Larry has annual return of $213 from $3000 invested at simple interest, some at 5% and the rest at 8%. How much money is invested at each rate? 

Let's denote the amount of money invested at 5% as $ x $, and the amount invested at 8% as $ y $.

Given:

1. The total amount invested is $3000:
   \[
   x + y = 3000
   \]

2. The annual return from the investments is $213:
   \[
   0.05x + 0.08y = 213
   \]

Now, we have a system of two equations:
\[
\begin{aligned}
1) \quad & x + y = 3000 \\
2) \quad & 0.05x + 0.08y = 213
\end{aligned}
\]

### Solving the system of equations:

1. **From the first equation** $ x + y = 3000 $, solve for $ y $:
   \[
   y = 3000 - x
   \]

2. **Substitute** $ y = 3000 - x $ into the second equation:
   \[
   0.05x + 0.08(3000 - x) = 213
   \]

3. **Expand and simplify**:
   \[
   0.05x + 240 - 0.08x = 213
   \]
   \[
   -0.03x + 240 = 213
   \]
   \[
   -0.03x = 213 - 240
   \]
   \[
   -0.03x = -27
   \]
   \[
   x = \frac{-27}{-0.03} = 900
   \]

So, $ x = 900 $.

4. **Substitute** $ x = 900 $ back into $ y = 3000 - x $ to find $ y $:
   \[
   y = 3000 - 900 = 2100
   \]

### Conclusion:
- Larry invested **$900** at 5%.
- Larry invested **$2100** at 8%.

Would you like any further explanation, or do you have any other questions?

---

Here are 5 related questions for further practice:

1. What would be the total return if Larry had invested the entire $3000 at 8%?
2. If Larry invested $100 more at 5%, how would that affect the total return?
3. How much would Larry need to invest at 5% to have a total return of $250?
4. What is the difference in the returns generated by the investments at 5% and 8%?
5. If the interest rates were reversed, how would the total return change?

**Tip:** When dealing with different interest rates, setting up a system of equations helps in efficiently solving the problem.

Explore how to solve investment problems using simple interest with a detailed step-by-step solution. This problem involves determining how much money Larry invested at different interest rates (5% and 8%) given his annual returns and total investment amount. Suitable for students in Grades 9-12.

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