Jean inherited 

​$78 comma 00078,000​, 

where the terms of the inheritance state that she is to receive 

​$760760 

at the end of each 

monthmonth​, 

starting in 

threethree 

​years, until the money is completely withdrawn. If the money is placed in a savings account earning 

3.53.5​% 

compounded 

annuallyannually​, 

how long will the inheritance​ last? State your answer in years and months​ (from 0 to 11​ months).

To determine how long the inheritance will last, we need to understand the situation clearly. Jean inherited $78,000, and she is to receive $760 at the end of each month starting three years from now. The savings account earns an interest rate of 3.5% compounded annually.

### Steps to Solve:

1. **Convert the annual interest rate to a monthly rate:**
   Since interest is compounded annually, we first need to convert the annual rate to an effective monthly rate.

\[ \text{Monthly rate} = (1 + \text{annual rate})^{1/12} - 1 \]
\[ \text{Monthly rate} = (1 + 0.035)^{1/12} - 1 \approx 0.002884 \]

2. **Determine the present value of the annuity:**
   Jean will start receiving payments after 3 years. To find out how long the inheritance will last, we need to calculate the present value of the annuity at the time she starts receiving payments.

The present value of the annuity (PV) is given by:
\[ PV = \frac{PMT \times (1 - (1 + r)^{-n})}{r} \]

where:
- $ PV $ is the present value we need to find.
- $ PMT $ is the monthly payment ($760).
- $ r $ is the monthly interest rate (0.002884).
- $ n $ is the number of payments (which we need to find).

However, the present value should equal the amount available at the start of the payments after 3 years of interest accumulation on the original inheritance.

3. **Calculate the future value (FV) of the $78,000 after 3 years:**
\[ FV = PV \times (1 + \text{annual rate})^3 \]
\[ FV = 78000 \times (1 + 0.035)^3 \]
\[ FV \approx 78000 \times 1.1087 \]
\[ FV \approx 86478.6 \]

This is the amount available at the start of the payments.

4. **Determine the number of monthly payments:**
Using the future value as the present value for our annuity formula:

\[ 86478.6 = \frac{760 \times (1 - (1 + 0.002884)^{-n})}{0.002884} \]
\[ 86478.6 \times 0.002884 = 760 \times (1 - (1 + 0.002884)^{-n}) \]
\[ 249.41 = 760 \times (1 - (1 + 0.002884)^{-n}) \]
\[ \frac{249.41}{760} = 1 - (1 + 0.002884)^{-n} \]
\[ 0.328 = 1 - (1 + 0.002884)^{-n} \]
\[ (1 + 0.002884)^{-n} = 0.672 \]
\[ -n \ln(1.002884) = \ln(0.672) \]
\[ -n \times 0.00288 \approx -0.397 \]
\[ n \approx \frac{0.397}{0.00288} \]
\[ n \approx 137.85 \]

So, the number of monthly payments is approximately 138 months. Converting this to years and months:

\[ 137.85 \text{ months} \approx 11 \text{ years and 6 months} \]

### Summary

The inheritance will last approximately 11 years and 6 months after Jean starts receiving the payments.

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. How is the present value of an annuity calculated?
2. What factors affect the duration of an inheritance payout?
3. How does the compounding frequency impact the interest calculation?
4. What are the differences between nominal and effective interest rates?
5. How can Jean maximize the longevity of her inheritance?
6. What other investment options are available for inherited money?
7. How does inflation impact the value of fixed monthly payments?
8. What are the tax implications of receiving an inheritance?

**Tip:** Always consider the impact of interest rates and compounding periods when planning long-term financial withdrawals.

Learn how to calculate how long an inheritance will last using compound interest and annuities. This problem involves understanding compound interest rates, annuity payments, and present value calculations. Suitable for advanced high school students and beyond.

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