Final answer:
To find the total number of solutions, set up a system of equations with the given information. The solution with the smallest number of five-dollar bills is (0, 0, 70), and the solution with the largest number of five-dollar bills is (70, 0, 0).
Step-by-step explanation:
To find the total number of solutions, we can use a system of equations. Let's say the number of five-dollar bills is x, the number of ten-dollar bills is y, and the number of twenty-dollar bills is z. We can set up the following equations:
x + y + z = 70 (equation 1)
5x + 10y + 20z = 960 (equation 2)
To solve this system of equations, we can use substitution or elimination. For simplicity, let's use substitution. We can solve equation 1 for x in terms of y and z: x = 70 - y - z. Substituting this into equation 2, we get:
5(70 - y - z) + 10y + 20z = 960
350 - 5y - 5z + 10y + 20z = 960
Combining like terms:
5z + 5y = 610
z + y = 122 (equation 3)
We now have a new equation that relates the number of ten-dollar bills and twenty-dollar bills. To find the total number of solutions, we need to find the possible values of y and z that satisfy equation 3. Since we have two variables and one equation, there is an infinite number of solutions.
b. To find the solution with the smallest number of five-dollar bills, we can minimize the value of x while satisfying equations 1 and 2. Let's start by setting x = 0 in equation 1:
0 + y + z = 70
y + z = 70
If we let y = 0, then z = 70. This means the solution with the smallest number of five-dollar bills is (0, 0, 70).
c. To find the solution with the largest number of five-dollar bills, we can maximize the value of x while satisfying equations 1 and 2. Let's start by setting x = 70 in equation 1:
70 + y + z = 70
y + z = 0
If we let y = 0, then z = 0. This means the solution with the largest number of five-dollar bills is (70, 0, 0).