We have two equations to solve this problem.

Let's call the number of dimes "d" and the number of quarters "q"

\(d+q=100\)

\(0.1d+0.25q=21.4\)

\(0.1(100-q)+0.25q=21.4\)

\(10-0.1q+0.25q=21.4\)

\(0.15q+10=21.4\)

\(0.15q=11.4\)

\(q=76\)

\(d=24\)

There are 76 quarters and 24 dimes.