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Solve question number 15 and explain step by step 

Let's solve the problem step by step.

### Problem Recap:
Carlos Periu invested some money at **3% simple interest** and $6000 more than that amount at **4.5% simple interest**. After 1 year, his total interest from the two accounts was $870. The question asks how much he invested at each rate.

### Step 1: Define Variables
Let:
- $ x $ be the amount invested at 3% interest.
- $ x + 6000 $ be the amount invested at 4.5% interest.

### Step 2: Set Up the Interest Formulas
The formula for simple interest is:
\[
\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time}
\]

Given that the time $ t = 1 $ year, the interest for each investment will be:
- Interest from the 3% investment: $ \text{Interest}_1 = x \times 0.03 \times 1 = 0.03x $
- Interest from the 4.5% investment: $ \text{Interest}_2 = (x + 6000) \times 0.045 \times 1 = 0.045(x + 6000) $

### Step 3: Set Up the Equation
The total interest from both investments is given as $870. So, the equation is:
\[
0.03x + 0.045(x + 6000) = 870
\]

### Step 4: Simplify and Solve the Equation
First, distribute the 0.045:
\[
0.03x + 0.045x + 0.045 \times 6000 = 870
\]

\[
0.03x + 0.045x + 270 = 870
\]

Next, combine like terms:
\[
0.075x + 270 = 870
\]

Now, subtract 270 from both sides:
\[
0.075x = 600
\]

Finally, solve for $ x $:
\[
x = \frac{600}{0.075} = 8000
\]

### Step 5: Find the Amount Invested at 4.5%
Since $ x = 8000 $, the amount invested at 4.5% is:
\[
x + 6000 = 8000 + 6000 = 14000
\]

### Conclusion:
- The amount invested at **3%** interest is **$8000**.
- The amount invested at **4.5%** interest is **$14000**.

Would you like more details on any of the steps, or do you have any other questions?

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Here are some related questions you might find helpful:
1. How would the solution change if the total interest was $1000?
2. What if the interest rates were swapped between the two investments?
3. How can the problem be solved using a system of linear equations?
4. What if Carlos invested $2000 more instead of $6000 more at 4.5% interest?
5. How does simple interest differ from compound interest in such problems?

**Tip:** Always double-check your calculations when dealing with percentages and interest rates, as small mistakes can lead to significantly different results.

Carlos Periu invested some money at 3% simple interest and $6000 more than that amount at 4.5% simple interest. After 1 year, his total interest from the two accounts was $870. How much did he invest at each rate?

Solve an investment problem where Carlos invests at 3% and 4.5% simple interest rates with a total interest of $870. Learn the step-by-step method to calculate the amount invested at each rate. This problem covers algebraic concepts and the simple interest formula, suitable for Grades 9-12.

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