Question

How do you convert 0.27 (27 repeating) to a fraction?

Answer 1

See explanation.

Explanation:

The fraction `0.bar(27)` can be written as an infinite sum:

`0.bar(27)=0.27+0.0027+0.000027+...`

The right hand side is a sum of a geometric sequence in which `a_1=0.27` and `r=0.01`. In the sequence the ratio satisfies condition `|r|<1`, so it is convergent and the sum can be calculated as:

`S=a_1/(1-r)=0.27/(1-0.01)=0.27/0.99=27/99=3/11`

So we can say that:

`0.bar(27)=3/11`