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

Jun 16, 2018

See explanation.

#### Explanation:

The fraction $0. \overline{27}$ can be written as an infinite sum:

$0. \overline{27} = 0.27 + 0.0027 + 0.000027 + \ldots$

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:

$S = {a}_{1} / \left(1 - r\right) = \frac{0.27}{1 - 0.01} = \frac{0.27}{0.99} = \frac{27}{99} = \frac{3}{11}$

So we can say that: