Betsy, a recent retiree, requires $5,000 per year in extra income. She has $70,000 to invest and can invest in B-rated bonds paying 13% per year or in a certificate of deposit (CD) paying 3% per year. How much money should be invested in each to realize exactly $5,000 in interest per year?

Answer 1

To find out how Betsy can invest her $70,000 to generate exactly $5,000 in income per year, we can set up an algebraic equation based on the amount she invests in each option.

Let \( x \) be the amount invested in the B-rated bonds paying 13%, and \( (70,000 - x) \) be the amount invested in the CD paying 3%.

The total income from both investments should equal $5,000. The income from the Bonds will be \( 0.13x \) and the income from the CD will be \( 0.03(70,000 - x) \).

So we can write the following equation:

\[ 0.13x + 0.03(70,000 - x) = 5,000 \]

Now we can simplify and solve for \( x \):

1. Distributing the 0.03:

\[ 0.13x + 0.03 \cdot 70,000 - 0.03x = 5,000 \]

\[ 0.13x + 2,100 - 0.03x = 5,000 \]

2. Combine like terms:

\[ (0.13 - 0.03)x + 2,100 = 5,000 \]

\[ 0.10x + 2,100 = 5,000 \]

3. Subtract 2,100 from both sides:

\[ 0.10x = 5,000 - 2,100 \]

\[ 0.10x = 2,900 \]

4. Divide by 0.10:

\[ x = \frac{2,900}{0.10} = 29,000 \]

Now we know that Betsy should invest \( x = 29,000 \) in the B-rated bonds.

To find the amount to invest in the CD:

\[ 70,000 - x = 70,000 - 29,000 = 41,000 \]

Conclusion: Betsy should invest $29,000 in B-rated bonds and $41,000 in the CD to generate exactly $5,000 in interest per year.

Verification:

• Interest from bonds: \( 0.13 \times 29,000 = 3,770 \)
• Interest from CD: \( 0.03 \times 41,000 = 1,230 \)

Total interest: \( 3,770 + 1,230 = 5,000 \) (which is correct).