Question

What is `0.75757575...` as a fraction?

Answer 1

`0.757575... = 25/33`

Explanation:

Note that `1/99 = 0.010101...`

So:

`0.757575... = 75*0.010101... = 75*1/99 = 75/99 = (25*color(red)(cancel(color(black)(3))))/(33*color(red)(cancel(color(black)(3)))) = 25/33`

Answer 2

`.7575757575...=25/33`

Explanation:

Another approach

`x=0.75757575........`

`100x=75 .color(red)(75757575...)`

`x = " " 0. color(red)(75757575....)`

subtract the two and the recurring decimals cancel, shown in red

`=>99x=75`

`:.x=75/99`

cancel with `3`

`=(cancel(3)xx25)/(cancel(3)xx33)`

`.7575757575...=25/33`

Answer 3

`0.75 = 75/99 = 25/33`

Explanation:

The complete explanation has been given elsewhere, but here is a nifty short cut.

If ALL the digits recur, write a fraction as:

`("the digits which recur")/("a 9 for each digit")`

`0.75757575.... = 0.dot7dot5 = 0.bar(75)`

Fraction = `75/99`

This simplies by `3` to give `25/33`

You can confirm with a calculator that `25/33 = 0.757575...`