difficult decimal to fractions - repeating

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

Example 1

Change `0.1dot6` to a fraction.

Take the non-repeating part out of the number.

`0.1dot6 = 0.1 + 0.0dot6`

       `= 0.1 + (0.0dot6)/10`

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

`= 1/10 + 6/90`

`= (9+6)/90`

`=15/90`

`15 ÷ 15 = 1`

`90 ÷ 15 = 6`

`= 1/6`

`0.1dot6` as a fraction `=1/6`

Example 2

Change `0.dot23dot4` into a fraction.

The way to do this is to say:

`x = 0.dot23dot4` (where `x` is the fraction equivalent)

Multiply both sides by `1000`

`1000 x = 0.dot23dot4  \text(x)  1000`

`1000 x = 234.dot23dot4`

Subtract `x = 0.dot23dot4` from both sides

`1000 x - x = 234.dot23dot4 - 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`