If she has N nickels then she has N+0.2N quarters=1.2N quarters. The total amount is $6.00 so:

0.05N+0.25×1.2N=6, 0.05N+0.3N=6, 0.35N=6, N=6/0.35=17.14. But this is not a whole number, so perhaps there were some dimes in the change. Suppose there were D dimes, value 0.1D, then 0.35N+0.1D=6.

Multiply through by 20: 7N+2D=120, and D=½(120-7N). Also 1.2N must be a whole number to give a whole number of quarters. N must also be even, so N must be a must be a multiple of 10. Let N=10 then there are 12 quarters and 25 dimes because D=½(120-70)=25. So we have 10 nickels, 12 quarters and 25 dimes=10×0.05+12×0.25+25×0.10=0.50+3.00+2.50=$6.00. Note that we can’t have N=20 because D would be negative so the only value for N=10. So she has 10 nickels. (She could have had some pennies in place of some of the dimes, but she has to have had 10 nickels so that the sum of nickels and quarters is 22 coins with a value of $3.50.)