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?

We can solve this problem by using two main concepts from finance: the **present value of an annuity** (to determine how much Grandma Sybil needs to deposit initially) and the **future value of the remaining balance** after Wei has received the monthly payments.

### Step 1: Calculate the Initial Deposit Needed

Wei will receive a monthly allowance of \$110 for 7 years, with an interest rate of 3.4% compounded monthly.

We can use the **present value of an annuity formula**:

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

Where:
- $PV$ is the present value (amount Grandma Sybil needs to deposit initially),
- $P = 110$ is the monthly payment,
- $r = \frac{3.4\%}{12} = \frac{0.034}{12}$ is the monthly interest rate,
- $n = 12$ is the number of months in a year,
- $t = 7$ is the number of years.

### Step 2: Calculate the Graduation Gift After 6 Years

Wei graduates after 6 years, so we need to calculate how much is left in the account after 6 years of receiving the monthly allowance. For this, we need to calculate the remaining balance in the account after 6 years of withdrawals using the **future value of an annuity** formula:

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

Where:
- $FV$ is the future value of the remaining amount,
- $t = 1$ (since there is 1 year left in the original 7-year term).

I'll calculate both parts now.1. **Initial Deposit Needed**: Grandma Sybil must initially deposit approximately **\$8,212.42** into the account so Wei can receive \$110 per month for 7 years.

2. **Graduation Gift**: After Wei graduates in 6 years, the amount remaining in the account (as a graduation gift) will be approximately **\$1,340.77**.

Would you like further details or have any specific questions?

Here are five related questions for deeper understanding:
1. How would the results change if the interest rate were higher?
2. What happens to the initial deposit if the allowance amount increases?
3. How does the number of compounding periods affect the future value?
4. What if Wei’s college duration was shorter, say 4 years instead of 7?
5. How would inflation affect the value of the allowance over time?

**Tip**: Understanding how compounding periods (monthly, quarterly, etc.) affect the growth of savings can help you make better financial decisions.

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 initial deposit required for a monthly allowance of $110 for 7 years with 3.4% interest compounded monthly. This step-by-step solution covers key financial concepts like the present value of an annuity and future value calculations. Suitable for advanced high school and undergraduate students.

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