Short Answer 
The problem involves defining variables for nickels, dimes, and quarters and setting up equations based on their quantities and values. After substituting and rearranging the equations, the solution reveals there are 7 nickels, 11 dimes, and 7 quarters.
Step 1: Define the Variables
Start by assigning variables for each type of coin. Let N represent the number of nickels, D the number of dimes, and Q for quarters. You will then set up three equations based on the information provided:
- The total number of coins: N + D + Q = 25
- The total value of the coins: 0.05N + 0.10D + 0.25Q = 3.20
- The relation between dimes and nickels: D = N + 4
Step 2: Substitute and Rearrange the Equations
Use the relationship from the third equation to simplify the first two equations. Substitute D in the first equation:
- Replace D in N + D + Q = 25:
N + (N + 4) + Q = 25 - Also substitute in the value equation:
0.05N + 0.10(N + 4) + 0.25Q = 3.20
Step 3: Solve the Equations
Now, simplify and solve the equations you have formed. Combine like terms and isolate the variables:
- From the first equation, simplify to 2N + Q = 21.
- From the value equation, simplify to find N and substitute to find Q.
- Upon solving, you will find that N = 7 (nickels), D = 11 (dimes), and Q = 7 (quarters).