How would you convert the repeating nondeterminating decimal into a fraction? 0.151515...

Answer 1

0.15151515... =

15 / 100 + 15 / 10000 + 15 / 1000000 + 15 / 100000000 + ... =

15 / 100 + 15 / 100 ^ 2 + 15 / 100 ^ 3 + 15 / 100 ^ 4 + ...

This is the geometric series :

a + a r + a r ^ 2 + a r ^ 3 + a r ^ 4 + ...

with a = 15 / 100 and r = 1 / 100

As n goes to infinity the sum of the geometric series are :

S = a / ( 1 - r )

In this case :

a = 15 / 100 , r = 1 / 100 so :

S = a / ( 1 - r )

S = ( 15 / 100 ) / ( 1 - 1 / 100 ) =

( 15 /100 ) / ( 100 / 100 - 1 / 100 ) =

( 15 / 100 ) / ( 99 / 100 ) =

15 * 100 / ( 99 * 100 ) = 15 / 99

0.15151515... = 15 / 99

Answer 2

0.15151515... = 15 / 99 = 3 * 5 / ( 3 * 33 ) = 5 / 33