Question

Question #65c13

Answer 1

`133/99`

Explanation:

Recall (Sum of Geometric Series Formula):

`a + ar + ar^2+ar^3+cdots=a/(1-r)` if `|r| < 1`

We can view a repeated decimal as the sum of a geometric series.

`1.343434...=1+[0.34+0.0034+0.000034+cdots]`

`=1+[34/100+34/100(1/100)+34/100(1/100)^2+cdots]`

By applying the formula above with `a=34/100` and `r=1/100` starting with the second term,

#=1+(34/100)/(1-1/100) =1+(34/100)/(99/100) =99/99+34/99=133/99#

Answer 2

`133/99`

Explanation:

Obtain 2 equations with the same repeating part then subtract them.

`"We can represent the repeated part by " 1.bar34`

`x=1.bar34to(1)larrcolor(red)" multiply by 100"`

`100x=134.bar34to(2)`

`"Subtract " (2)-(1)`

`(100x-x)=(134.bar34-1.bar34)`

`rArr99x=133larrcolor(red)" repeating part eliminated"`

`rArrx=133/99larrcolor(red)"in simplest form"`