Let's call the number of dimes Luke has "D" and the number of quarters "Q."
We can set up a system of equations to represent the given information:
1. D + Q = 21 (because Luke has 21 coins in total).
2. 10D + 25Q = 390 (because the value of dimes is 10 cents each, and the value of quarters is 25 cents each, and the total value is $3.90, which is 390 cents).
Now, you can use these equations to solve for D and Q. You can start by solving the first equation for one of the variables. Let's solve it for D:
D = 21 - Q
Now, substitute this expression for D into the second equation:
10(21 - Q) + 25Q = 390
Now, distribute the 10 on the left side:
210 - 10Q + 25Q = 390
Combine like terms:
15Q = 390 - 210
15Q = 180
Now, divide by 15 to solve for Q:
Q = 180 / 15
Q = 12
Now that we know Q (the number of quarters) is 12, we can find D (the number of dimes) using the first equation:
D + 12 = 21
Subtract 12 from both sides:
D = 21 - 12
D = 9
So, Luke has 9 dimes and 12 quarters.