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

In summary, the conversation is about expressing a number as a ratio of integers, specifically the number 10.1353535... The attempted solution involved using geometric series, but resulted in an incorrect answer. The correct solution involved setting up two equations and solving for the repeating part of the number, resulting in the ratio 5017/495.
  • #1
Jbreezy
582
0

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
 
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  • #2
jbreezy said:

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.
 
  • #3
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.
 
  • #4
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
 
  • #5
Jbreezy said:
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
Jbreezy said:
Why did you have 10S I just had x
So that the repeating part lines up in both numbers.
 
  • #6
1000S = 10135.353535...
10S = 101.353535..

Subtract

990S = 10034 ---> S = 5017/495
 
  • #7
That's more like it...

And of course you can check by doing the division.
 

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

1. How do I represent a series as a ratio of two integers?

To represent a series as a ratio of two integers, you need to find the common difference between each term in the series. Then, the numerator of the ratio will be the first term in the series, and the denominator will be the common difference. For example, if the series is 2, 4, 6, 8, the ratio would be 2/2, or 1.

2. Why is it helpful to represent a series as a ratio of two integers?

Representing a series as a ratio of two integers can help make calculations and comparisons easier. It can also provide a simplified and more concise way of expressing a series.

3. How can I use the ratio of two integers to find the sum of a series?

To find the sum of a series using the ratio of two integers, you can use the formula S = n/2(a1 + an), where S is the sum, n is the number of terms, and a1 and an are the first and last terms in the series, respectively. This formula is derived from the arithmetic series formula.

4. Can a series always be represented as a ratio of two integers?

No, not all series can be represented as a ratio of two integers. Some series may have a pattern that cannot be expressed in a simplified ratio form. In these cases, other methods may be needed to represent the series.

5. How can I check if a series is represented correctly as a ratio of two integers?

To check if a series is represented correctly as a ratio of two integers, you can use the ratio to find the next term in the series. If the next term matches the actual next term in the series, then the ratio is correct. You can also use the ratio to find the sum of the series and compare it to the actual sum.

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