Question

etsy, a recent retiree, requires $5,000 per year in extra income. she has $60,000 to invest and can invest in B-rated bonds at 13% per year or in certificate of deposit paying 3% per year. How much money should be invested in each realize exactly $5,000 per year in interest?

Answer 1

To determine how much Etsy should invest in both B-rated bonds and certificates of deposit (CDs) to achieve exactly $5,000 per year in interest, we can set up a system of equations.

Let:
- \( x \) = amount invested in B-rated bonds (at 13% interest)
- \( y \) = amount invested in certificates of deposit (at 3% interest)

We have the following two conditions:

1. The total investment:
\[
x + y = 60,000
\]

2. The total interest earned from both investments:
\[
0.13x + 0.03y = 5,000
\]

Now we can solve this system of equations step-by-step.

**Step 1: Solve the first equation for one variable**

From the first equation:

\[
y = 60,000 - x
\]

**Step 2: Substitute into the second equation**

Now, substitute \( y \) in the second equation:

\[
0.13x + 0.03(60,000 - x) = 5,000
\]

Distributing \( 0.03 \):

\[
0.13x + 1,800 - 0.03x = 5,000
\]

Combine like terms:

\[
(0.13 - 0.03)x + 1,800 = 5,000
\]
\[
0.10x + 1,800 = 5,000
\]

**Step 3: Isolate \( x \)**

Subtract 1,800 from both sides:

\[
0.10x = 5,000 - 1,800
\]
\[
0.10x = 3,200
\]

Now divide by 0.10:

\[
x = \frac{3,200}{0.10} = 32,000
\]

**Step 4: Solve for \( y \)**

Substituting \( x \) back to find \( y \):

\[
y = 60,000 - x = 60,000 - 32,000 = 28,000
\]

**Conclusion:**

Etsy should invest:
- **$32,000** in B-rated bonds at 13%
- **$28,000** in certificates of deposit at 3%

To verify:

- Interest from B-rated bonds:
\[
0.13 \times 32,000 = 4,160
\]

- Interest from CDs:
\[
0.03 \times 28,000 = 840
\]

- Total interest:
\[
4,160 + 840 = 5,000
\]

The calculations confirm that these investments result in exactly $5,000 per year in interest.