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## Dividing Fractions

#### Introduction

Dividing two fractions can be made easier by multiplying one fraction by the reciprocal of the other.

#### Terms

**Fraction**- Fractions are used when using numbers to express parts of whole.

**Divisor/Denominator**- the ‘bottom’ number of a fraction. It is the total number of parts.

**Dividend/Numerator**- the ‘top’ number of a fraction. It is the number of parts being taken from the whole.

**Reciprocal**- this is the multiplicative inverse of a fraction. Simply, it is the fraction flipped upside down (so the numerator becomes the denominator and vice versa).

## Lesson

To divide $rac{1}{2}$ by $rac{3}{4}$, we take the reciprocal of the fraction doing the dividing – i.e. $rac{3}{4}$. The inverse of $rac{3}{4}$ is $rac{4}{3}$. Then we multiply the first fraction by the reciprocal of the second: $rac{1}{2} * rac{4}{3} = rac{4}{6} = rac{2}{3}$.

Here is another example:

$dfrac{2}{3} div dfrac{1}{4} = dfrac{2}{3} * dfrac{4}{1} = dfrac{8}{3}$

## Examples

#### Dividing Fractions (Example #1)

$\dfrac{2}{5} \div \dfrac{1}{5}$

Dividing fractions is the same as multiplying by the reciprocal

The reciprocal of the second fraction is $\dfrac{5}{1}$

$\dfrac{2}{5}$ / $\dfrac{1}{5}$ = $\dfrac{2}{5}$ * $\dfrac{5}{1}$

$\dfrac{2}{5} * \dfrac{5}{1}$

$\dfrac{(2 * 5)}{(5 * 1)}$

$\dfrac{(2 * 5^1)}{(5^1 * 1)}$

$2$

#### Dividing Fractions (Example #2)

$\dfrac{3}{4} \div \dfrac{4}{5}$

Dividing fractions is the same as multiplying by the reciprocal

The reciprocal of the second fraction is $\dfrac{5}{4}$

$\dfrac{3}{4}$ / $\dfrac{4}{5}$ = $\dfrac{3}{4}$ * $\dfrac{5}{4}$

$\dfrac{3}{4} * \dfrac{5}{4}$

$\dfrac{(3 * 5)}{(4 * 4)}$

$\dfrac{15}{16}$

#### Dividing Fractions (Example #3)

$\dfrac{9}{10} \div \dfrac{4}{5}$

Dividing fractions is the same as multiplying by the reciprocal

The reciprocal of the second fraction is $\dfrac{5}{4}$

$\dfrac{9}{10}$ / $\dfrac{4}{5}$ = $\dfrac{9}{10}$ * $\dfrac{5}{4}$

$\dfrac{9}{10} * \dfrac{5}{4}$

$\dfrac{(9 * 5)}{(10 * 4)}$

$\dfrac{(9 * 5^1)}{(10^2 * 4)}$

$1\dfrac{1}{8}$

#### Dividing Fractions (Example #4)

$\dfrac{1}{2} \div \dfrac{1}{4}$

Dividing fractions is the same as multiplying by the reciprocal

The reciprocal of the second fraction is $\dfrac{4}{1}$

$\dfrac{1}{2}$ / $\dfrac{1}{4}$ = $\dfrac{1}{2}$ * $\dfrac{4}{1}$

$\dfrac{1}{2} * \dfrac{4}{1}$

$\dfrac{(1 * 4)}{(2 * 1)}$

$\dfrac{(1^1 * 4^2)}{(2^1 * 1^1)}$

$2$