To solve these problems, we can use a system of equations to represent the given information and find the unknowns. Let's tackle each problem separately.
1. Cindy's problem:
Let's represent the number of pennies and dimes with the variables 'p' and 'd', respectively.
Since Cindy had a total of 9 coins, we can form the equation p + d = 9. (Equation 1)
The value of her coins is $1.08, which can be written as 0.01p + 0.10d = 1.08. (Equation 2)
Finally, we know that she had the same number of pennies as dimes, so p = d. (Equation 3)
To solve this system of equations, we can substitute Equation 3 into Equation 1 and Equation 2.
Substituting p = d into Equation 1, we get d + d = 9, which simplifies to 2d = 9. Solving for d, we find d = 4.5.
Since coins cannot be fractions, we know that d must be a whole number. However, since it's not possible for Cindy to have 4.5 dimes, we can conclude that this system of equations has no solution.
2. Danielle's problem:
Let's represent the number of half dollars, quarters, dimes, nickels, and pennies with the variables 'h', 'q', 'd', 'n', and 'p', respectively.
Since Danielle had a total of 96 coins, we can form the equation h + q + d + n + p = 96. (Equation 4)
We also know that she had the same number of nickels as all the other coins combined, so n = h + q + d + p. (Equation 5)
To solve this system of equations, we can substitute Equation 5 into Equation 4.
Substituting n = h + q + d + p into Equation 4, we get h + q + d + (h + q + d + p) + p = 96, which simplifies to 2h + 2q + 2d + 2p = 96.
Dividing both sides of the equation by 2, we have h + q + d + p = 48. (Equation 6)
Now, we have a new system of equations:
Equation 4: h + q + d + n + p = 96
Equation 6: h + q + d + p = 48
To further solve this, we need additional information or another equation to directly solve for the values of h, q, d, n, and p. Without that information, this system of equations has no unique solution either.
In both cases, it appears that there is missing information or inconsistencies in the given problems.