Question

Write 2.3181818... as a mixed number

Answer 1

let's take a peek at 2.3181818.... and let's firstly move the 3 to the left-side

`2.3\overline{1818}\implies \cfrac{23.\overline{1818}}{10}\implies \cfrac{23 + 0.\overline{1818}}{10}`

now, let's take a peek at the 0.1818... part, make it a variable and move the repeating part to the left-side.

`x=0.\overline{1818}\hspace{5em} \begin{array}{llll} 100x&=&18.\overline{1818}\\ &&18+0.\overline{1818}\\ &&18 + x\\\\ 100x&=&18+x\\\\ 99x&=&18\\\\ x&=&\cfrac{18}{99}\\\\ x&=&\cfrac{2}{11} \end{array}`

now, let's plug that "x" value in the fraction above.

`\cfrac{23 + 0.\overline{1818}}{10}\implies \cfrac{23 + x}{10}\implies \cfrac{23 + (2)/(11)}{10}\implies \cfrac{ ~~ (253+2)/(11) ~~ }{10} \\\\\\ \cfrac{ ~~ (255)/(11) ~~ }{(10)/(1)}\implies \cfrac{255}{11}\cdot \cfrac{1}{10}\implies \cfrac{255}{110}\implies \cfrac{51}{22}\implies 2(7)/(22)`