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