# How to work out this easy fraction?

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

The question is:

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

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

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

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?

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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.

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Thanks, that makes sense. – PerfectNutter Dec 16 '13 at 20:29
This can also be remembered by the saying "Keep, Change, Flip", where you keep the fraction in the numerator, change the division to multiplication, and then flip the fraction in the denominator. – Bob Dec 17 '13 at 6:33

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.

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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 $5$.

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Though if you want a reduced answer, it's better to leave the fraction product form factored and look for possible cancellations before multiplying it out, isn't it? – MPW Dec 16 '13 at 20:56
@MPW: I was starting from where OP left off. Doing it on my own, I would follow Eckhard's approach – Ross Millikan Dec 16 '13 at 21:05

5/2 * 9/5

Multiply by what you're dividing by's reciprocal.

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