A collection of nickels, dimes and pennies has an average value of 7 cents per coin. If a nickel were replaced by five pennies, the average would drop to 6 cents per coin. What is the number of dimes in the collection?

The number of dimes is:

Dimes = -(Nickels + Pennies) + 24.

Such that Nickels + Pennies < 24, and each quantity refers to the initial number of coins of the given type.

Step-by-step explanation:

Let's call N = number of nickels, D =  number of dimes, and  P = number of pennies.

The total number of coins is N + D + P

We know that the total value in nickels is:

N*$0.05

The total value in dimes is:

D*$0.10

And the total value on Pennies is:

P*$0.01

Then, the "mean" value of the coins will be equal to the total value of all the coins, divided the total number of coins.

$0.07 = (N*$0.05 + D*$0.10 + P*$0.01)/(N + D + P)

we can mutiply at both sides by the total number of coins, and we have:

$0.07*(N + D + P) = N*$0.05 + D*$0.10 + P*$0.01

Now, if we remove one nickel by five penies, we have:

$0.06 = ((N-1)*$0.05 + D*$0.10 + (P+5)*$0.01)/(N - 1 + D + P + 5)

Again, we multiply in both sides by the total number of coins and:

$0.06*(N - 1 + D + P + 5) = ((N-1)*$0.05 + D*$0.10 + (P+5)*$0.01)

Ok, now we have two equations:

$0.07*(N + D + P) = N*$0.05 + D*$0.10 + P*$0.01

$0.06*(N + D + P + 4) = ((N-1)*$0.05 + D*$0.10 + (P+5)*$0.01)

We can take the second equation and write the right side as:

$0.06*(N + D + P + 4) = ((N-1)*$0.05 + D*$0.10 + (P+5)*$0.01)

                                    = N*$0.05 + D*$0.10 + P*$0.01 - 1*$0.05 + 5*$0.01

                                    =  N*$0.05 + D*$0.10 + P*$0.01

Now, the two equations are:

$0.07*(N + D + P) = N*$0.05 + D*$0.10 + P*$0.01

$0.06*(N + D + P + 4) = N*$0.05 + D*$0.10 + P*$0.01

Then we have that:

$0.07*(N + D + P) = $0.06*(N + D + P + 4)

D*($0.07 - $0.06) = ($0.06 - $0.07)*(N + P)) + 4*$0.06

D*$0.01 = -$0.01(N + P) + 4*$0.06

D = (-$0.01(D + P) + 4*$0.06)/$0.01 = -(N + P) + 24

So the number of Dimes is related to the number of pennies and nickels that we have at the begginig.

Such that in both cases the number of dimes is equal:

D = -(N + P) + 24

Notice that N + P can not be larger or equal than 24, so we have the rule:

N + P < 24.

D

step-by-step explanation:

answer: the answer is ann