We can solve this investment problem using the method of linear programming to maximize your annual interest earned. Let's define the variables for the amounts you will invest in each of the three stocks:

• Let $x$ be the amount invested in MarkyB Inc..
• Let $y$ be the amount invested in JohnJohn Ltd..
• Let $z$ be the amount invested in Garretts Spaghetti House.

Objective:

The goal is to maximize the annual interest earned, which can be written as the function: $\text{Maximize: } P = 0.16x + 0.15y + 0.09z$ where $P$ is the total annual interest earned.

Constraints:

1. Total investment constraint: $x + y + z = 99,000$
2. Investment limit in Garretts Spaghetti House: $z \leq 11,000$
3. Investment constraint for MarkyB Inc. (the amount invested in MarkyB Inc. should be at least the combined investment in JohnJohn Ltd. and Garretts Spaghetti House): $x \geq y + z$
4. Non-negativity constraints (since you cannot invest a negative amount): $x \geq 0, \quad y \geq 0, \quad z \geq 0$

Solving the Problem:

We will now solve the system of constraints to find the optimal values of $x$, $y$, and $z$ that maximize the interest.

Step 1: Substituting the total investment constraint

From the equation $x + y + z = 99,000$, we can express $x$ as: $x = 99,000 - y - z$ This allows us to substitute into other constraints.

Step 2: Substituting into the MarkyB Inc. constraint

Using $x \geq y + z$ and substituting $x = 99,000 - y - z$, we get: $99,000 - y - z \geq y + z$ Simplifying: $99,000 \geq 2y + 2z$ $49,500 \geq y + z$ Thus, we have another constraint: $y + z \leq 49,500$

Step 3: Analyzing Garretts Spaghetti House constraint

We know that $z \leq 11,000$. Combining this with $y + z \leq 49,500$, we can deduce the maximum allowable $y$.

Now, let's proceed to solve this system to maximize the objective function $P = 0.16x + 0.15y + 0.09z$.

Let me calculate the optimal values for $x$, $y$, and $z$.The optimal investment strategy is to:

• Invest $99,000 in MarkyB Inc..
• Invest $0 in JohnJohn Ltd..
• Invest $0 in Garretts Spaghetti House.

This strategy will yield the maximum annual interest of $15,840.

It appears that the best option is to invest entirely in MarkyB Inc., as it offers the highest interest rate at 16%.

Would you like more details on the solution, or do you have any questions?

Here are 5 related questions for further exploration:

1. How would the solution change if Garretts Spaghetti House had a higher interest rate?
2. What if there were a minimum investment requirement for JohnJohn Ltd.?
3. How would the investment strategy change if you were required to diversify among all three stocks?
4. What is the impact of reducing the total amount available for investment?
5. How would taxes or fees affect the final interest earned?

Tip: In linear programming, maximizing returns often leads to investing heavily in the option with the highest rate, provided constraints allow it.