The following pair of investment plans are identical except for a small difference in interest rates. Compute the balance in the accounts after 10 and 30 years. Discuss the difference. Chang invests $1200 in a savings account that earns 4.0% compounded annually. Kio invests ​$1200 in a different savings account that earns 4.25% compounded annually. Part 1: After 10 years Chang will have a balance of approximately ​$_____ After 30 years Chang will have a balance of approximately $_____

To solve this problem, we'll use the formula for compound interest:

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

where:
- $ A $ is the amount of money accumulated after $ n $ years, including interest.
- $ P $ is the principal amount (the initial amount of money).
- $ r $ is the annual interest rate (decimal).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the time the money is invested or borrowed for, in years.

For Chang's investment:
- $ P = 1200 $
- $ r = 0.04 $ (4.0%)
- $ n = 1 $ (compounded annually)
- $ t = 10 $ and $ t = 30 $ years

### After 10 years:
\[
A = 1200 \left(1 + \frac{0.04}{1}\right)^{1 \times 10} = 1200 \times \left(1.04\right)^{10}
\]

Let's calculate this.

### After 30 years:
\[
A = 1200 \left(1 + \frac{0.04}{1}\right)^{1 \times 30} = 1200 \times \left(1.04\right)^{30}
\]

Now let's compute both values.

### Calculation
1. For 10 years:
   \[
   A = 1200 \times (1.04)^{10} \approx 1200 \times 1.48024 \approx 1776.29
   \]
   So, after 10 years, Chang will have approximately **$1776.29**.

2. For 30 years:
   \[
   A = 1200 \times (1.04)^{30} \approx 1200 \times 3.2434 \approx 3892.08
   \]
   So, after 30 years, Chang will have approximately **$3892.08**.

### Summary:
- After 10 years, Chang will have a balance of approximately **$1776.29**.
- After 30 years, Chang will have a balance of approximately **$3892.08**.

Would you like to see the detailed calculation for Kio's investment or discuss the differences? 

### Related Questions:
1. How much more will Kio's account be worth after 10 years compared to Chang's?
2. What is the difference in the interest earned by Chang and Kio after 30 years?
3. How would the results change if both investments were compounded semi-annually instead of annually?
4. What is the effective annual rate (EAR) for Chang’s investment compared to Kio's?
5. If Chang had invested $2000 instead of $1200, what would be his balance after 30 years?

### Tip:
Always remember that even small differences in interest rates can significantly impact the final balance, especially over long periods of time due to the power of compound interest.

Explore how small differences in interest rates can significantly affect investment outcomes over time. Learn to calculate compound interest and compare the balances for two investment plans after 10 and 30 years. Suitable for Grades 10-12 students.

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