# The keep-change-flip rule for fractions

## Statement

When dividing by a fraction, you multiply by the reciprocal.

$$a \div \frac{b}{c} = a * \frac{c}{b}$$

## Proof

Write the fraction using negative exponents.

$$a \div \frac{b}{c} = a \div (b * c^{-1})$$

Now write the division using negative exponents as well.

$$a \div \frac{b}{c} = a * (b * c^{-1})^{-1}$$

Append the powers to the factors.

$$a \div \frac{b}{c} = a * b^{-1} * (c^{-1})^{-1}$$

$$a \div \frac{b}{c} = a * b^{-1} * c^1$$

Finally, write $b^{-1} * c$ as a fraction again.

$$a \div \frac{b}{c} = a * \frac{c}{b}$$