Suppose that portfolios I and II in Problem 58 are unchanged and portfolio III consists of 2 blocks of common stock, 2 municipal bonds, and 3 blocks of preferred stock. A customer wants 12 blocks of common stock, 6 municipal bonds, and 6 blocks of preferred stock. How many units of each portfolio should be offered?
To determine the number of units of each portfolio that should be offered, we need to set up a system of equations based on the given information.
Let's denote:
x = number of units of portfolio I
y = number of units of portfolio II
z = number of units of portfolio III
According to the information given:
Portfolio I: 1 block of common stock, 1 municipal bond, and 2 blocks of preferred stock
Portfolio II: 1 block of common stock, 3 municipal bonds, and 1 block of preferred stock
Portfolio III: 2 blocks of common stock, 2 municipal bonds, and 3 blocks of preferred stock
The customer wants:
12 blocks of common stock, 6 municipal bonds, and 6 blocks of preferred stock
Based on this, we can set up the following equations:
Equation 1: x + y + 2z = 12 (for common stock)
Equation 2: x + 3y + z = 6 (for municipal bonds)
Equation 3: 2x + 2y + 3z = 6 (for preferred stock)
Now, we can solve this system of equations to find the values of x, y, and z.
To solve this, you can use various methods like substitution, elimination, or matrix methods. Let's use the elimination method:
Multiply Equation 1 by 2 and subtract Equation 3 from it:
2(x + y + 2z) - (2x + 2y + 3z) = 2(12) - 6
2x + 2y + 4z - 2x - 2y - 3z = 24 - 6
z = 18
Substitute the value of z into Equation 2:
x + 3y + 18 = 6
x + 3y = -12 (subtract 18 from both sides)
Substitute the values of z from above and solve the equation:
x + y + 2z = 12
x + y + 2(18) = 12
x + y + 36 = 12
x + y = -24 (subtract 36 from both sides)
So, we have two equations:
x + 3y = -12
x + y = -24
Now solve this system of equations to find the values of x and y.