Question

What is 0.166 (repeating) as a fraction?

Answer 1

It can be written as `166/999`. See expanation for details.

Explanation:

The task is not complete because you did not indicate which part of the number is repeating. I solve it as if `166` was the period.

Note: to indicate the period of such decimals you can either put it in brackets: `0.(166)` or write a horizontal bar over the fraction's period: `0.bar(166)` without hashtag it would be 0.bar(166)

Solution

`0.bar(166)=0.166166166166...`, so it can be written as an infinite sum:

`0.bar(166)=0.166+0.000166+0.000000166+...`

From the last sum you can see that it is a sum of an infinite geometrical sequence, where: `a_1=0.166, q=0.001`

Since `q in (-1;1)` the sequence is convergent, so you can use the formula to calculate the sum:

`S=a_1/(1-q)`

`S=0.166/(1-0.001)`

`S=0.166/0.999`

Now we have to expand the fraction by 1000 to make both numerator and denominator integer numbers:

`S=166/999`