An actor invests some money at 8​%, and ​$23000 more than twice the amount at 12 %. The total annual interest earned from the investment is ​$35400. How much did he invest at each​ amount?


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!

An actor invests some money at 8​%, and ​$23000 more than twice the amount at 12 %. The total annual interest earned from the investment is ​$35400. How much did he invest at each​ amount?

Solve an algebra problem where an actor invests money at 8% and 12%, with a total annual interest of $35,400. Learn how to set up and solve linear equations involving interest calculations for both investments, suitable for students in Grades 9-12.

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