There are a total of 81 coins that equal $5.75. There are only dimes and nickels in the container. How many dimes are there? How many nickels are there?

X dimes.

81-x nickels.

0.1x + 0.05(81-x) = 5.75
Multiply both sides by 100:
10x + 5(81-x) = 575.
x = ?

To find the number of dimes and nickels, we can set up a system of equations. Let's denote the number of dimes as 'D' and the number of nickels as 'N'.

We know that there are 81 coins in total, so we have one equation:
D + N = 81

We also know that the total value of the coins is $5.75. Each dime is worth $0.10, and each nickel is worth $0.05. So, the equation for the total value is:
0.10D + 0.05N = 5.75

Now, we can solve this system of equations to find the values of D and N.

One way to solve this system of equations is by substitution. Solve the first equation for D and substitute it into the second equation:

D = 81 - N

0.10(81 - N) + 0.05N = 5.75

8.10 - 0.10N + 0.05N = 5.75

Combine like terms:

-0.05N = 5.75 - 8.10

-0.05N = -2.35

Divide both sides by -0.05:

N = -2.35 / -0.05

N = 47

Now that we have the value for N, we can substitute it back into the first equation to find D:

D + 47 = 81

D = 81 - 47

D = 34

Therefore, there are 34 dimes and 47 nickels in the container.