Yumi's grandparents presented her with a gift of $18,000 when she was 12 years old to be used for her college education. Over the next 5 years, until she turned 17, Yumi's parents had invested her money in a tax-free account that had yielded interest at the rate of 3.5%/year compounded monthly. Upon turning 17, Yumi now plans to withdraw her funds in equal annual installments over the next 4 years, starting at age 18. If the college fund is expected to earn interest at the rate of 4%/year, compounded annually, what will be the size of each installment? (Assume no interest is accrued from the point she turns 17 until she makes the first withdrawal. Round your answer to the nearest cent.)

We can solve this problem in two parts:

### Part 1: Calculating the balance in Yumi's account at age 17.

The initial gift was $18,000, which was invested for 5 years at an annual interest rate of 3.5% compounded monthly. The formula for compound interest is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

where:
- $ A $ is the amount after interest,
- $ P $ is the principal ($18,000),
- $ r $ is the annual interest rate (3.5% or 0.035),
- $ n $ is the number of times the interest is compounded per year (12 for monthly),
- $ t $ is the number of years (5 years).

Substitute the values:

\[
A = 18,000 \left(1 + \frac{0.035}{12}\right)^{12 \times 5}
\]

### Part 2: Calculating the size of each installment.

Yumi will withdraw the money in 4 equal annual installments over the next 4 years, starting at age 18. The account will earn 4% interest compounded annually during this period. The formula for the future value of an annuity (which gives equal withdrawals over time) is:

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

where:
- $ A $ is the amount of each installment,
- $ P $ is the total balance at age 17 (calculated in Part 1),
- $ r $ is the interest rate per period (4% or 0.04),
- $ n $ is the number of periods (4 years).

Now let's solve this step-by-step. I'll calculate the total amount at age 17 first.At age 17, Yumi's account balance is approximately $21,436.97.

### Now, let's calculate the size of each installment over the next 4 years, assuming the balance earns 4% interest compounded annually.

We will use the formula for annuity withdrawals, where:
\[
A = \frac{P \times (1 - (1 + r)^{-n})}{r}
\]
with:
- $ P = 21,436.97 $,
- $ r = 0.04 $ (4% annual interest),
- $ n = 4 $ (4 years).

Now, let's calculate the installment amount.The size of each annual installment Yumi can withdraw is approximately $5,905.67.

Would you like any further explanation or details?

Here are five related questions to expand on this topic:
1. How would the result change if the interest rate during the withdrawal phase was 3% instead of 4%?
2. What happens if Yumi decides to withdraw the money in monthly installments instead of yearly?
3. How would the result differ if the interest was compounded quarterly instead of annually during the withdrawal period?
4. What if Yumi's parents invested at a higher rate of 5% for the first 5 years?
5. How is compound interest different from simple interest in long-term savings?

**Tip:** Regularly contributing small amounts to a tax-free account can significantly increase savings over time due to the power of compound interest.

Learn how to calculate the size of each installment Yumi can withdraw from her college fund using compound interest and annuity formulas. This problem covers key financial concepts suitable for students in Grades 11-12, focusing on compound interest, annuity withdrawals, and time value of money.

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