• We’re currently investigating an issue related to the forum theme and styling that is impacting page layout and visual formatting. The problem has been identified, and we are actively working on a resolution. There is no impact to user data or functionality, this is strictly a front-end display issue. We’ll post an update once the fix has been deployed. Thanks for your patience while we get this sorted.

Math help...Expressing repeating decimals into fractions

pray4mojo

pray4mojo

Diamond Member
How do you write
1.060606...
1.4272727...
0.4999999...
as fractions?

I also came across expressing 0.9999.... as a fraction 😉
 
Originally posted by: pray4mojo
How do you write
1.060606...
1.4272727...
0.4999999...
as fractions?

I also came across expressing 0.9999.... as a fraction 😉
Click to expand...

Think geometric series 😉

-silver
 
It comes from a well-known result in series. If you have the series

1 1 1 1
x = - + -- + -- + ... + --
n a 2 3 n
a a a

as n increases, the sum gets closer to 1/(a-1). So as you
keep going through the digits of .111111..., which is

1 1 1 1
--- + --- + --- + --- + ...
10 2 3 4
10 10 10

your sum gets closer and closer to 1/(10-1) = 1/9. Since .5555 is 5
times that, the sum is 5 times (1/9) or 5/9.
 
Originally posted by: pray4mojo
that didn't make any more sense than my book did.
Click to expand...

lol, I know how to do this and I have no idea what fatbaby just said.

Ok, here you go.

Take .060606 repeating. Split it up into .06 + .0006 + .000006 + etc, etc These terms all have something in common (hint: divide a term by the previous one) and you will see there is a common ratio.

You can express this sum as a series (a geometric one actually). Isn't it convenient that a geometric series converges to a / (1-r) where a = the first term and r = the number to the nth power.

😉

-silver
 
Example: 2.838383838

2.838383838=2 + 83/100 + 83/100^2 + 83/100^3 + 83/100^4 +83/100^n

Sum (from 1 to infinity) of 83/100^n

This is a geometric series so a=83/100 and r=1/100

sum=a/(1-r)
=2+(83/100)/(1-(1/100))=281/99
 
A lowtech version 😉 :

x = 1.060606...

100x = 106.060606...
10000x = 10606.060606...

10000x - 100x = 10606.060606... - 106.060606... (notice how the decimal parts cancel each other out, which leaves you with an integer)
9900x = 10500
x = 10500 / 9900 = 35 / 33

------

x = 1.4272727...
10x = 14.272727... (the repeating part is 27, the 4 is not part of that).
1000x = 1427.272727...
100000x = 142727.272727...

100000x - 1000x = 142727.272727... - 1427.272727...
99000x = 141300
x = 141300 / 99000 = 157 / 110

Is the idea clear to you?
 
Originally posted by: Muzzan
x = 1.060606...

100x = 106.060606...
10000x = 10606.060606...

10000x - 100x = 10606.060606... - 106.060606... (notice how the decimal parts cancel each other out, which leaves you with an integer)
9900x = 10500
x = 10500 / 9900 = 35 / 33

------

x = 1.4272727...
10x = 14.272727... (the repeating part is 27, the 4 is not part of that).
1000x = 1427.272727...
100000x = 142727.272727...

100000x - 1000x = 142727.272727... - 1427.272727...
99000x = 141300
x = 141300 / 99000 = 157 / 110

Is the idea clear to you?
Click to expand...


your first ATOT post! and it was constructive...it's all down hill from here 🙂

 
Basically, if you don't have to show any work or anything, it's just the number divided by 10^n - 1 where n is the length of each repeating string.

For example, 1.060606...

would be 1 + 6/99

.354354354...

would be 354/999
 
You must log in or register to reply here.

TRENDING THREADS

Back
Top