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 ​$1776.29. After 30 years Chang will have a balance of approximately ​$3892.08.

Part 2: After 10 years Kio will have a balance of approximately ​$1822.85  After 30 years Kio will have a balance of approximately ​$_____

Let's compute the balance for Kio's investment after 30 years using 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 for in years.

For Kio:
- $ P = 1200 $
- $ r = 4.25\% = 0.0425 $
- $ n = 1 $ (since the interest is compounded annually)
- $ t = 30 $ years

Substitute the values into the formula:

\[
A = 1200 \left(1 + \frac{0.0425}{1}\right)^{1 \times 30}
\]
\[
A = 1200 \left(1 + 0.0425\right)^{30}
\]
\[
A = 1200 \left(1.0425\right)^{30}
\]

Now, let's calculate this step by step:

\[
A \approx 1200 \times 3.281030071
\]
\[
A \approx 3937.24
\]

Thus, after 30 years, Kio will have approximately **$3937.24** in the account.

### Discussion of the Difference:
After 10 years:
- Chang: **$1776.29**
- Kio: **$1822.85**

After 30 years:
- Chang: **$3892.08**
- Kio: **$3937.24**

Even though the difference in interest rates is small (4.0% vs. 4.25%), over time it significantly impacts the final balances due to the effect of compounding. After 30 years, Kio's balance is about **$45.16** higher than Chang's. This demonstrates how even a slight increase in the interest rate can lead to substantial differences in long-term savings.

Do you want more details or have any questions?

Here are five related questions you might find useful:
1. How would the balances change if the interest was compounded quarterly instead of annually?
2. What would the balances be if both had invested $1500 instead of $1200?
3. How would a decrease in the interest rate by 0.5% affect the final amount?
4. What is the formula for continuous compounding, and how would it affect these scenarios?
5. How does inflation impact the real value of these investments over 30 years?

**Tip:** When comparing savings plans, always consider how the interest rate and compounding frequency affect long-term growth. Small differences can lead to significant disparities over time.

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 ​$1776.29. After 30 years Chang will have a balance of approximately ​$3892.08. Part 2: After 10 years Kio will have a balance of approximately ​$1822.85  After 30 years Kio will have a balance of approximately ​$_____

Explore how small differences in interest rates impact long-term savings. Compare the compound interest outcomes of investing $1200 at 4.0% and 4.25% annually over 10 and 30 years. Understand the formula and concepts behind exponential growth in savings.

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