Jack and Jill are saving for a rainy day and decide to put ​$65 away in their local bank every year for the next 20 years.  The local​ Up-the-Hill Bank will pay them 66​% on their account.

**a.**  If Jack and Jill put the money in the account faithfully at the end of every​ year, how much will they have in it at the end of 20 years?

**b.** ​ Unfortunately, Jack had an accident in which he sustained head injuries after only 10 years of savings.  The medical bill has come to ​$800. Is there enough in the​ rainy-day fund to cover​ it?

Part 1

**a.**  If Jack and Jill put the money in the account faithfully at the end of every​ year, how much will they have in it at the end of 20 years?

To solve the problem, we'll calculate the future value of an annuity where Jack and Jill deposit $65 at the end of each year for 20 years, and the interest rate is 66% annually.

The formula for the future value of an annuity is:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $FV$ is the future value of the annuity.
- $P$ is the annual payment (in this case, $65).
- $r$ is the annual interest rate (in decimal form, so 66% becomes 0.66).
- $n$ is the number of years (in this case, 20).

### Step-by-step:

1. **Annual payment $P$**: $65
2. **Interest rate $r$**: 66% = 0.66
3. **Number of years $n$**: 20

Substitute the values into the formula:

\[
FV = 65 \times \frac{(1 + 0.66)^{20} - 1}{0.66}
\]

Let's calculate this.The future value of the annuity after 20 years, with annual deposits of $65 and an interest rate of 66%, will be approximately **$2,486,073.58**.

Would you like to proceed to part (b) or need any further clarifications?

Here are 5 related questions to expand the topic:

1. How would the future value change if the interest rate were lower, say 50%?
2. What if Jack and Jill made deposits at the beginning of each year instead of at the end?
3. How does the formula differ for continuous compounding interest?
4. What would be the result if they only saved for 15 years instead of 20?
5. How much would they save without interest over the same period?

**Tip:** The higher the interest rate, the more compounding significantly boosts savings over time.

Jack and Jill are saving for a rainy day and decide to put ​$65 away in their local bank every year for the next 20 years. The local​ Up-the-Hill Bank will pay them 66​% on their account.

**a.** If Jack and Jill put the money in the account faithfully at the end of every​ year, how much will they have in it at the end of 20 years?

**b.** ​Unfortunately, Jack had an accident in which he sustained head injuries after only 10 years of savings. The medical bill has come to ​$800. Is there enough in the​ rainy-day fund to cover​ it?

Learn how to calculate the future value of an annuity where Jack and Jill deposit $65 annually with a 66% interest rate for 20 years. Detailed steps and formulas are provided for students in Grades 9-12.

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