How to work out this easy fraction?

Question

I need help working out this fraction, I know it seems quite easy but I'm a bit stuck.

The question is:

$$\frac{\frac {5}{2}}{\frac{5}{9}}$$

My attempt was changing the denominators by multiplying by $2$ to make $18$ and then changing the same way the nominators:

$$\frac{\frac {45}{18}}{\frac{10}{18}}$$

The thing is that I'm not sure how to simplify it as no numbers go into all of them. Thank you, I know this is quite easy.

Question: Why is it considered a division if the easiest solution is by multiplication?

Answer 1

Remember that dividing by a fraction is the same as multiplying by its inverse.

In this case

$$ \frac52 \Big/ \frac59 = \frac52 \times \frac95 = \frac92 $$ because the $5$ cancels.

Answer 2

From $\dfrac {\ \frac {45}{18}\ }{\frac {10}{18}}$ you can multiply by $\frac {18}{18}$ to get $\frac {45}{10}$, then (if you want) divide out the common factor $5$.

Answer 3

Let's derive the answer, assuming that rules for such are completely unknown. We will use only the basic fraction definition $\ x = \dfrac{a}b \color{#c00}\iff b\, x\, =\, a\,$ and standard fraction arithmetic. We have

$$\ \ \ x = \dfrac{5/2}{5/9}\ \overset{\times\, 5/9}{\color{#c00}\Rightarrow}\ \dfrac{5x}9 = \dfrac{5}2\overset{\,\times\,1/5}\Rightarrow \dfrac{x}9 = \dfrac{1}2\ \overset{\times 9}\Rightarrow\ x = \dfrac{9}2$$

In the same way, using only the definition of a fraction and fraction arithmetic, you can easily deduce the rules used in the other answers, as well as other handy fraction rules.

Remark $\ $ Notice how the above works. By using the fraction definition, we have eliminated the nested fraction (fraction of fractions), yielding an equation involving only unnested fractions - to which known fraction arithmetic applies. The same technique works generally: to grok hairy composite objects, it often helps to unwind the definitions of the hairy objects, i.e. replace them by the simpler constituent objects that define them.