A trust account client has $295,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $22,300.
b) If $18,000 is invested at 10%, how much should be invested at each of the other rates?
8%	$   
Incorrect: Your answer is incorrect.
7%

To solve this problem, we need to determine how much money should be invested at 8% and 7% such that the total interest earned is $22,300, given that $18,000 is already invested at 10%.

### Step-by-Step Solution:

1. **Identify the known quantities:**
   - Total available for investment: \$295,000
   - Interest earned goal: \$22,300
   - Amount invested at 10%: \$18,000

2. **Define the unknowns:**
   Let:
   - $ x $ = amount to be invested at 8%
   - $ y $ = amount to be invested at 7%

3. **Interest equation:**
   The total interest earned from all the investments should equal \$22,300:
   \[
   0.08x + 0.07y + 0.10(18,000) = 22,300
   \]

4. **Simplify the equation:**
   First, calculate the interest earned from the \$18,000 invested at 10%:
   \[
   0.10(18,000) = 1,800
   \]
   So, the equation becomes:
   \[
   0.08x + 0.07y + 1,800 = 22,300
   \]
   Subtract 1,800 from both sides:
   \[
   0.08x + 0.07y = 20,500
   \]

5. **Total investment equation:**
   The total investment should add up to \$295,000. Since \$18,000 is already invested at 10%, the remaining amount is:
   \[
   x + y = 295,000 - 18,000 = 277,000
   \]

6. **Solve the system of equations:**
   Now we have two equations:
   1. $ 0.08x + 0.07y = 20,500 $
   2. $ x + y = 277,000 $

   Solve equation 2 for $ y $:
   \[
   y = 277,000 - x
   \]

7. **Substitute $ y $ into the first equation:**
   \[
   0.08x + 0.07(277,000 - x) = 20,500
   \]
   Expand:
   \[
   0.08x + 0.07 \times 277,000 - 0.07x = 20,500
   \]
   \[
   0.08x + 19,390 - 0.07x = 20,500
   \]
   Combine like terms:
   \[
   0.01x + 19,390 = 20,500
   \]
   Subtract 19,390 from both sides:
   \[
   0.01x = 1,110
   \]
   Solve for $ x $:
   \[
   x = \frac{1,110}{0.01} = 111,000
   \]

8. **Find $ y $:**
   Substitute $ x = 111,000 $ into the equation $ y = 277,000 - x $:
   \[
   y = 277,000 - 111,000 = 166,000
   \]

### Final Answer:
- Amount invested at 8%: **\$111,000**
- Amount invested at 7%: **\$166,000**

Would you like more details or have any questions?

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Here are 5 related questions you could explore:
1. What if the investment goal was \$25,000 in interest? How would the amounts change?
2. How would the distribution change if the interest rate on one of the options increased by 1%?
3. What if only \$250,000 was available for investment?
4. How long would it take to reach the investment goal of \$22,300 if compounded annually?
5. How does changing the amounts invested in the 10% option impact the overall interest?

**Tip:** When solving for multiple investments, always check that the total interest earned meets the goal by summing up the contributions from each rate.

A trust account client has $295,000 to be invested. The investment choices have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $22,300. If $18,000 is invested at 10%, how much should be invested at each of the other rates?

This problem demonstrates how to calculate the amount to invest at 8% and 7% to reach an interest goal of $22,300, given a $295,000 total investment with $18,000 already invested at 10%. Learn how to solve the system of linear equations using interest rates and total investment constraints.

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