Problem converting repeating decimals?

Question

Here's my problem. I have learned that to convert a repeating decimal into a fraction requires these steps:

Example: to convert the repeating decimal .333333333...

First I let x=.333333333 (that's as many as my calculator can show)

Next I let n=the number of repeating numbers. In this case it's 1.

Then I multiply both sides of the equations x=.333333333 by 10^n or 10^1 (which equals 100)

So the second equation is now 100x=33.333333333

I then subtract the first equation from the second to get:
99x=33

I then divide the 99 to get 33/99 which equals 1/3.

But now if I use the same process to try to find say .416666666 (which is 5/12)
x=.41666666
n=1
multiply 10^1 to both sides to get
100x=41.66666666 subtract the first equation to get
99x=41.249999999
Neither of these numbers is divisible by 5 or 12! Why doesn't this formula work? What am I doing wrong. Please show examples where n=1 as well as n=more than 1.

Thank you.

Answer 1

x=41.66666666
100x=41.6666666................(1)
1000x=416.66666................(2)
(2)-(1) 900x=375
x=375/900=25/60=5/12

Answer 2

Yes, you made many mistakes, and the relevant one is didn't read the pre-condition for the method to begin with.

I guess that the method works when all the decimal digits are form by some repeating pattern. However, this is not the case in .4166666...

So, before applying the method, change it to one that does, e.g. 41.666... or 416.666..., of course 41.666... will do, and the rest is in the first answer.

Personally, I don't like the description of the method, because it doesn't help you to get the idea behind the method, which is to remove the repeating decimal places.

For those who applied directly, and then calculate the answer, you miss the point. It is like using the right tool the improper way.

Another mistake: it is a contradiction to use calculator for this method!!!

Answer 3

First of all, 10^1=10, 10^2=100.

Using your steps:

x=.416666
n=1

10x=4.1666

9x=3.75

Answer 4

.416666666... starts repeating in the thousanths place. You need to multiply by 1000 instead of 100

Answer 5

Actually it does work...

so take x= .4166666....

then multiply by 100 gives

100x= 41.666666.....

subract away an x then gives

99x= 41.25 (you did some funky arithmetic here I think)

41.25/99= 5/12

Hope this helps