Question #6ccf9

Mar 11, 2016

Dividing fractions is done by multiplying the reciprocal of the divisor.

Explanation:

First let's make sure we know which is the divisor:

When using fractions, we simply convert the division into a multiplication by taking the reciprocal of the divisor, for example:

$2 \div \frac{1}{2} = 2 \cdot \frac{2}{1} = 4$

If both are fractions, we do the same thing:

$\frac{1}{2} \div \frac{1}{2} = \frac{1}{2} \cdot \frac{2}{1} = 1$

which makes sense, since anything divided by itself is one!

PROOF:

to prove this, lets set $x$ equal to the quotient that we want to calculate:

$x = \frac{a}{b} \div \frac{c}{d} = \frac{\frac{a}{b}}{\frac{c}{d}}$

now mulitply both sides by $\frac{c}{d}$

$\frac{c}{d} \cdot x = \frac{\frac{a}{b}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{\frac{c}{d}}}}} \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{\frac{c}{d}}}} = \frac{a}{b}$

multiply both sides by $\frac{d}{c}$

$x = \frac{a}{b} \cdot \frac{d}{c}$

therefore

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$