# Thread: Series with repeating decimal

1. ## Series with repeating decimal

10.9090909091.....

I don't really know what to do with this problem, I know that I should set it up like this:

$\displaystyle \frac{1090}{100} + (\frac {1090}{100}* \frac {1}{100})+ (\frac {1090}{100}*(\frac{1}{100})^2)....$

But whats next?

So A=1090/100 and R=1/100?

2. Originally Posted by redman223
10.9090909091.....

I don't really know what to do with this problem, I know that I should set it up like this:

$\displaystyle \frac{1090}{100} + (\frac {1090}{100}* \frac {1}{100})+ (\frac {1090}{100}*(\frac{1}{100})^2)....$

But whats next?

So A=1090/100 and R=1/100?
Here's a short cut:

S = 10.909090.....

100S = 1090.9090.....

Therefore 100S - S = 1080 => 99S = 1080 => S = 1080/99.

Otherwise, what you do next is sum the infinite geometric series.

3. Will that always work? Do you basically just take the original repeating number s, multiply it by 100, subtract s from it to make it 99s and then divide by 99?

Also, the number was 10.9090909091, the 1 at the end is throwing me off, because I don't know if that trick still applies or not.

4. The trick is to count how many repeating decimals you have,

For instance in

$\displaystyle x=0.94444$

Muliply both sides of this equation by $\displaystyle 10^n$, where n is the number of digits in the repeating string. Here, the single digit 4 repeats infinitely, so n = 1

$\displaystyle 10^1(x)=10^1(.9444444...)$

$\displaystyle 10x = 9.4444444...$

Subtract the original equation (x=0.9444444...) from the modified equation (10x = 9.444444...) which equals

9x= 8.5

$\displaystyle x= \frac{17}{18}$

5. Ok, that makes sense. But what about the number in my problem, I am confused by the way its written.

10.9090909091... I don't know what is repeated. Is it the 1? Is is the whole thing? The problem is that there is no teacher or anything to help me clarify that problem.

So I will have to do it multiple times until the computer accepts the answer.

So would I take 10.9090909091 and multiply it by what? By 1? Or by 10?

6. But you may be required to use a series solution.
$\displaystyle \begin{array}{rcl} {10.90\overline {90} } & = & {10 + .9 + .009 + .00009 \cdots } \\ {} & = & {10 + 9\left( {10^{ - 1} } \right) + 9\left( {10^{ - 3} } \right) + \cdots } \\ {} & = & {10 + 9\sum\limits_{k = 1}^\infty {10^{ - (2k - 1)} } } \\ {} & = & {10 + 9\frac{{10^{ - 1} }} {{1 - 10^{ - 2} }}} \\ \end{array}$

7. I am still stuck on what to do, because of the 1 at the end of

10.9090909091

8. Originally Posted by redman223
I am still stuck on what to do, because of the 1 at the end of 10.9090909091
I think that is a misprint.
If not the the other methods do not work either.
If it is not a misprint then it is a pointless question.

9. I think so, too, or they rounded the number by accident

10. Express as a rational number, in the form
where and are positive integers with no common factors.

This is how the problem is written out on webwork (I despise webwork, I wish our school would stop using it). I tried using 1080 and 99 but those didn't work.

11. $\displaystyle 10 + \frac{90}{100} + \frac{90}{10000} + \frac{90}{1000000} + \frac{90}{100000000} +\frac{91}{10000000000} = \frac{109090909091}{10000000000}$

12. I tried that earlier and it didn't work. I am stumped.

13. Originally Posted by redman223
I tried that earlier and it didn't work. I am stumped.
Hmm so am i

14. Originally Posted by redman223
I tried that earlier and it didn't work. I am stumped.
As Plato remarked earlier, it's probably an error. Can you check whether or not it's an error in the question.

Otherwise it is simple but tedious. And ridiculous (an observation also earlier made).

15. Originally Posted by redman223
Express as a rational number, in the form
where and are positive integers with no common factors.

This is how the problem is written out on webwork (I despise webwork, I wish our school would stop using it). I tried using 1080 and 99 but those didn't work.
If it's not a typo in the question here is an outline of the ridiculous solution:

10 + 0.909090909 + 0.0000000001111........ (I assume)

Obviously 0.909090909 = 909090909/1000000000

Let S = 0.00000000011111.....

Then 1000000000S = 0.1111111 and 10000000000S = 1.1111111

Therefore 9000000000S = 1 => S = 1/9000000000.

So your number is 10 + (909090909/1000000000) + (1/9000000000).

Now add the fractions in the usual way to get the required representation.