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 15% per year or in a certificate of deposit (CD) paying 7% 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 allocate her $70,000 investment between B-rated bonds and a certificate of deposit (CD) to obtain $5,000 in interest per year, we can set up a system of equations.

Let:

• \( x \) = amount invested in B-rated bonds
• \( y \) = amount invested in the certificate of deposit (CD)

From the information given, we know the following:

1. The total investment is $70,000: \[ x + y = 70,000 \]

2. The total interest earned from both investments should equal $5,000:

   • The B-rated bonds pay 15% interest, hence the interest from bonds will be \( 0.15x \).
   • The CD pays 7% interest, hence the interest from the CD will be \( 0.07y \). Therefore, the interest equation is: \[ 0.15x + 0.07y = 5,000 \]

Now we have the following system of equations:

1. \( x + y = 70,000 \)
2. \( 0.15x + 0.07y = 5,000 \)

We can solve this system step-by-step.

Step 1: Solve for \( y \) in terms of \( x \) from the first equation. \[ y = 70,000 - x \]

Step 2: Substitute \( y \) into the second equation. \[ 0.15x + 0.07(70,000 - x) = 5,000 \] \[ 0.15x + 4,900 - 0.07x = 5,000 \] \[ 0.15x - 0.07x + 4,900 = 5,000 \] \[ 0.08x + 4,900 = 5,000 \] \[ 0.08x = 5,000 - 4,900 \] \[ 0.08x = 100 \] \[ x = \frac{100}{0.08} = 1,250 \]

Step 3: Substitute \( x \) back to find \( y \). \[ y = 70,000 - 1,250 = 68,750 \]

So, Betsy should invest:

• \( x = 1,250 \) in B-rated bonds
• \( y = 68,750 \) in the certificate of deposit (CD)

Final Answer: Betsy should invest \( \text{ } $1,250 \text{ in B-rated bonds} \) and \( \text{ } $68,750 \text{ in the certificate of deposit (CD)} \) to achieve exactly \( $5,000 \) in interest per year.