Final answer:
To solve this problem, set up a system of equations to represent the number of dimes and quarters and their total value. Use elimination to find the values of x and y. The collection has 12 dimes and 20 quarters.
Step-by-step explanation:
To solve this problem, we can set up a system of equations with two variables. Let x represent the number of dimes and y represent the number of quarters.
We know that the total number of coins is 32, so we have the equation: x + y = 32.
We also know that the value of the coins is $6.20, which can be expressed as 0.10x + 0.25y = 6.20.
We can solve this system of equations using substitution or elimination to find the values of x and y.
Let's use elimination:
Multiply the first equation by -0.10 to get -0.10x - 0.10y = -3.20.
Add this equation to the second equation: -0.10x - 0.10y + 0.10x + 0.25y = -3.20 + 6.20.
Simplify and combine like terms: 0.15y = 3.00.
Divide both sides by 0.15: y = 20.
Substitute this value back into the first equation: x + 20 = 32.
Solve for x: x = 12.
Therefore, there are 12 dimes and 20 quarters in the collection.