Question

write the following number as ratios of integers

Answer 1

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

$x=-2.0\overline{42}$

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

$-x=2.0\overline{42}$

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

$-10x=20.\overline{42}$

and the other by multiplying either side by 1000:

$-1000x=2042.\overline{42}$

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

$-1000x-(-10x)=2042.\overline{42}-20.\overline{42}\\ -1000x+10x=2022\\ -990x=2022$

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

$x= \frac{2022}{-990}=-\frac{2022}{990}\div \frac{6}{6}=-\frac{337}{165}$