You have $98,000 you would like to invest in three different stocks: MarkyB Inc., JohnJohn Ltd., and Garretts Spaghetti House. You would like to invest no more than $18,000 in Garretts Spaghetti House, and you want the amount you invest in MarkyB Inc., to be at least the amount invested in JohnJohn Ltd. and Garretts Spaghetti House combined.

If MarkyB Inc. earns 18% annual interest, JohnJohn Ltd. earns 14% annual interest, and Garretts Spaghetti House earns 9% annual interest, how much money (in dollars) should you invest in each stock to maximize your annual interest earned? What is the maximum annual interest earned?

You should invest $  into MarkyB Inc., $  into JohnJohn Ltd., and $  into Garretts Spaghetti House, which will give you give you maximum annual interest $  .

To solve this problem, we need to maximize the total annual interest earned from the investments, while meeting the given constraints. Let's break it down step by step:

### Variables:
Let:
- $ x_1 $ be the amount invested in MarkyB Inc.
- $ x_2 $ be the amount invested in JohnJohn Ltd.
- $ x_3 $ be the amount invested in Garretts Spaghetti House.

### Constraints:
1. Total amount invested should be $98,000:
   \[
   x_1 + x_2 + x_3 = 98,000
   \]
2. You want to invest no more than $18,000 in Garretts Spaghetti House:
   \[
   x_3 \leq 18,000
   \]
3. The amount invested in MarkyB Inc. should be at least the amount invested in JohnJohn Ltd. and Garretts Spaghetti House combined:
   \[
   x_1 \geq x_2 + x_3
   \]

### Objective Function:
The total annual interest earned is based on the interest rates of the stocks:
- MarkyB Inc. earns 18% interest, so the interest earned on $ x_1 $ is $ 0.18 \times x_1 $.
- JohnJohn Ltd. earns 14% interest, so the interest earned on $ x_2 $ is $ 0.14 \times x_2 $.
- Garretts Spaghetti House earns 9% interest, so the interest earned on $ x_3 $ is $ 0.09 \times x_3 $.

Thus, the total interest earned is:
\[
\text{Total Interest} = 0.18x_1 + 0.14x_2 + 0.09x_3
\]

### Goal:
Maximize the total interest $ 0.18x_1 + 0.14x_2 + 0.09x_3 $ subject to the constraints.

### Steps to Solve:
1. We need to express $ x_1 $, $ x_2 $, and $ x_3 $ in such a way that satisfies the constraints.
2. Then we can maximize the objective function.

Let me calculate the solution.To maximize your annual interest earned, you should invest the entire $98,000 in **MarkyB Inc.** and nothing in JohnJohn Ltd. or Garretts Spaghetti House. This investment strategy will give you the maximum annual interest of **$17,640**.

### Summary:
- **MarkyB Inc.**: $98,000
- **JohnJohn Ltd.**: $0
- **Garretts Spaghetti House**: $0
- **Maximum Annual Interest**: $17,640

Would you like more details on how the constraints influenced the solution, or do you have any other questions?

### Here are 5 related questions to expand your understanding:
1. How would the solution change if you removed the constraint on Garretts Spaghetti House?
2. What would happen if the interest rate for JohnJohn Ltd. increased to 18%?
3. How would you allocate the funds if MarkyB Inc. had a lower interest rate?
4. What is the effect of increasing the maximum allowed investment in Garretts Spaghetti House?
5. How would the investment strategy change if you wanted to minimize risk by diversifying across all three stocks?

### Tip:
In optimization problems like this, it's often beneficial to focus on the investment with the highest return, subject to constraints.

You have $98,000 you would like to invest in three different stocks: MarkyB Inc., JohnJohn Ltd., and Garretts Spaghetti House. You would like to invest no more than $18,000 in Garretts Spaghetti House, and you want the amount you invest in MarkyB Inc. to be at least the amount invested in JohnJohn Ltd. and Garretts Spaghetti House combined. If MarkyB Inc. earns 18% annual interest, JohnJohn Ltd. earns 14% annual interest, and Garretts Spaghetti House earns 9% annual interest, how much money should you invest in each stock to maximize your annual interest earned?

Optimize your investment of $98,000 across three stocks, ensuring no more than $18,000 is placed in Garretts Spaghetti House. Maximize your annual interest using 18%, 14%, and 9% interest rates. Solve this linear programming problem with constraints, yielding a maximum annual interest of $17,640. Suitable for advanced algebra and optimization students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation