How to do this using series. Rep as ratio of two integers

[Jbreezy]

Homework Statement

Express the number as a ratio of integers.
10.1(35) = 10.135353535 the part in the left in () is where is is over lined to indicate it is repeating

Homework Equations

Geometric series

The Attempt at a Solution

10.1(35) = 10.1 + .035 = (101)/ (1000) + (35/10^3 + 35/10^6...)

= 101/1000 + ((35/1000)/(1-(1000)) = .1360350
Clearly wrong. What did I mess up

 

[Mark44]

Homework Statement

express the number as a ratio of integers.
10.1(35) = 10.135353535 the part in the left in () is where is is over lined to indicate it is repeating

Homework Equations

geometric series

The Attempt at a Solution

10.1(35) = 10.1 + .035 = (101)/ (1000) + (35/10^3 + 35/10^6...)

= 101/1000 + ((35/1000)/(1-(1000)) = .1360350
clearly wrong. What did i mess up

10.1 ≠ 101/1000.

 

[Mark44]

There's another way to go at this, as well.
Let S = 10.1353535...

Then 1000S = 10135.353535...
and 10S = 101.353535...

Subtract the 2nd equation from the first and solve for S.

 

[Jbreezy]

Yeah that is actually what I did but I did it different. I said that it was Because I thought that you had to move the decimal to the end of the first repeating number.So in my case I said 10.1(35) the I have to move it to after the 5 so I multiply by 1000. My two equations will read.

S = 10.135353535
1000x = 10135.353535
Do the subtraction and you get 999x = 10125 So I ended up with 375/37
Why did you have 10S I just had x

 

[Mark44]

Yeah that is actually what I did but I did it different. I said that it was Because I thought that you had to move the decimal to the end of the first repeating number.So in my case I said 10.1(35) the I have to move it to after the 5 so I multiply by 1000. My two equations will read.

S = 10.135353535
1000x = 10135.353535
Do the subtraction and you get 999x = 10125 So I ended up with 375/37

No you don't. 10135.353535... - 10.135353535... ≠ 10125

Why did you have 10S I just had x

So that the repeating part lines up in both numbers.

 

[Jbreezy]

1000S = 10135.353535...
10S = 101.353535..

Subtract

990S = 10034 ---> S = 5017/495

 

[Mark44]

That's more like it...

And of course you can check by doing the division.