Let the number of dimes and nickels be D and N. D+N=30 because there are 30 coins.
Their combined value is 10D+5N=260 (working in cents).
Subtract 10D+5N=260 and we get 5N=40 so N=8 (worth 40c). There are 8 nickels.
And D=30-8=22 (worth 220c). Combined value is 220+40=260c=$2.60.