Two ways to do it.

1. Without algebra:

Suppose Frank started out with all 25 coins being nickels.  He'd 
then have only $1.25.  But Frank has 60 cents more than that, so 
he would need 12 more nickels, so he'd need to trade 12 of those 
nickels for dimes.  Therefore the answer is 12 dimes, and so the 
other 13 coins are nickels.

2.  With algebra:

        N + D = 25
0.05N + 0.10D = 1.85 

Multiply the second one through by 100

        N +   D =  25 
       5N + 10D = 185

Solve that system by substitution or elimination, and you'll
get N=13, D=12.

Edwin