Given that
10 dimes = 1 dollar, this is also known as 10 cents.
4 quarters = 1 dollar, this is also known as 25 cents
20 nickels = 1 dollar, this is also known as 5 cents
Let d represent dime
Let q represent quarter
Let n represent nickel
From the first statement,
[tex]d=3n+1[/tex]From the second statement,
[tex]q=5n[/tex]The total of the coins will be,
[tex]d+q+n=\text{ \$41.70}[/tex]Also,
[tex]0.1d+0.25q+0.05n=41.70[/tex]From the first two equations above,
[tex]\begin{gathered} 0.1(3n+1)+0.25(5n)+0.05n=41.70 \\ 0.3n+0.1+1.25n+0.05n=41.70 \\ 0.3n+1.25n+0.05n=41.70-0.1 \\ 1.6n=41.60 \\ \frac{1.6n}{1.6}=\frac{41.60}{1.6} \\ n=26 \\ \therefore n=\text{26} \end{gathered}[/tex]Let us now get the quantities of the remaining coins
[tex]\begin{gathered} d=(3n+1)=(3(26)+1)=(78+1)=79 \\ \therefore d=79 \end{gathered}[/tex][tex]\begin{gathered} q=5n=5(26)=130 \\ \therefore q=130 \end{gathered}[/tex]Hence, the numbers of each coins are
[tex]\begin{gathered} d=79=0.1(79)=\text{ \$7.9} \\ q=130=0.25(130)=\text{ \$32.5}0 \\ n=26=0.05(26)=\text{ \$1.30} \end{gathered}[/tex]