difficult decimal to fractions - repeating

Here are some examples of converting decimal to fractions - repeating, but are difficult.

Example 1

Change `0.1\dot{6}` to a fraction.

Take the non-repeating part out of the number.

`0.1\dot{6} = 0.1 + 0.0\dot{6}`

       `= 0.1 + (0.0\dot{6})/10`

We know that `0.dot6` repeating is = `6/9` and `0.0\dot{6}` repeating is `6/90`

`= 1/10 + 6/90`

`= (9+6)/90`

`=15/90`

`15 ÷ 15 = 1`

`90 ÷ 15 = 6`

`= 1/6`

`0.1\dot{6}` as a fraction `=1/6`

Example 2

Change `0.\dot{2}3\dot{4}` into a fraction.

The way to do this is to say:

`x = 0.\dot{2}3\dot{4}` (where `x` is the fraction equivalent)

Multiply both sides by `1000`

`1000 x = 0.\dot{2}3\dot{4}  \text(x)  1000`

`1000 x = 234.\dot{2}3\dot{4}`

Subtract `x = 0.\dot{2}3\dot{4}` from both sides

`1000 x - x = 234.\dot{2}3\dot{4} - x`

The repeating part cancels.

`1000 x - x = 234`

`999 x =234`

`x = 234/999`

Simplify

`234 ÷ 3 = 78`

`999 ÷ 3 = 333`

`x = 78/333`

`78 ÷ 3 = 26`

`333 ÷ 3 = 11`

`x = 26/111`