In a professional cooperative a physician, an attorney, and a financial planner trade services. At the end of the period the physician had spent 30% of his time on his own care, 40% on the attorney's care, and 30% on the financial planner's care. The attorney had spent 50% of her time on the physician's affairs, 20% on her own affairs, and 30% on the financial planner's affairs. Finally, the financial planner spent 20% of his time managing the physician's portfolio, 20% managing the attorney's portfolio, and 60% managing his own portfolio. If the financial planner earns $20,000, how should each of the other two professionals be paid for their work?
Carry out all calculations exactly and round the final answer to two decimal places.

This problem involves determining the value of the services exchanged between the three professionals using a system of linear equations. We know the time each professional spent on the others' services, and we also know the financial planner's income ($20,000). The task is to find how much the physician and attorney should be paid.

### Step 1: Define Variables
Let:
- $ x $ be the physician's income,
- $ y $ be the attorney's income,
- $ z = 20,000 $ (the financial planner's income).

### Step 2: Set Up the Equations
Each professional's total income should be equal to the sum of the payments they received based on the time spent on their services.

1. **Physician's income** $ x $:
   - 30% of the physician’s work was on his own care, 40% on the attorney's care, and 30% on the financial planner’s care. Hence:
   \[
   x = 0.3x + 0.4y + 0.3z
   \]

2. **Attorney's income** $ y $:
   - 50% of the attorney's work was on the physician's care, 20% on her own care, and 30% on the financial planner's care. Hence:
   \[
   y = 0.5x + 0.2y + 0.3z
   \]

3. **Financial planner's income** $ z = 20,000 $:
   - 20% of the financial planner's work was on the physician's care, 20% on the attorney's care, and 60% on his own care. Hence:
   \[
   z = 0.2x + 0.2y + 0.6z
   \]
   But since $ z = 20,000 $, we can substitute this into the equation.

### Step 3: Simplify the Equations
Let’s rearrange and simplify the system of equations.

1. From the first equation:
   \[
   x = 0.3x + 0.4y + 0.3(20,000)
   \]
   \[
   x - 0.3x = 0.4y + 6,000
   \]
   \[
   0.7x = 0.4y + 6,000
   \]
   \[
   x = \frac{0.4y + 6,000}{0.7}
   \]

2. From the second equation:
   \[
   y = 0.5x + 0.2y + 0.3(20,000)
   \]
   \[
   y - 0.2y = 0.5x + 6,000
   \]
   \[
   0.8y = 0.5x + 6,000
   \]
   \[
   y = \frac{0.5x + 6,000}{0.8}
   \]

3. From the third equation, substitute $ z = 20,000 $:
   \[
   20,000 = 0.2x + 0.2y + 0.6(20,000)
   \]
   \[
   20,000 = 0.2x + 0.2y + 12,000
   \]
   \[
   0.2x + 0.2y = 8,000
   \]
   \[
   x + y = 40,000
   \]

### Step 4: Solve the System of Equations
We now have two equations:
1. $ x + y = 40,000 $
2. $ x = \frac{0.4y + 6,000}{0.7} $

Substitute $ x = 40,000 - y $ into the second equation:
\[
40,000 - y = \frac{0.4y + 6,000}{0.7}
\]
Multiply through by 0.7:
\[
0.7(40,000 - y) = 0.4y + 6,000
\]
\[
28,000 - 0.7y = 0.4y + 6,000
\]
Move terms with $ y $ to one side:
\[
28,000 - 6,000 = 0.4y + 0.7y
\]
\[
22,000 = 1.1y
\]
\[
y = \frac{22,000}{1.1} = 20,000
\]

Substitute $ y = 20,000 $ into $ x + y = 40,000 $:
\[
x + 20,000 = 40,000
\]
\[
x = 20,000
\]

### Step 5: Conclusion
- The physician should be paid $ x = 20,000 $.
- The attorney should be paid $ y = 20,000 $.
- The financial planner already earns $ z = 20,000 $.

### Final Answer:
Each professional should be paid \$20,000 for their work.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the payments change if the time spent by the professionals was different?
2. What would happen if one professional charged a different rate per hour of service?
3. How can linear algebra help solve complex systems of professional service exchanges?
4. What would be the result if the financial planner earned more than \$20,000?
5. How would additional professionals in the cooperative affect the system of payments?

**Tip:** Linear systems like this can be solved efficiently using matrix methods such as Gaussian elimination.

In a professional cooperative a physician, an attorney, and a financial planner trade services. At the end of the period the physician had spent 30% of his time on his own care, 40% on the attorney's care, and 30% on the financial planner's care. The attorney had spent 50% of her time on the physician's affairs, 20% on her own affairs, and 30% on the financial planner's affairs. Finally, the financial planner spent 20% of his time managing the physician's portfolio, 20% managing the attorney's portfolio, and 60% managing his own portfolio. If the financial planner earns $20,000, how should each of the other two professionals be paid for their work? Carry out all calculations exactly and round the final answer to two decimal places.

Learn how to solve a system of linear equations involving a physician, attorney, and financial planner who exchange services. This problem explains how to determine their income distribution using linear algebra. The financial planner earns $20,000, and the calculations show how much each professional should be paid. Suitable for Grades 11-12.

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