator switched: $rac{3}{19}$

Multiplying together two reciprocals will give the answer 1:

$rac{2}{3}$ x $rac{3}{2}$ = $rac{6}{6}$ = 1

## Examples

#### Finding the Reciprocal (Example #1)

$\dfrac{2}{3}$
$\dfrac{2}{3} * \dfrac{3}{2} = 1$
Therefore the reciprocal (multiplicative inverse) of $\dfrac{2}{3}$ is $\dfrac{3}{2}$

#### Finding the Reciprocal (Example #2)

$-\dfrac{3}{4}$
$-\dfrac{3}{4} * -\dfrac{4}{3} = 1$
Therefore the reciprocal (multiplicative inverse) of $-\dfrac{3}{4}$ is $-\dfrac{4}{3}$

#### Finding the Reciprocal (Example #3)

$\dfrac{7}{5}$
$\dfrac{7}{5} * \dfrac{5}{7} = 1$
Therefore the reciprocal (multiplicative inverse) of $\dfrac{7}{5}$ is $\dfrac{5}{7}$

#### Finding the Reciprocal (Example #4)

$\dfrac{1}{11}$
$\dfrac{1}{11} * \dfrac{11}{1} = 1$
Therefore the reciprocal (multiplicative inverse) of $\dfrac{1}{11}$ is $\dfrac{11}{1}$