Expert: Josh - 3/26/2010

Question
How do i convert a repeating decimal into a fraction?? For example...  .3533... and the last three has the bar over it and just continues! Please help me I'm going insane trying to figure it out?

Answer
Hi Brad,

The key to this is understanding the role of geometric series. Let me explain.

First, observe that 0.3533333333... may be written as 0.35 +0.003 +0.0003 +0.00003 + ....

The first term (non-recurring part) is simply 35/100. [#1]
The subsequent terms equals 3/1000 + 3/10000 + 3/100000 + ... and so forth.  [#2]
The important thing is to recognize that this actually forms a geometric series.

To illustrate this, note that each successive term in the sum of [#2] differ by a ratio of 1/10.
i.e., if we compare the 2nd term to the 1st term, 3/10000 is smaller than 3/1000 by a factor of 1/10. In fact, when we divide any term by the previous term, we always get a ratio of 1/10 (in this case). This is consistent with the definition of a geometric series, which follows the pattern {a, a*r, a*r^2, a*r^3,....}, where "a" is the value of the first term, "r" is the geometric ratio [r=1/10 in this question].

Using the formula for a geometric sum,
a + a*r + a*r^2 + a*r^3 +...
= a [1+r+r^2+r^3+...]
= a (1-r^n)/(1-r)   if we have n terms, when r<1.
For a recurring figure, it will comprise an infinite number of terms, so the formula becomes
a/(1-r) in the limit as n tends toward infinity.

Plugging in the numbers (a=3/1000 and r=1/10),
3/1000 + 3/10000 + 3/100000 + ...
= (3/1000)/[1-(1/10)]            [using the formula, geometric sum = a/(1-r)]
= (3/1000)/(9/10)
= (3/1000)*(10/9)
= 1/300   ....[#3]

Combining the expressions from [#1] and [#3], we get
0.3533333....
=0.35 + (0.003 +0.0003 +0.00003 +...)
=(35/100) + (1/300)
=(35/100)*(3/3) + (1/300)
=(35*3/300) + (1/300)
=(3*35 + 1)/300
= 106/300

Makes sense?
No need to pull your hair out, lol:P