Charles invested $2000 in gold and a company working on prosthetics. Over the course of the investment, the gold earned a 1.8% annual return and the prosthetics earned 1.2%. If the total return after one year on the investment was $31.20, how much was invested in each? Assume simple interest.

Let $ x $ be the amount Charles invested in gold, and let $ 2000 - x $ be the amount invested in the prosthetics company. 

The annual returns on the investments can be expressed as follows:

- Return from gold: $ 0.018 \times x $
- Return from prosthetics: $ 0.012 \times (2000 - x) $

According to the problem, the total return from both investments after one year is $31.20. Therefore, we set up the equation:

\[
0.018x + 0.012(2000 - x) = 31.20
\]

Now, let's solve this equation step by step.

### Step 1: Expand the equation

\[
0.018x + 0.012 \times 2000 - 0.012x = 31.20
\]

\[
0.018x + 24 - 0.012x = 31.20
\]

### Step 2: Combine like terms

\[
(0.018 - 0.012)x + 24 = 31.20
\]

\[
0.006x + 24 = 31.20
\]

### Step 3: Isolate $ x $

\[
0.006x = 31.20 - 24
\]

\[
0.006x = 7.20
\]

\[
x = \frac{7.20}{0.006}
\]

\[
x = 1200
\]

### Step 4: Find the amount invested in prosthetics

\[
2000 - x = 2000 - 1200 = 800
\]

### Conclusion

- Amount invested in gold: **$1200**
- Amount invested in prosthetics: **$800**

Would you like more details or have any questions?

Here are some related questions you might consider:
1. How would the investments change if the rates were reversed?
2. What if the total return was different, how would that affect the amounts invested?
3. How does compound interest differ from simple interest in this scenario?
4. What would be the return if both investments had the same rate of return?
5. How does changing the total investment amount affect the returns?

**Tip:** When dealing with multiple investments, setting up equations for returns helps isolate the contribution of each investment.

Solve a simple interest investment problem where Charles invests $2000 in gold and prosthetics. Learn how to calculate the amounts invested using a linear equation based on 1.8% and 1.2% returns, with a total return of $31.20. Suitable for Grades 8-10 students.

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