# What is 0.66666... as a fraction?

Oct 24, 2017

$\frac{6}{10}$ or $\frac{6}{99}$

#### Explanation:

I would say from your sentence the possible fractions is;

$\frac{6}{99}$

Which gives the repeated sequence of $6$

That's for geometric progression in finding the least fraction..

But in converting the decimal $0.6$ into fraction is following due process..

$0.6$, $\Rightarrow 6$ there is representing tenth, in other words its $\frac{6}{10}$

Hence,

$0.6 = \frac{6}{10} \to \text{As a fraction}$

But with $6$ as a repeating fraction is $\frac{6}{99}$

Oct 24, 2017

$0.66666 \ldots = \frac{2}{3}$

#### Explanation:

I think you intended $0.66666 \ldots$ which can be written $0. \overline{6}$

In which case see what happens when we multiply by $10$:

$10 \cdot 0.66666 \ldots = 6.66666 \ldots$

So if we subtract the original, we get an integer. That is:

$\left(10 - 1\right) \cdot 0.66666 \ldots . = 6.66666 \ldots - 0.66666 \ldots = 6$

Dividing both ends by $\left(10 - 1\right)$ we find:

$0.66666 \ldots = \frac{6}{10 - 1} = \frac{6}{9} = \frac{2}{3}$