Question

A brokerage house offers 3 stock portfolios. Portfolio I consists of 2 blocks of common stock and 1 municipal bond. Portfolio II consists of 4 blocks of common stock, 2 municipal bonds and 3 blocks of preferred stock. 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 preferred stocks. How many portfolio III should be offered?

• A. 1
• B. 2
• C. 3
• D. None of the above

Hint:

Write one equation per stock type (common, municipal, preferred) in terms of the number of each portfolio, then eliminate variables to find how many Portfolio III units are needed.

Answer 1

The Correct Option is D

Step 1: Set up variables.
Let \(x\), \(y\), \(z\) be the number of Portfolio I, II and III sold.

Step 2: Write one equation per stock type.
Common stock: Portfolio I gives 2, II gives 4, III gives 2, and the customer wants 12:
\[ 2x + 4y + 2z = 12 \]
Municipal bonds: Portfolio I gives 1, II gives 2, III gives 2, and the customer wants 6:
\[ x + 2y + 2z = 6 \]
Preferred stock: Portfolio I gives 0, II gives 3, III gives 3, and the customer wants 6:
\[ 3y + 3z = 6 \implies y + z = 2 \]

Step 3: Simplify the common stock equation.
Divide the first equation by 2:
\[ x + 2y + z = 6 \]

Step 4: Compare with the municipal bond equation.
\[ x + 2y + 2z = 6 \]
\[ x + 2y + z = 6 \]
Subtracting the second from the first gives:
\[ z = 0 \]

Step 5: Find x and y, then check every equation.
From \(y+z=2\) and \(z=0\), \(y=2\). From \(x+2y+z=6\), \(x+4+0=6\), so \(x=2\).
Check: common stock \(2(2)+4(2)+2(0)=4+8+0=12\), bonds \(2+2(2)+2(0)=6\), preferred \(3(2)+3(0)=6\). All three match, so this solution is correct.

Step 6: Read off the answer.
Portfolio III should be offered \(z=0\) times. This value is not 1, 2 or 3, so none of those numeric options is correct.

Final Answer:
\[ \boxed{\text{None of the above (0 units of Portfolio III)}} \]