 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 17​% 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?                                                                                                                                The amount of money invested at 17​% =
The amount of money invested at 7​% =


Let's denote:
- $ x $ as the amount invested in B-rated bonds at 17% per year.
- $ y $ as the amount invested in the certificate of deposit (CD) at 7% per year.

Betsy wants to earn exactly $5,000 in interest per year, and she has a total of $70,000 to invest. This gives us two equations:

1. The total amount invested:
\[
x + y = 70,000
\]

2. The total interest earned from both investments:
\[
0.17x + 0.07y = 5,000
\]

### Solving the System of Equations

Let's solve for $ x $ and $ y $.

**Step 1: Solve the first equation for one of the variables, say $ y $:**

\[
y = 70,000 - x
\]

**Step 2: Substitute this expression for $ y $ in the second equation:**

\[
0.17x + 0.07(70,000 - x) = 5,000
\]

Simplify and solve for $ x $:

\[
0.17x + 4,900 - 0.07x = 5,000
\]

Combine like terms:

\[
0.10x = 100
\]

Solve for $ x $:

\[
x = \frac{100}{0.10} = 1,000
\]

**Step 3: Find $ y $ using the value of $ x $:**

\[
y = 70,000 - x = 70,000 - 1,000 = 69,000
\]

### Conclusion

- The amount of money invested at 17% is **$1,000**.
- The amount of money invested at 7% is **$69,000**.

Would you like more details on how these equations were derived, or have any other questions? 

Here are 5 related questions you might consider:
1. How does the interest rate affect the amount needed to invest to achieve a specific income?
2. What would happen if the total desired income was higher or lower?
3. How could Betsy maximize her earnings while still limiting her risk?
4. How does compound interest differ from simple interest in investments?
5. What are other investment options that provide a similar interest rate?

**Tip:** Diversifying investments can help mitigate risk while potentially achieving desired returns.

Learn how to calculate the investment distribution for Betsy, who needs $5,000 per year in extra income. This problem involves solving a system of linear equations to determine the optimal investment amounts in B-rated bonds and a certificate of deposit (CD), considering different interest rates. Suitable for advanced high school students.

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