jmc

Recall that a simplified fraction has a terminating decimal representation if and only if the denominator is divisible by no primes other than 2 or 5.

The prime factorization for $30$ is $3\cdot 2\cdot 5$. Therefore, the decimal representation for $n/30$ terminates if and only if $n$ has a factor of 3 to cancel the 3 in the denominator. There are $9$ multiples of 3 less than 30 (namely $3(1),3(2)3(3),\dots ,3(9)$) so there are $29-9=\boxed{20}$ integers $n$ between 1 and 29 for which $n/30$ is a repeating decimal.