Grandma Sybil wants to help Wei while she's in college by giving her a $110 monthly allowance for 7 years of college out of an account that earns 3.4% interest compounded monthly.

How much must Sybil have in the account for Wei to receive the $110 payments for 7 years?

When Wei graduates after 6 years, Grandma Sybil gives Wei the amount remaining in the account as a graduation gift. How much is the gift?

To determine how much Grandma Sybil needs to have in the account initially, and how much will be left after 6 years as a graduation gift, we can solve this in two parts:

### Part 1: Amount Sybil must have in the account for Wei to receive $110 monthly for 7 years
This is a classic annuity problem where payments are made monthly, and interest is compounded monthly. The formula for the present value $ PV $ of an annuity is:

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

Where:
- $ P $ = payment per period = \$110
- $ r $ = monthly interest rate = $ \frac{3.4\%}{12} = 0.0028333 $
- $ n $ = number of payments per year = 12
- $ t $ = total time in years = 7

Substituting the values into the formula, we can find the present value, or how much Sybil needs in the account initially.

### Part 2: Amount remaining in the account after 6 years
To find the remaining amount after 6 years, we need to find the future value of the account after 6 years of payments. The formula for the future value of an annuity after $ t $ years of payments is:

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

This will give us the amount left in the account after 6 years, which Sybil will give to Wei as a graduation gift.

Let's calculate these values.### Results:
1. **Initial Amount Sybil must have in the account**: Sybil needs to have approximately **$8,212.42** in the account to provide Wei with $110 monthly for 7 years.

2. **Graduation Gift after 6 years**: After 6 years of payments, the remaining amount in the account, which Sybil gives Wei as a graduation gift, will be approximately **$1,296.01**.

Would you like more details on the calculations or have any questions?

### Related Questions:
1. What if the interest rate was 4% instead of 3.4%?
2. How would the monthly payment change if the duration was 5 years instead of 7?
3. How much would Sybil need if the payments increased to $150 per month?
4. What happens if interest is compounded annually instead of monthly?
5. How would the graduation gift change if Wei graduates after 5 years instead of 6?

### Tip:
To make the most of compound interest, starting earlier or increasing the compounding frequency results in higher returns over time.

Grandma Sybil wants to help Wei while she's in college by giving her a $110 monthly allowance for 7 years of college out of an account that earns 3.4% interest compounded monthly. How much must Sybil have in the account for Wei to receive the $110 payments for 7 years? When Wei graduates after 6 years, Grandma Sybil gives Wei the amount remaining in the account as a graduation gift. How much is the gift?

Learn how to calculate the present value required for monthly payments of $110 over 7 years, and the future value left after 6 years with a 3.4% compounded monthly interest rate. Explore key concepts such as annuities and time value of money, perfect for college-level financial mathematics.

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