Hi Emma,

I am going to try to explain using the repeating decimal 0.15151515...

Write 0.15151515... as

15/100 + 15/10000 + 15/1000000 + 15/100000000 + ...
=15/100 + 15/100² + 15/100³ + 15/100⁴ + ...

This is the geometric series

a + ar + ar² + ar³ + ar⁴ + ...

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

I don't know if you have seen geometric series in your algebra classes yet but if you haven't you will see them soon. In a response to a previous question Claude and I showed that if |r| < 1 then the series

a + ar + ar² + ar³ + ar⁴ + ...

approaches a/(1 - r) as the number of terms approaches infinity. In the case of 0.15151515.. this means that

15/100 + 15/10000 + 15/1000000 + 15/100000000 + ...
=15/100 + 15/100² + 15/100³ + 15/100⁴ + ...
=(15/100)/(1 -1/100) = 0.15/99
=(15/100)/(99/100)
=15/99

A similar argument works for all the examples you sent.

Penny