Roberto invested some money at 5%, and then invested $2000 more than twice this amount at 5%. His total annual income from the two investments was $2200. How much was invested at 5%?

Let’s denote the amount Roberto invested at 5% as $ x $. According to the problem, he then invested $2000 more than twice this amount at the same 5% rate. Therefore, the second investment can be expressed as:

\[
\text{Second investment} = 2x + 2000
\]

Given that his total annual income from both investments was $2200, we can write the income equations for each investment and set up the total income equation:

1. Income from the first investment:  
   \[
   0.05x
   \]

2. Income from the second investment:  
   \[
   0.05(2x + 2000)
   \]

Adding these two incomes gives the total annual income:

\[
0.05x + 0.05(2x + 2000) = 2200
\]

Simplify this equation:

\[
0.05x + 0.1x + 100 = 2200
\]

Combine like terms:

\[
0.15x + 100 = 2200
\]

Subtract 100 from both sides:

\[
0.15x = 2100
\]

Now, solve for $ x $:

\[
x = \frac{2100}{0.15} = 14000
\]

So, the amount invested at 5% is:

\[
\boxed{14000}
\]

### Verification:
- First investment: $ 14000 $
- Second investment: $ 2(14000) + 2000 = 30000 $
- Income from first investment: $ 0.05 \times 14000 = 700 $
- Income from second investment: $ 0.05 \times 30000 = 1500 $
- Total income: $ 700 + 1500 = 2200 $, which matches the given condition.

Would you like further details or have any questions?

### Related Questions:
1. What if the total income was $2400 instead of $2200?
2. How would the solution change if the second investment had a different interest rate?
3. What if Roberto had invested a different amount more than twice the first investment?
4. How much would Roberto need to invest if the total income needed to be $3000?
5. How does the investment ratio affect the overall income?

**Tip:** Always verify your solution by substituting the values back into the original conditions to ensure they satisfy all requirements.

Explore Roberto's investment problem where he calculates returns on investments at 5%. Understand how to solve for the amount invested at 5% with a step-by-step solution. Suitable for Grades 10-12 students and learners interested in financial mathematics.

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