A principal of ​$5500 is invested in an account paying an annual rate of 5​%. Find the amount in the account after 4 years if the account is compounded​ semiannually, quarterly, and monthly.

To find the amount in the account after 4 years with different compounding periods, we can use the formula for compound interest:

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

where:
- $ A $ is the amount in the account after time $ t $,
- $ P $ is the principal amount (initial investment),
- $ r $ is the annual interest rate (as a decimal),
- $ n $ is the number of times interest is compounded per year,
- $ t $ is the time the money is invested for in years.

Given:
- $ P = 5500 $ dollars,
- $ r = 0.05 $ (since 5% as a decimal is 0.05),
- $ t = 4 $ years.

### 1. Semiannually (n = 2)
\[
A = 5500 \left(1 + \frac{0.05}{2}\right)^{2 \times 4}
\]
\[
A = 5500 \left(1 + 0.025\right)^{8}
\]
\[
A = 5500 \left(1.025\right)^{8}
\]
\[
A \approx 5500 \times 1.2184 = 6701.80 \, \text{dollars}
\]

### 2. Quarterly (n = 4)
\[
A = 5500 \left(1 + \frac{0.05}{4}\right)^{4 \times 4}
\]
\[
A = 5500 \left(1 + 0.0125\right)^{16}
\]
\[
A = 5500 \left(1.0125\right)^{16}
\]
\[
A \approx 5500 \times 1.2194 = 6706.70 \, \text{dollars}
\]

### 3. Monthly (n = 12)
\[
A = 5500 \left(1 + \frac{0.05}{12}\right)^{12 \times 4}
\]
\[
A = 5500 \left(1 + 0.0041667\right)^{48}
\]
\[
A = 5500 \left(1.0041667\right)^{48}
\]
\[
A \approx 5500 \times 1.2204 = 6711.33 \, \text{dollars}
\]

### Summary
- **Semiannually**: $6701.80
- **Quarterly**: $6706.70
- **Monthly**: $6711.33

Would you like any further details or explanations?

Here are some related questions you might consider:
1. What would the final amount be if compounded daily?
2. How does increasing the number of compounding periods affect the final amount?
3. What would the final amount be with continuous compounding?
4. What would happen if the interest rate were 6% instead of 5%?
5. How would the results change if the investment period was 6 years instead of 4?

**Tip:** The more frequently interest is compounded, the greater the final amount will be, but the difference becomes smaller as the frequency increases.

Learn how to calculate the amount in an account after 4 years with a principal of $5500 and an annual interest rate of 5%, compounded semiannually, quarterly, and monthly. Get detailed solutions and comparisons for each compounding period.

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