Let the amount invested at 8% be $x$.

Then, according to the problem, the amount invested at 12% is $2x + 23000$.

The total annual interest earned is the sum of the interest from both investments. The interest from the investment at 8% is $0.08x$, and the interest from the investment at 12% is $0.12(2x + 23000)$.

We are given that the total interest is $35,400. So, the equation becomes:

$0.08x + 0.12(2x + 23000) = 35400$

Let's solve this step-by-step:

Step 1: Expand the equation

$0.08x + 0.12(2x) + 0.12(23000) = 35400$ $0.08x + 0.24x + 2760 = 35400$

Step 2: Combine like terms

$0.32x + 2760 = 35400$

Step 3: Subtract 2760 from both sides

$0.32x = 35400 - 2760$ $0.32x = 32640$

Step 4: Divide by 0.32

$x = \frac{32640}{0.32} = 102000$

So, the amount invested at 8% is $x = 102000$.

Step 5: Find the amount invested at 12%

The amount invested at 12% is $2x + 23000 = 2(102000) + 23000 = 204000 + 23000 = 227000$.

Final answer:

• The amount invested at 8% is $102,000.
• The amount invested at 12% is $227,000.

Would you like a detailed explanation of any steps, or do you have any other questions?

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Here are 5 related questions to think about:

1. How would the solution change if the interest rates were swapped?
2. What if the total interest was $40,000 instead of $35,400?
3. How does the interest earned from both investments compare?
4. If the actor had invested $300,000 total, what would the interest amounts be?
5. What is the difference in interest earned between the two investments?

Tip: Always check your work by plugging your solution back into the original equation to ensure accuracy!