There are two unknown values:
Let d represent the number of dimes.
Let q represent the number of quarters.
Since there are two unknown values, we need two equations.
Equation #1: value of the coins
Knowing that a dime is worth $0.10 and a quarter is worth $0.25, we can set up the equation:
0.10d + 0.25q = 22.50
Equation #2: number of coins
We are told that there are 15 more dimes than quarters:
d = q + 15
This gives us a system to solve.
0.10d + 0.25q = 22.50
d = q + 15
Since the second equation is solved for d, we can use substitution. Substitute q+15 in for d in the first equation:
0.10(q+15) + 0.25q = 22.50
Then solve as usual: distribute, combine like terms, isolate the d-term, solve for d.
0.10q + 1.50 + 0.25q = 22.50 (distribute)
0.35q + 1.50 = 22.50 (combine like terms)
0.35q = 21.00 (subtract 1.50)
q = 60 (divide by 0.35)
Now substitute 60 back in for q in the second equation:
d = 60 + 15 = 75
Make sure to check:
0.10(75) + 0.25(60) = 7.50 + 15 = 22.50
There are 60 quarters and 75 dimes.
