Let's use a system of equations to solve the problem. Let x be the number of dimes and y be the number of nickels.
From the problem, we know that:
The total value of the coins is $5.20, or 520 cents: 10x + 5y = 520
Roy has more nickels than dimes: y > x
To solve for x and y, we can use substitution. Rearrange the first equation to solve for x:
10x + 5y = 520
10x = 520 - 5y
x = 52 - (1/2)y
Substitute this expression for x into the second equation:
y > x
y > 52 - (1/2)y
Solve for y:
y + (1/2)y > 52
(3/2)y > 52
y > 34.67
Since y must be a whole number, the smallest possible value for y is 35.
Substitute y = 35 into the expression for x:
x = 52 - (1/2)y
x = 52 - (1/2)(35)
x = 34.5
Since x must be a whole number, the closest integer value for x is 35.
Therefore, Roy has 35 dimes and 35 + k nickels, where k is some positive integer.
To check, we can calculate the total value of the coins:
Total value = 10x + 5y
Total value = 10(35) + 5(35 + k)
Total value = 350 + 175 + 5k
Total value = 525 + 5k
This value is equal to 520 cents, so the solution is correct.
In summary, Roy has 35 dimes and 40 nickels