jmc

The prime factorization of $60$ is $60=2^2\cdot 3\cdot 5$. This means that the odd positive divisors of $60$ are the divisors of $3\cdot 5=15$ which are $1$, $3$, $5$, and $15$. The sum of these numbers is $1+3+5+15=\boxed{24}$.