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leonard has 23 coins consisting of quarters and nickels worth $4.15. how many quarters and how many nickels does he have?

8 nickels and 15 quarters

Step-by-step explanation:

There are two unknowns in this problem (the number of quarters and the number of nickels Leonard has), so in order to generate algebraic expressions (equations) that can lead us to the answer, let's assign to them letters:

N to represent the number of Nickels (each worth 5 cents or $0.05)

Q to represent the number of quarters (each worth 25 cents or $0.25)

The first equation we can create is associated with the first piece of information: "Leonard has 23 coins"

We can write then that the number of nickels he has plus the number of quarters, should total 23. This in algebraic form is given by:

N + Q = 23

which is the first equation we have created.

Now to the second equation that takes in consideration the second piece of information about the total value of the collection. If he has N nickels each with a value of $0.05, then the amount coming from the nickels would be the product of the number of nickels (N) times $0.05: "0.05 N"

Similarly, we consider the value that the "Q" quarters (each worth $0.25) contribute to the collection:  Q times $0.25: "0.25 Q"

So the total value of the collection ($4.15) must be the amount from the nickels plus the amount from the quarters:

0.05 N + 0.25 Q = 4.15

This is our second equation.

Now to solve this system of two equations we created, we solve for one of the unknowns in terms of the other in the first equation which is the simplest one (let's say we solve for N):

N + Q = 23  then N = 23 - Q

We use this "substitution" for N writing "23- Q" instead of N in the second equation that involves the value:

0.05 (23 - Q) + 0.25 Q = 4.15

Now operate to remove the parenthesis:

1.15 - 0.05 Q + 0.25 Q = 4.15

Now combine the terms in "Q":

1.15 + 0.20 Q = 4.15

Now isolate the term in Q on one side of the equal sign:

0.20 Q = 4.15 - 1.15

0.20 Q = 3.0

And finally solve for Q by dividing both sides by 0.20:

Q = 3.0 / 0.20

Q = 15

So there are 15 quarters in the collection.

Finding the number of nickels is simple by using that the addition of quarters and nickels should be 23, which leads us to using the substitution for N that we did earlier:

N = 23 - Q

N = 23 - 15

N = 8

So there are 8 nickels in the collection