Question

How would you convert a repeating decimal like 1.27 or 0.6 into a fraction?

Answer 1

The examples in your question are not examples of repeating decimals. A repeating decimal is one in which the same number is repeated infinitely. An example would be `0.666...`, also written as `0.bar6`, or `0.0909...`, also written as `0.bar09`.

`1.27=1 (27)/(100)`

`0.6=6/10=3/5`

Answer 2

The algorythm is given in the explanation.

Explanation:

The fractions you wrote in the questions are not repeating. You probably meant fractions like `1.(27)` or `0.(6)`.

If you want to sugest that a decimal fraction has a period, you have to put it (the period) in brackets. Other way of indicating a period may be a horizontal bar over the period as suggested in the previous answer. So you can write the repeating fraction as: `1.(27)` or `1.bar(27)`

To show how such decimals can be changed into a fraction let's look closer at the decimals. Let's take `1.(27)`. It can be written as:

`1.(27)=1+0.27+0.0027+0.000027+...`. So you can see, that the fraction can be written as `1` plus the sum of an infinite geometrical sequence, for which: `a_1=0.27`, and `q=0.01`.

Since `|q|<1` the sequence is convergent, so you can calculate the finite sum of all terms.

`S=a_1/(1-q)`

`S=0.27/(1-0.01)=0.27/0.99=27/99=3/11`

So now you can write, that `1.(27)=1+3/11=1 3/11`