Sammy has fifty coins in his pocket that add up to one dollar. how many coins of each denomination are in sammy's pocket?

Assuming some of each coin must be used, I get

1 quarter
2 dimes
2 nickels
45 pennies

Otherwise, you could also have just

2 dimes
8 nickels
40 pennies

To find out how many coins of each denomination are in Sammy's pocket, we can use a system of equations. Let's assume Sammy has x number of quarters, y number of dimes, and z number of nickels.

We know that Sammy has a total of 50 coins, so our first equation is:
x + y + z = 50

We also know that the sum of the values of these coins is equal to one dollar, which is 100 cents. Considering the denominations, we can write a second equation:
25x + 10y + 5z = 100

Now we have a system of two equations with three variables. We need to solve this system to find the values of x, y, and z.

One approach to solve the equations is by using substitution or elimination. However, in this case, we can simplify the equations by dividing both sides of the second equation by 5, which gives us:

5x + 2y + z = 20

Now we have two equations with three variables:
x + y + z = 50
5x + 2y + z = 20

Now we can solve this system of equations using any method we prefer. I will use the substitution method here:

From the first equation, we can express x as:
x = 50 - y - z

Substitute the value of x in the second equation:
5(50 - y - z) + 2y + z = 20

Simplify:
250 - 5y - 5z + 2y + z = 20
-3y - 4z = -230

Now we have a simplified system of equations:
x + y + z = 50
-3y - 4z = -230

From the first equation, we can express x as:
x = 50 - y - z

Now, we can substitute the value of x in the remaining equations and continue solving to find the values of y and z.