Results for "Nicholas invested $31,300 in a mutual fund at 3.5% compounded quarterly. After 4 years, the interest rate was changed to 6.7% compounded semi-annually. a) How much was the value of the fund 6 years after the rate change? b) How much was the total compound interest earned during the whole term?"


Let's break down the problem and solve each part step by step.

### Given Information:
- Initial investment (principal) $ P_0 = 31,300 $
- Interest rate for the first 4 years = 3.5% compounded quarterly
- Time for the first period = 4 years
- After 4 years, the interest rate changes to 6.7% compounded semi-annually
- Time for the second period = 6 years

### Formula for compound interest:
The general formula for compound interest is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- $ A $ = final amount
- $ P $ = principal amount (initial investment)
- $ r $ = annual interest rate (decimal)
- $ n $ = number of times the interest is compounded per year
- $ t $ = time in years

### Part (a): Value of the fund 6 years after the rate change

#### Step 1: Calculate the value of the investment after the first 4 years.
For the first 4 years, the rate is 3.5% compounded quarterly. Thus:
- $ P_0 = 31,300 $
- $ r = 0.035 $
- $ n = 4 $ (since it's compounded quarterly)
- $ t = 4 $ years

The amount after the first 4 years is:

\[
A_1 = 31,300 \left(1 + \frac{0.035}{4}\right)^{4 \times 4}
\]

\[
A_1 = 31,300 \left(1 + 0.00875\right)^{16}
\]

\[
A_1 = 31,300 \left(1.00875\right)^{16}
\]

\[
A_1 \approx 31,300 \times 1.148883
\]

\[
A_1 \approx 35,944.07
\]

After 4 years, the value of the investment is approximately **$35,944.07**.

#### Step 2: Calculate the value after 6 more years at the new rate (6.7% compounded semi-annually).

Now the interest rate is 6.7% compounded semi-annually, and we use the value from the first period ($ A_1 $) as the new principal:

- $ P_1 = 35,944.07 $
- $ r = 0.067 $
- $ n = 2 $ (since it's compounded semi-annually)
- $ t = 6 $ years

The amount after 6 years is:

\[
A_2 = 35,944.07 \left(1 + \frac{0.067}{2}\right)^{2 \times 6}
\]

\[
A_2 = 35,944.07 \left(1 + 0.0335\right)^{12}
\]

\[
A_2 = 35,944.07 \left(1.0335\right)^{12}
\]

\[
A_2 \approx 35,944.07 \times 1.491933
\]

\[
A_2 \approx 53,609.97
\]

After 6 years, the value of the fund is approximately **$53,609.97**.

### Part (b): Total compound interest earned during the whole term

Total compound interest earned is the difference between the final amount and the initial investment.

\[
\text{Total interest} = A_2 - P_0
\]

\[
\text{Total interest} = 53,609.97 - 31,300
\]

\[
\text{Total interest} = 22,309.97
\]

So, the total compound interest earned during the whole term is approximately **$22,309.97**.

### Summary of results:
1. The value of the fund 6 years after the rate change is approximately **$53,609.97**.
2. The total compound interest earned during the whole term is approximately **$22,309.97**.

---

Would you like more details on any part of this solution or have any specific questions?

Here are some related questions to consider:
1. How would the outcome change if the interest rates were compounded monthly?
2. What would happen if the interest rate never changed and stayed at 3.5% for 10 years?
3. How would the final value be affected if the second rate was 5% instead of 6.7%?
4. What is the impact of increasing the compounding frequency on the total interest earned?
5. How would you calculate the future value if Nicholas made additional contributions during the term?

**Tip:** The more frequently interest is compounded, the more interest you will earn over time. This is because the interest is calculated and added back into the principal more often.

Nicholas invested $31,300 in a mutual fund at 3.5% compounded quarterly. After 4 years, the interest rate was changed to 6.7% compounded semi-annually. a) How much was the value of the fund 6 years after the rate change? b) How much was the total compound interest earned during the whole term?

Discover how to calculate the compound interest on a $31,300 mutual fund investment, starting at 3.5% compounded quarterly, then switching to 6.7% compounded semi-annually after 4 years. Learn step-by-step how to find the value of the fund after 6 years and the total interest earned, using the compound interest formula.

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