​Betsy, a recent​ retiree, requires ​$5 comma 000 per year in extra income.  She has ​$50 comma 000 to invest and can invest in​ B-rated bonds paying 15​% per year or in a certificate of deposit​ (CD) paying 5​% per year.  How much money should be invested in each to realize exactly ​$5 comma 000 in interest per​ year?

To solve this problem, we need to set up a system of linear equations. Let's define the variables first:

- Let $ x $ be the amount invested in the B-rated bonds.
- Let $ y $ be the amount invested in the certificate of deposit (CD).

Given:
- The B-rated bonds pay 15% interest per year.
- The CD pays 5% interest per year.
- Betsy needs a total of $5,000 in interest per year.
- She has $50,000 to invest in total.

### Step 1: Set up the equations

1. **Interest Equation:** 
   The total interest earned from both investments must equal $5,000:
   \[
   0.15x + 0.05y = 5000
   \]

2. **Total Investment Equation:** 
   The total amount invested should equal $50,000:
   \[
   x + y = 50000
   \]

### Step 2: Solve the system of equations

1. **Solve the second equation for $ y $:**
   \[
   y = 50000 - x
   \]

2. **Substitute $ y $ in the interest equation:**
   \[
   0.15x + 0.05(50000 - x) = 5000
   \]

3. **Simplify and solve for $ x $:**
   \[
   0.15x + 2500 - 0.05x = 5000
   \]
   \[
   0.10x + 2500 = 5000
   \]
   \[
   0.10x = 2500
   \]
   \[
   x = 25000
   \]

So, Betsy should invest $25,000 in the B-rated bonds.

4. **Find $ y $ using $ x = 25000 $:**
   \[
   y = 50000 - 25000 = 25000
   \]

### Conclusion:
Betsy should invest $25,000 in B-rated bonds and $25,000 in the certificate of deposit (CD) to achieve exactly $5,000 in interest per year.

Would you like more details, or do you have any questions?

Here are five related questions:
1. What would happen if Betsy needed $6,000 per year instead of $5,000?
2. How would the investment amounts change if the interest rate on the CD was 6% instead of 5%?
3. What is the total interest earned over 5 years with the current investment plan?
4. What is the effect of risk if Betsy decides to invest more in the B-rated bonds?
5. How does compound interest affect the total interest if the CD compounds annually?

**Tip:** When dealing with investments, always consider the risk associated with higher interest rates, as they often come with higher risk.

Discover how to allocate investments in B-rated bonds and certificates of deposit (CD) to meet specific annual income goals. This problem involves solving linear equations and understanding interest calculations. Suitable for high school students and above.

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