Question

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

How do I express `0.dot6dot9` in the form of `p/q` where p and q are integers?

Your help is greatly appreciated!

Answer 1

I'm assuming it is 0.69 recurring

Explanation:

Let `x=0.6969...`

So `100x=69.6969...`

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

`99x=69`

Divide by 99

`x=69/99`

Divide by 3

`x=23/33`

Answer 2

`0.dot6dot9=23/33`

Explanation:

Let `0.dot6dot9=0.6969696969....................=x`

Then `100x=69.6969696969..............................`

Subtracting former from latter we get

`99x=69`

Hence, `x=69/99=23/33`

i.e. `0.dot6dot9=23/33`

Answer 3

`0.6969....=23/33`

Explanation:

Let

`x=0.69696969............\ \quad (1)`

`100x=69.69696969............\ \quad (2)`

subtracting (1) from (2) as follows

`100x-x=(69.696969...)-(0.696969....)`

`99x=69`

`x=69/99`

`x=23/33`