You have $31,000 you would like to invest in two different stocks, Stock Boll and Stock Coff. The buying limit on Stock Coff is $13,500 and you want to spend at least four times as much money on Stock Boll as Stock Coff.
If Stock Boll earns 7% annual interest and Stock Coff earns 4% 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 (in dollars)?

Let:
- $ x $ = the amount invested in Stock Boll
- $ y $ = the amount invested in Stock Coff

We know the following:

1. The total investment is $31,000:
   \[
   x + y = 31,000
   \]

2. The buying limit on Stock Coff is $13,500:
   \[
   y \leq 13,500
   \]

3. You want to spend at least four times as much money on Stock Boll as Stock Coff:
   \[
   x \geq 4y
   \]

4. The annual interest earned from Stock Boll is 7%, and from Stock Coff is 4%. The objective is to maximize the total annual interest:
   \[
   \text{Interest} = 0.07x + 0.04y
   \]

### Step 1: Set up the constraints

The constraints can be rewritten as:
\[
x + y = 31,000
\]
\[
y \leq 13,500
\]
\[
x \geq 4y
\]

### Step 2: Express $ x $ in terms of $ y $

From the total investment equation:
\[
x = 31,000 - y
\]

Substituting $ x \geq 4y $:
\[
31,000 - y \geq 4y
\]
\[
31,000 \geq 5y
\]
\[
y \leq 6,200
\]

Combining this with the constraint $ y \leq 13,500 $, we have:
\[
y \leq 6,200
\]

### Step 3: Substitute $ y $ to maximize interest

To maximize the interest, we substitute the maximum value of $ y $:
\[
y = 6,200
\]

Then:
\[
x = 31,000 - 6,200 = 24,800
\]

### Step 4: Calculate the maximum annual interest

The total annual interest is:
\[
\text{Interest} = 0.07 \times 24,800 + 0.04 \times 6,200
\]

Let's compute this: 

\[
\text{Interest} = 0.07 \times 24,800 + 0.04 \times 6,200 = 1,736 + 248
\]

\[
\text{Interest} = 1,984
\]

### Final Answer

- You should invest **$24,800** in Stock Boll and **$6,200** in Stock Coff.
- The maximum annual interest is **$1,984**.

---

Would you like a detailed step-by-step verification or any other questions? Here are some related questions:

1. How would the investment strategy change if the interest rates were reversed?
2. What if the investment limit on Stock Coff was higher?
3. How would a minimum investment constraint on Stock Boll affect the solution?
4. What is the impact on the maximum interest if Stock Coff earns 5% instead of 4%?
5. How would you adjust the investments if you wanted to limit the risk exposure?

**Tip:** Always set up clear inequalities when dealing with constraints to ensure accurate maximization or minimization problems!

You have $31,000 you would like to invest in two different stocks, Stock Boll and Stock Coff. The buying limit on Stock Coff is $13,500 and you want to spend at least four times as much money on Stock Boll as Stock Coff. If Stock Boll earns 7% annual interest and Stock Coff earns 4% 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 (in dollars)?

Discover how to invest $31,000 between Stock Boll and Stock Coff to maximize your annual interest. This problem involves solving linear equations and inequalities while taking into account buying limits and investment ratios. Perfect for students in Grades 10-12, this solution includes step-by-step calculations, including how to achieve a maximum annual interest of $1,984.

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