If you are asked to convert a repeating decimals to a fraction, you might draw a blank. You might be given a repeating decimal number that you know right off the bat what the fractional equivalent is, but just in case it is not one that you know of, here is one of the easy method that you can always use. In this example, we will derive a fraction for the decimal number of 0.232323232323..... You cannot estimate this decimal number and convert it to a fraction of 23/100, because you ended up rounding off by a lot of numbers after 0.23.
So what we need to do is to setup an equation such as
equation 1: x = 0.232323232323 (repeating)
Setup another equation (equation 2) similar to equation 1 which you will use later to subtract with.
So for equation 2, I will multiply equation 1 by 100.
equation 2 = 100 x equation 1
equation 2: 100x = 23.232323232323 (repeating)
If you subtract equation 1 from equation 2 (you will see that all the repeating decimals ended up disappearing), and then solve for x:
equation 2 - equation 1 =
100x - x = 23.232323232323 - 0.232323232323
99x = 23
x = 23/99
So to summarize, what we ended doing is to setup 2 equations. First equation is to set x to be equal to the initial repeating decimal number. Then we will multiply the first equation by some multiples of 10, so that we can subtract the equations and ending up eliminating the repeating decimals. Once all the repeating decimals have been eliminated, all you need to do is to solve for x, which provides you with the fractional equivalence of the repeating decimal number.
For our last example, we will find the fractional equivalent to the number of 0.00131313...
equation 1: x = 0.00131313 (repeating)
equation 2: 100x = 0.13131313
equation 2 - equation 1
100x - x = 0.13131313 - 0.00131313
99x = 0.13
9900x = 13
x = 13/9900
If you have any question regarding this type of problems, please feel free to reach out to me or any of the instructors in my center.
Michael Huang
Center Director
Mathnasium of Glen Rock/Ridgewood
T: 201-444-8020
E: glenrock@mathnasium.com
www.mathnasium.com/glenrock
