prove that the recurring decimal 0.39 = 13/33 by the way the 0.39 is supposed to have a dot on 3 and a dot on 9

Question

prove that the recurring decimal 0.39 = 13/33 by the way the 0.39 is supposed to have a dot on 3 and a dot on 9

Answer 1

An alternative solution to the one that Mark provided is shown below.

Let S = 0.39393939........

100S = 39.393939......

 

Subtract the first equation from the second.

99S = 39

    S = 39/99 = 13/33

Answer 2

0.393939393939... = 0.39 + 0.0039 + 0.000039 + 0.00000039 + ...

 

The expression on the right side of the equal sign is a geometric series with common ratio r = 0.01. Since r is in the interval (-1,1), the sum of the series is given by the formula a₁/(1-r), where a₁ is the first term of the series.  So, we have:

 

0.3939393939393... = 0.39 + 0.0039 + 0.000039 + ...

 

                             = 0.39/(1-0.01)

 

                             = 0.39/0.99

 

                             = 39/99 = 13/33