Question

Use geometric series to find the fraction of 0.8967898989

Answer 1

I'm guessing the repeating part is 89 at the end, so that

`x=0.8967\overline89\implies10^4x=8967.\overline{89}`

Then

`10^4x=8967+\displaystyle89\sum_(i=1)^\infty\frac1{100^i}`

`10^4x=8967+89\left(\frac1{1-\frac1{100}}-1\right)`

`10^4x=8967+(89)/(99)`

`x=(8967)/(10^4)+(89)/(99\cdot10^4)`

`x=(443911)/(495000)`

###

An arguably quicker way without using geometric series:

`10^4x=8967.\overline{89}`

`10^6x=896789.\overline{89}`

`10^6x-10^4x=887822`

`x=(887822)/(10^6-10^4)=(443911)/(495000)`