Yumi's grandparents presented her with a gift of $20,000 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 2.5%/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 3%/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 solve this problem, we'll break it into two parts:

### Part 1: Calculate the value of the investment at age 17.
Yumi's grandparents gave her $20,000, which was invested for 5 years at an interest rate of 2.5% compounded monthly.

The formula to calculate the future value of an investment compounded monthly is:

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

Where:
- $ A $ is the future value of the investment,
- $ P $ is the principal amount (initial investment),
- $ r $ is the annual interest rate (in decimal form),
- $ n $ is the number of times interest is compounded per year,
- $ t $ is the time in years.

Given:
- $ P = 20,000 $,
- $ r = 2.5\% = 0.025 $,
- $ n = 12 $ (monthly compounding),
- $ t = 5 $.

Substituting the values:

\[
A = 20000 \left(1 + \frac{0.025}{12}\right)^{12 \times 5}
\]

We'll calculate the future value $ A $.

### Part 2: Determine the equal annual withdrawals.
Once we have the future value at age 17, Yumi plans to withdraw the funds over 4 years with an interest rate of 3%, compounded annually. We will use the annuity formula to determine the equal annual installment amount.

The formula for the present value of an annuity is:

\[
PVA = \frac{C}{r} \left( 1 - \frac{1}{(1 + r)^t} \right)
\]

Where:
- $ PVA $ is the present value of the annuity (the amount available at age 17),
- $ C $ is the annual installment (what we're solving for),
- $ r $ is the interest rate per period (3% = 0.03),
- $ t $ is the number of periods (4 years).

Rearranging the formula to solve for $ C $:

\[
C = PVA \times \frac{r}{1 - (1 + r)^{-t}}
\]

We'll calculate the value of each annual installment $ C $.# Given values for Part 1
P = 20000  # Initial investment
r = 0.025  # Annual interest rate for first 5 years (2.5%)
n = 12     # Compounded monthly
t = 5      # Duration of investment in years

# Future value calculation (A)
A = P * (1 + r/n)**(n*t)
AThe value of Yumi's investment at age 17 is approximately $22,660.02.

### Part 2: Determine the Equal Annual Withdrawals

Now, we need to calculate the size of each annual installment over 4 years, with an interest rate of 3% compounded annually. We will use the annuity formula for this.

Given:
- Present value $ PVA = 22,660.02 $,
- Interest rate $ r = 0.03 $ (3% annually),
- Number of periods $ t = 4 $.

We'll now calculate the annual withdrawal amount $ C $.The size of each annual installment Yumi can withdraw over the next 4 years is approximately $6,096.16.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the installment change if the interest rate was higher?
2. What happens if Yumi wanted to withdraw the money over 5 years instead of 4?
3. How does compounding frequency affect the growth of the investment?
4. What if Yumi made withdrawals semi-annually instead of annually?
5. How would inflation affect the real value of the installments?

**Tip:** When dealing with interest calculations, always be clear about the compounding frequency as it can significantly impact your final results.

Discover how to calculate the size of each annual installment from a college fund that was invested at 2.5% compounded monthly for 5 years and is to be withdrawn over 4 years at 3% interest compounded annually. This detailed explanation is perfect for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation