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? 

This problem can be solved in two parts: first, determining how much Grandma Ruby needs in the account initially to give Jia the $140 monthly for 5 years, and second, calculating how much money will remain after 4 years (the time Jia graduates) to give as a graduation gift.

### Part 1: Initial Amount Calculation

This is an **annuity** problem because Ruby makes regular monthly payments, and the account earns interest compounded monthly. We can use the **present value of an annuity** formula:

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

Where:
- $ PV $ is the present value (amount Ruby must initially deposit),
- $ P = 140 $ is the monthly payment,
- $ r = \frac{0.039}{12} = 0.00325 $ is the monthly interest rate (3.9% annually compounded monthly),
- $ n = 12 $ is the number of compounding periods per year,
- $ t = 5 $ is the total number of years.

Let’s compute this first.

### Part 2: Graduation Gift Calculation

Ruby will stop making payments after 4 years, but the account will still have some money left. To find out how much is left after 4 years of Jia receiving payments, we need to:
1. Calculate the remaining balance in the account at the end of 4 years.
2. Use the **future value of an annuity** formula, adjusting it to compute the remaining balance.

The formula for the **future value of an annuity** is:

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

Where we use $ t = 1 $ year to calculate how much remains in the account after 4 years of payments.

I'll calculate both values now.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can still guide you through the manual calculations. Would you like to proceed with that or ask about something else?

Here are five related questions to consider:
1. What is the difference between simple interest and compound interest?
2. How does the frequency of compounding affect the total interest earned on an account?
3. What would happen if Ruby increased her monthly allowance to $200 instead of $140?
4. How would the solution change if the interest was compounded quarterly instead of monthly?
5. What would be the future value of Jia's allowance if the interest rate were 5%?

**Tip**: When solving financial problems involving regular payments, understanding how annuities work is essential for calculating both present and future values.

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?

This page explains how to calculate the present value of a $140 monthly allowance over 5 years at a 3.9% interest compounded monthly. It also provides a method for determining the remaining balance as a graduation gift after 4 years of payments. Key concepts include annuities, compound interest, and present/future value calculations.

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