1. ## Recurring decimals

Could someone please remind me how to convert a recurring decimal into a fraction

1) 0.8888888888888....
2) 0.1888888888888....
3) 0.656565656565....
4) 0.8333333333333...
5) 0.369369369369...
6) 0.416666666666...

2. Let $N=0.416\overline{6}$

$100N=41.6\overline{6}$

$1000N=416.6\overline{6}$

$900N=375$

3. Oh I understand but how would it work with number 3?

Let N = 0.65656565
100N = 65.656565
1000N = 656.565
900N = ?

4. Hello, Natasha1!

I'll do a few of them . . . It should trigger a memory.

Please remind me how to convert a recurring decimal into a fraction.

$1)\;0.8888\hdots$

$\begin{array}{cccccc}\text{We have:} & N &=& 0.8888\hdots & [1] \\
\text{Multiply by 10:} & 10N &=& 8.8888\hdots & [2] \end{array}$

$\text{Subtract [2] - [1]: }\;9N \:=\:8 \quad\Rightarrow\quad N \:=\:\dfrac{8}{9}$

$3)\;0.656565\hdots$

$\begin{array}{ccccc}\text{We have:} & N &=& \;\;0.656565\hdots & [1] \\
\text{Multiply 100:} & 100N &=& 65.656565\hdots & [2] \end{array}$

$\text{Subtract [2] - [1]: }\;99N \:=\:65 \quad\Rightarrow\quad N \;=\;\dfrac{65}{99}$

$6)\;0.41666\hdots$

$\begin{array}{cccccc}
\text{We have:} & N &=& \qquad 0.41666\hdots & [1] \\ \\[-3mm]
\text{Multiply by 100:} & 100N &=& \;\;41.666\hdots & [2] \\
\text{Multiply by 1000:} & 1000N &=& 416.666\hdots & [3] \end{array}$

$\text{Subtract [3] - [2]: }\;900N \:=\:375 \quad\Rightarrow\quad N \:=\:\dfrac{375}{900} \:=\:\dfrac{5}{24}$

5. Notice that $99N=65$.

6. ah yes brilliant!! Thank you