Let us assume number of quarters = q.
Let us assume number of nickels = n.
Total number of coins = 69.
We can setup first eqation,
Number of quarters + number of nickels = 69.
q + n = 69 -------------- equation(1).
Total value of all the coins = $9.05
We can setup second equation as,
0.25*number of quarters + 0.05*number of nickels - $9.05.
0.25q + 0.05n =9.05 ........................equation(2).
We got two equations of a system of equation.
Let us apply substitution mehod of solving system of linear equations.
Solving first equation q+n=69 for n.
Adding q on both sides of the equation, we get
q-q+n = 69-q.
n= 69-q.
We need substitute n=69-q in second equation 0.25q + 0.05n =9.05.
0.25q +0.05(69-q) = 9.05
Distributing 0.05 over (69-q).
0.25q +0.05*69 -0.05*q =9.05.
0.25q +3.45 -0.05q =9.05
Combining like terms 0.025q -0.05q, we get
0.20q +3.45 =9.05
Subtracting 3.45 from both sides.
0.20q +3.45-3.45 =9.05-3.45
0.20q = 5.6
Dividing both sides by 0.20
0.20q/0.20 = 5.6/0.20
q = 28.
Plugging q=28 in equation(1)
q+n =69 => 28 +n =69
Subtracting 28 from both sides, we get
28-28 +n =69-28.
n = 41.
Therefore, number of quarters 28 and number of nickels 41.
