# How do I express 0.6969... in the form of p/q where p and q are integers?

## How do I express $0. \dot{6} \dot{9}$ in the form of $\frac{p}{q}$ where p and q are integers? Your help is greatly appreciated!

Jul 22, 2018

I'm assuming it is 0.69 recurring

#### Explanation:

Let $x = 0.6969 \ldots$

So $100 x = 69.6969 \ldots$

Subtract the first from the second and all the decimal numbers will disappear to leave

$99 x = 69$

Divide by 99

$x = \frac{69}{99}$

Divide by 3

$x = \frac{23}{33}$

Jul 22, 2018

$0. \dot{6} \dot{9} = \frac{23}{33}$

#### Explanation:

Let $0. \dot{6} \dot{9} = 0.6969696969 \ldots \ldots \ldots \ldots \ldots \ldots . . = x$

Then $100 x = 69.6969696969 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

Subtracting former from latter we get

$99 x = 69$

Hence, $x = \frac{69}{99} = \frac{23}{33}$

i.e. $0. \dot{6} \dot{9} = \frac{23}{33}$

$0.6969 \ldots . = \frac{23}{33}$

#### Explanation:

Let

$x = 0.69696969 \ldots \ldots \ldots \ldots \setminus \setminus \quad \left(1\right)$

$100 x = 69.69696969 \ldots \ldots \ldots \ldots \setminus \setminus \quad \left(2\right)$

subtracting (1) from (2) as follows

$100 x - x = \left(69.696969 \ldots\right) - \left(0.696969 \ldots .\right)$

$99 x = 69$

$x = \frac{69}{99}$

$x = \frac{23}{33}$