Repeating New Year

It's a recurring decimal, you can take all of the digits into consideration, it's only when the decimal is either terminating (limited number of decimal places) or irrational (no pattern to an infinite number of decimal places) that you need to take into account the number of digits you need to be working with.

For recurring decimals they can be represented by sums, for instance $0.1\overline{6}$ (otherwise known as $\frac{1}{6}$ ) can be written as:

$0.1\overline{6} = 0.1 + 0.06 + 0.006 + 0.0006 + \cdots$

Ignoring the $0.1$ it's easy to see the pattern the sum is taking and from that we can find the fraction that sum represents.

$0.06 + 0.006 + 0.0006 + \cdots = \frac{6}{100} + \frac{6}{1000} + \frac{6}{10000} + \cdots$

First we multiply by $10$ , we'll call the original sum $x$ for now

$10x = \frac{6}{10} + \frac{6}{100} + \frac{6}{1000} + \frac{6}{10000} + \cdots$

We know the righthand side of the equation contains $x$ due to the simple fact that $x$ is an infinite sum.

$10x = \frac{6}{10} + x$

From here we subtract $x$ and divide by $9$

$9x = \frac{6}{10}$

$x = \frac{6}{90}$

From here we just simplify

$x = \frac{3}{45}$

$x = \frac{1}{15}$

This isn't $\frac{1}{6}$ of course but that's only because we haven't added the $0.1$ back in from when we took it out earlier

$0.1\overline{6} = 0.1 + \frac{1}{15}$

$0.1\overline{6} = \frac{1}{10} + \frac{1}{15}$

$0.1\overline{6} = \frac{15}{150} + \frac{10}{150}$

$0.1\overline{6} = \frac{25}{150}$

Finally we simplify

$0.1\overline{6} = \frac{5}{30}$

$0.1\overline{6} = \frac{1}{6}$

And there we have it.

If you get stuck on questions like these in the future try and remember this method as it works for every recurring decimal you might come across.

If you need anything else explaining just respond below this comment and I'll try to respond back, I'm not as active on here as I used to be so it might be a while before you get another response.

Hope I helped!