Using Geometric Series to Solve Rewrite a Repeating Decimal as a Fraction

Whenever you are working with a decimal that repeats forever, that decimal could be written as an infiinte geometric sum.

For example: = 46.84444.... = 46.8 + 0.04 + 0.004 + 0.0004 + 0.00004 + ... since when you add these vertically, you get:

This equals:

The summation in the last part of this equation is is a geometric series with a first term of 4/100.  So its sum is . 

Now if you add this to the 46.8 you end up with: .

Any repeated decimal could be converted to a fraction this way.  The common ratio will always be either 0.1 or 0.01 or 0.001, etc depending on how many digits are being repeated.

Another way to solve this problem doesn't use the idea of geometric series at all.

All you need to do is set a variable equal to the repeating decimal:  x = 46.8444...

Then, multiply both sides of the equation by 10, 100, or 1000 etc, depending on how many digits are repeating.  In this example, only one digit is repeating so multiplying both sides by 10 will work.  We get 10x = 468.4444...

Now line these two equations up so we can subtract both sides and solve for x as a fraction.

As you can see, this method works just as well.