Finding the rational expression of a repeating decimal

[OnceKnown]

Homework Statement

Express the repeating decimal as a series, and find the rational number that it represents
1) 3.2\overline{394}

Homework Equations

Geometric Series (a/1-r)

The Attempt at a Solution

I tried putting the value into a series by having n=1 as it goes to infinity
Ʃ3.2 + 394(0.0001)ⁿ
but I have a feeling that is wrong. Is there anyway to just isolate the 0.0394 as the repeating value?

I also tried using (3.2 + 394) as "a", and (0.0001) as "r"

putting this into the geometric series formula to find the rational number I got

((16/5) +394)/(1-0.0001) which came out to be 1986/0.9999

The Correct answer for the rational expression is 16,181/4,995

but I don't know how to get there. I have a feeling that I chose my "a" and "r" incorrectly

 

[SteveL27]

Homework Statement

Express the repeating decimal as a series, and find the rational number that it represents
1) 3.2\overline{394}

Do you know the standard trick for converting repeating decimals? For example if I give you .123123123... where "123" is the repeating block, do you know an *EASY* way to immediately represent that repeating decimal as n/m? If yes, this is an application of that idea; if not ... well, there's a really EASY trick for this. Probably in your class notes somewhere.

(note -- of course if you don't know the trick, my post wasn't helpful ... but I can't think of a hint that only goes part of the way there.)

 

[tiny-tim]

Hi OnceKnown!

I also tried using (3.2 + 394) as "a", and (0.0001) as "r"

putting this into the geometric series formula to find the rational number …

No, just the 394 …

you can add the 3.2 later! ​

 

[OnceKnown]

Do you know the standard trick for converting repeating decimals? For example if I give you .123123123... where "123" is the repeating block, do you know an *EASY* way to immediately represent that repeating decimal as n/m? If yes, this is an application of that idea; if not ... well, there's a really EASY trick for this. Probably in your class notes somewhere.

(note -- of course if you don't know the trick, my post wasn't helpful ... but I can't think of a hint that only goes part of the way there.)

Hi Steve,

I wouldn't know about this "Easy trick" you're speaking of sorry lol. Thanks for the help though.

 

[OnceKnown]

Hi OnceKnown!


No, just the 394 …

you can add the 3.2 later! ​

Hi Tim,

so just the 394 as "a"

would represent 394/(1-0.0001), but that would bring me to 394/0.9999 still, which I'm still stuck.

 

[HallsofIvy]

Hi Steve,

I wouldn't know about this "Easy trick" you're speaking of sorry lol. Thanks for the help though.

It's pretty straightforward. The original number is x= 3.2394394394... so 10x= 32.394394..., the multiplication moving the decimal point one place. Multiplying by another 10^3= 1000 moves the decimal point another three places: 10000x= 32394.394394394...

Now subtract: 10000x- 10x= 9990x= 32362. The "decimal part" cancels because of that repetition.

 

[OnceKnown]

Thank you Halls,

This is different approach from what we are learning in calc class but it works.

 

[SteveL27]

Thank you Halls,

This is different approach from what we are learning in calc class but it works.

Once you know that trick you can shortcut it by just putting the repeating block over the same number of 9's. So .123123123... = 123/999, etc.