Question

What is `1.9791666...` as a fraction ?

Answer 1

`95/48`

Explanation:

First let me introduce you to some notation in case you have not met it. When a decimal representation repeats, you can write it using a bar over the repeating digits to indicate that they repeat.

So for our example:

`1.979166666... = 1.9791bar(6)`

Departing from my usual method for such problems, first note that:

`2/3 = 0.bar(6)`

So let us try to simplify the given decimal, by mutiplying by `3` first.

`color(blue)(3)*1.9791bar(6) = 5.9375`

Since this ends with a digit `5`, multiply by `2` to find:

`color(blue)(2)*5.9375 = 11.875`

Since this ends with a digit `5`, multiply by `2` to find:

`color(blue)(2)*11/875 = 23.75`

Since this ends with a digit `5`, multiply by `2` to find:

`color(blue)(2)*23.75 = 47.5`

Since this ends with a digit `5`, multiply by `2` to find:

`color(blue)(2)*47.5 = 95`

Collecting the multipliers we used, we have:

`95 = 2*2*2*2*3*1.9791bar(6) = 48*1.9791bar(6)`

Dividing both ends by `48`, we find:

`1.9791bar(6) = 95/48`

Due to the way we found this, we have introduced no common factors, so this is automatically in lowest terms.