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%.