Question

Whats the answer to that question?

Answer 1

Answer:

-0.9090... can be written as $\frac{10}{11}$.

Explanation:

Any repeating decimal can be written as a fraction by dividing the section of the pattern to be repeated by 9's.

We can start by listing out

0.909090... = 9/10 + 0/100 + 9/1000 + 0/10000 + 9/100000 + 0/1000000 + ...

Now. we let this series be equal to x, that is

$x$ = 9/10 + 0/100 + 9/1000 + 0/10000 + 9/100000 + 0/1000000 + ...

Now, we'll multiply both sides by 100 .

$100x$ = 90 + 0 + 9/10 + 0/100 + 9/1000 + 0/10000 + ...

Then, subtract the 1st equation from the second like so:

$100x$ = 90 + 0 + 9/10 + 0/100 + 9/1000 + 0/10000 + 9/100000 + 0/1000000 + ...

$-x$ = - 9/10 - 0/100 - 9/1000 - 0/10000 - 9/100000 - 0/1000000 - ...

And we end up with this:

$99x=90$

Finally, we divide both sides by 99 in order to isolate x and get the fraction we're looking for.

$x=\frac{90}{99}$

Which can be reduced and simplified to

$x=\frac{10}{11}$

Hope this helps!