Grandma Ruby wants to help Jia while she's in college by giving her a $140 monthly allowance for 5 years of college out of an account that earns 3.9% interest compounded monthly.

How much must Ruby have in the account for Jia to receive the $140 payments for 5 years? Correct

When Jia graduates after 4 years, Grandma Ruby gives Jia the amount remaining in the account as a graduation gift. How much is the gift? 

To solve this problem, we need to break it into two parts:

1. **How much Ruby needs in the account initially** for Jia to receive $140 monthly for 5 years (60 months) with a 3.9% interest compounded monthly.
2. **The graduation gift after 4 years**, which is the remaining balance in the account after 48 months of payments, while Jia continues to receive monthly allowances.

### Step 1: Amount Ruby needs initially for the monthly payments

This is a typical problem of **present value of an annuity** with monthly payments. The formula to calculate the present value of an annuity is:

\[
PV = PMT \times \left(1 - \left(1 + \frac{r}{n}\right)^{-nt}\right) \div \left(\frac{r}{n}\right)
\]

Where:
- $PV$ = Present value (the amount Ruby must deposit initially)
- $PMT$ = $140 (the monthly payment)
- $r$ = 3.9% annual interest rate = 0.039
- $n$ = 12 (since it's compounded monthly)
- $t$ = 5 years

Now, plugging in the values:

\[
PV = 140 \times \left(1 - \left(1 + \frac{0.039}{12}\right)^{-12 \times 5}\right) \div \left(\frac{0.039}{12}\right)
\]

I'll calculate this step now.The initial amount Ruby must deposit in the account for Jia to receive $140 monthly for 5 years is approximately **$7,620.53**.

### Step 2: Graduation Gift (Remaining Amount After 4 Years)

After 4 years, 48 months' worth of payments will have been made. We now need to calculate the remaining balance in the account after these payments. 

To do this, we calculate the future value of the remaining payments at that point (after 48 months). The formula for the **future value of an annuity** is:

\[
FV = PMT \times \left(\frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}\right)
\]

Where:
- $PMT$ = $140 (the monthly payment)
- $r$ = 3.9% annual interest rate = 0.039
- $n$ = 12 (compounded monthly)
- $t$ = 1 year (since 4 years of payments have been made, only 1 year remains)

Let's calculate the future value at the end of 4 years.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you have any further questions or need additional clarifications, feel free to ask! Meanwhile, here are five follow-up questions that might help expand on this problem:

1. How does the interest rate affect the amount Ruby needs to deposit initially?
2. What happens if Jia decides to extend her education by one year? How would this impact Ruby's calculations?
3. If the interest were compounded quarterly instead of monthly, how would Ruby’s initial deposit change?
4. How would Ruby’s total payments change if the monthly allowance was increased to $200?
5. What would happen to the final gift amount if Jia graduated after only 3 years instead of 4?

**Tip:** When dealing with long-term financial planning, small changes in the interest rate or payment frequency can significantly impact the required savings and future balance.

Grandma Ruby wants to help Jia while she's in college by giving her a $140 monthly allowance for 5 years of college out of an account that earns 3.9% interest compounded monthly. How much must Ruby have in the account for Jia to receive the $140 payments for 5 years? When Jia graduates after 4 years, Grandma Ruby gives Jia the amount remaining in the account as a graduation gift. How much is the gift?

Learn how to calculate the present value required for Grandma Ruby to give Jia a $140 monthly allowance for 5 years, with a 3.9% interest compounded monthly. This step-by-step solution also explains how to calculate the remaining balance after 4 years, which Ruby gives as a graduation gift. Suitable for advanced high school students in Grades 11-12.

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