Write the following number as ratios of integers

We're going to "cut" the repeating part here in a few steps. First, we're going to put the number in a variable:

Next, to get rid of the negative, we can multiply either side by -1 to get

Now, we won't actually use this -x directly; instead, we want to create two new values, one by multiplying either side by 10:

and the other by multiplying either side by 1000:

Next, we can get rid of the repeated part of the number by subtracting -10x from -1000x:

And finally, we can divide either side of the equation by -990 to find that

both equations and proportions are better.