Answer:
-0.9090... can be written as [tex]\frac{10}{11}[/tex].
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
[tex]x[/tex] = 9/10 + 0/100 + 9/1000 + 0/10000 + 9/100000 + 0/1000000 + ...
Now, we'll multiply both sides by 100
.
[tex]100x[/tex] = 90 + 0 + 9/10 + 0/100 + 9/1000 + 0/10000 + ...
Then, subtract the 1st equation from the second like so:
[tex]100x[/tex] = 90 + 0 + 9/10 + 0/100 + 9/1000 + 0/10000 + 9/100000 + 0/1000000 + ...
[tex]-x[/tex] = - 9/10 - 0/100 - 9/1000 - 0/10000 - 9/100000 - 0/1000000 - ...
And we end up with this:
[tex]99x=90[/tex]
Finally, we divide both sides by 99 in order to isolate x and get the fraction we're looking for.
[tex]x=\frac{90}{99}[/tex]
Which can be reduced and simplified to
[tex]x=\frac{10}{11}[/tex]
Hope this helps!
