Yumi's grandparents presented her with a gift of $18,000 when she was 10 years old to be used for her college education. Over the next 7 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 4.5%/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 5%/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.)

To determine the size of each installment Yumi will withdraw from her college fund, we need to approach the problem in two steps:

1. **Calculate the value of the investment at the end of 7 years** (when Yumi turns 17).
2. **Determine the size of the equal annual installments** she can withdraw over the next 4 years.

### Step 1: Calculate the Future Value of the Investment at Age 17

The initial amount invested is $18,000. The investment earns an interest rate of 4.5% per year, compounded monthly, for 7 years.

The future value $FV$ of the investment can be calculated using the formula for compound interest:

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

where:
- $P = 18000$ (the principal amount),
- $r = 0.045$ (the annual interest rate),
- $n = 12$ (the number of compounding periods per year),
- $t = 7$ (the number of years).

Substituting the values:

\[
FV = 18000 \left(1 + \frac{0.045}{12}\right)^{12 \times 7}
\]

Let's calculate this.

\[
FV = 18000 \left(1 + \frac{0.045}{12}\right)^{84}
\]

\[
FV = 18000 \left(1 + 0.00375\right)^{84}
\]

\[
FV = 18000 \times \left(1.00375\right)^{84}
\]

Now calculate the future value.

### Step 2: Calculate the Size of Each Installment

After calculating the future value, we will determine the size of the equal annual installments over 4 years. The formula for calculating the installment amount, $A$, using the future value and the given interest rate (5% per year compounded annually), is:

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

Here:
- $FV$ is the future value calculated from Step 1,
- $r = 0.05$ (annual interest rate),
- $n = 4$ (number of installments).

Let's solve this using the calculated future value and determine $A$.# Given values for the first step
P = 18000  # Principal amount
r = 0.045  # Annual interest rate
n = 12     # Compounding periods per year
t = 7      # Number of years

# Calculate future value FV using the compound interest formula
FV = P * (1 + r/n)**(n*t)
FVThe future value of Yumi's investment at the end of 7 years, when she turns 17, is approximately **$24,650.14**.

### Step 2: Calculate the Size of Each Installment

Now, we need to calculate the size of each annual installment that Yumi can withdraw over the next 4 years. The future value $FV = 24650.14$, the interest rate is 5% per year compounded annually, and the number of installments is 4. We use the formula for the present value of an annuity to find the installment amount $A$:

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

Rearranging the formula to solve for $A$:

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

Let's calculate $A$.The size of each annual installment Yumi can withdraw over the next 4 years is approximately **$6,951.63**.

### Summary:
- **Future Value at Age 17**: $24,650.14
- **Annual Installment for 4 Years**: $6,951.63

Would you like further details or have any questions?

Here are five related questions you might consider:
1. How would the installment change if the interest rate during withdrawals was different?
2. What if the funds were withdrawn in monthly installments instead of annual?
3. How does the length of the withdrawal period affect the installment amount?
4. What would be the final amount if Yumi left the funds untouched until the end of college?
5. How does compounding frequency affect the future value of an investment?

**Tip:** Understanding how compounding works can significantly impact investment growth. The more frequent the compounding, the greater the growth over time.

Learn how to calculate the annual installments Yumi can withdraw from her $18,000 college fund, with interest rates compounded monthly and annually. This financial mathematics problem involves compound interest and annuity calculations, suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation