Question

Esraa invested $10,000 from which she earns an annual income of $550 per year. She invested part of the 10,000 at 5% annual interest, and the remainder in bonds paying 6% interest. How much did she invest at each rate?

Answer 1

Let \( x \) be the amount invested at 5%, and \( y \) be the amount invested at 6%. According to the problem, we have the following two equations:

1. The total investment is $10,000:
\[
x + y = 10000
\]

2. The total annual income from both investments is $550:
\[
0.05x + 0.06y = 550
\]

Now, we can solve these equations step by step.

**Step 1: Solve for \( y \) in terms of \( x \) using the first equation:**
\[
y = 10000 - x
\]

**Step 2: Substitute \( y \) into the second equation:**
\[
0.05x + 0.06(10000 - x) = 550
\]

**Step 3: Expand and simplify the equation:**
\[
0.05x + 600 - 0.06x = 550
\]
\[
600 - 0.01x = 550
\]

**Step 4: Solve for \( x \):**
\[
600 - 550 = 0.01x
\]
\[
50 = 0.01x
\]
\[
x = 50 / 0.01
\]
\[
x = 5000
\]

**Step 5: Calculate \( y \):**
Using \( y = 10000 - x \):
\[
y = 10000 - 5000 = 5000
\]

**Final Answer:**
Esraa invested **$5,000 at 5%** and **$5,000 at 6%**.

**Verification:**
- Income from 5% investment: \( 0.05 \times 5000 = 250 \)
- Income from 6% investment: \( 0.06 \times 5000 = 300 \)
- Total income: \( 250 + 300 = 550 \) (correct)

Thus, the amounts are verified.
Esraa invested **$5,000 at 5%** and **$5,000 at 6%**.