Division by fractions

Question

Why is it that to divide by a fraction you need to multiply by the reciprocal of that fraction? For instance:

$\eqalign{ & 1 \div \frac{1}{3} \cr & = 1 \times 3 \cr & = 3 \cr} $

Thank you.

Answer 1

Maybe thinking of the opposite case might you.$$1\div3=1\times\frac{1}{3}$$ This can be re-expressed as:$$1\div\frac{3}{1}=1\times\frac{1}{3}$$

Answer 2

Note that

$$1 \div \frac{1}{3} = \frac{1}{\frac{1}{3}} = \frac{1}{\frac{1}{3}}\times 1 = \frac{1}{\frac{1}{3}}\times \frac{3}{3} = \frac{1\times 3}{\frac{1}{3}\times 3} = \frac{3}{1} = 3.$$

More generally, you can use the above manipulations to show $\dfrac{a}{b}\div\dfrac{c}{d} = \dfrac{ad}{bc}$.

Answer 3

Division can (should?) be understood in terms of multiplication. That is, if $a/b = c$, that's because $a=b \cdot c$.

So if $b = \dfrac{1}{p}$, we can solve for $c$ in $a = b \cdot c$ by multiplying through by $\dfrac{p}{1}$ ...