If 24,200 dollars is invested at an interest rate of 7 percent per year, find the value of the investment at the end of 5 years for the following compounding methods.

Round to the nearest penny.


(a) Annual:
Your answer is 
Correct  
(b) Semiannual:
Your answer is 
Correct  
(c) Monthly:
Your answer is 
(d) Daily:
Your answer is

The formula to calculate the future value of an investment with compound interest 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), which is 24,200 dollars.
- $r$ is the annual interest rate (7% or 0.07).
- $n$ is the number of times the interest is compounded per year.
- $t$ is the number of years the investment is held (5 years in this case).

Let's calculate the future value for each compounding method.

### (a) Annual compounding ($n = 1$):

\[
A = 24,200 \left( 1 + \frac{0.07}{1} \right)^{1 \times 5}
\]
\[
A = 24,200 \times (1.07)^5
\]

Calculating this:

\[
A \approx 24,200 \times 1.402551
\]
\[
A \approx 33,948.40
\]

So, the value of the investment with annual compounding is **\$33,948.40**.

### (b) Semiannual compounding ($n = 2$):

\[
A = 24,200 \left( 1 + \frac{0.07}{2} \right)^{2 \times 5}
\]
\[
A = 24,200 \times \left( 1 + 0.035 \right)^{10}
\]
\[
A = 24,200 \times (1.035)^{10}
\]

Calculating this:

\[
A \approx 24,200 \times 1.410598
\]
\[
A \approx 34,169.47
\]

So, the value of the investment with semiannual compounding is **\$34,169.47**.

### (c) Monthly compounding ($n = 12$):

\[
A = 24,200 \left( 1 + \frac{0.07}{12} \right)^{12 \times 5}
\]
\[
A = 24,200 \times \left( 1 + 0.0058333 \right)^{60}
\]
\[
A = 24,200 \times (1.0058333)^{60}
\]

Calculating this:

\[
A \approx 24,200 \times 1.419006
\]
\[
A \approx 34,370.34
\]

So, the value of the investment with monthly compounding is **\$34,370.34**.

### (d) Daily compounding ($n = 365$):

\[
A = 24,200 \left( 1 + \frac{0.07}{365} \right)^{365 \times 5}
\]
\[
A = 24,200 \times \left( 1 + 0.00019178 \right)^{1825}
\]
\[
A = 24,200 \times (1.00019178)^{1825}
\]

Calculating this:

\[
A \approx 24,200 \times 1.421012
\]
\[
A \approx 34,396.49
\]

So, the value of the investment with daily compounding is **\$34,396.49**.

---

Would you like further explanation or have any questions?

Here are 5 related questions to consider:
1. How does increasing the compounding frequency affect the final value?
2. What would the future value be if the interest rate were higher?
3. How would the investment change if compounded continuously instead of daily?
4. What is the formula for continuous compounding and how does it differ from discrete compounding?
5. How does the duration of the investment (t) impact the final amount?

**Tip:** The more frequently the interest is compounded, the greater the investment's future value, but the effect diminishes as the frequency increases.

If 24,200 dollars is invested at an interest rate of 7 percent per year, find the value of the investment at the end of 5 years for the following compounding methods.

(a) Annual:
(b) Semiannual:
(c) Monthly:
(d) Daily:

Find out how much a $24,200 investment grows over 5 years at a 7% annual interest rate using different compounding methods: annual, semiannual, monthly, and daily. Learn how compound interest calculations affect investment value based on compounding frequency.

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