Find the fractional representation $p/q$...

Question

Been trying to get some sort of solution for this for hours now, with no avail.

Find the fractional representation $p/q$, with $p \in \mathbb{N}$ and $q \in \mathbb{N}$, of the rational number whose decimal representation is:

$22.521111...$

Any help would be greatly appreciated.

Thanks

Answer 1

let $x = 22.52111111\cdots,$ then $10x = 225.2111111\cdots$ subtracting one from the other gives you $9x = 202.69$ and $$x = \frac{202.69}{9} = \frac{20269}{900}. $$

Answer 2

The repeating decimal is broken down to: $$22.521111...=22.52+0.001111...=\frac{2252}{100}+\frac{1}{900}$$ Can you add up the fractions?

Answer 3

Here is the general method: $$22.5211111\dots=\frac{2252}{100}+\frac1{100}\times 0.11111\dots$$ Now $\enspace 0.11111=\displaystyle\sum_{k=1}^{\infty}\frac1{10^k}=\frac{\cfrac1{10}}{1-\cfrac1{10}}=\frac19$, so

$$22.5211111\dots=\frac{2252}{100}+\frac1{100}\frac19=\frac{9\cdot2252+1}{900}=\frac{20\,269}{900},$$ which is an irreducible fraction.

High school version:

Set $x=0.11111\dots$. Then $10x=1.11111\dots=1+0.11111\dots=1+x$, so $9x=1$.