If 19900 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, to the nearest cent.

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

To calculate the future value of the investment using different compounding methods, we can 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 number of years the money is invested or borrowed.

Given:
- $P = 19900$
- $r = 0.07$
- $t = 5$

### (a) Annual Compounding
For annual compounding, $n = 1$.

\[
A = 19900 \left(1 + \frac{0.07}{1}\right)^{1 \times 5} = 19900 \left(1 + 0.07\right)^{5} = 19900 \left(1.07\right)^{5}
\]

Calculating:

\[
A \approx 19900 \times 1.402552 \approx 27845.28
\]

**Annual Compounding: $27,845.28**

---

### (b) Semiannual Compounding
For semiannual compounding, $n = 2$.

\[
A = 19900 \left(1 + \frac{0.07}{2}\right)^{2 \times 5} = 19900 \left(1 + 0.035\right)^{10} = 19900 \left(1.035\right)^{10}
\]

Calculating:

\[
A \approx 19900 \times 1.410737 \approx 28147.72
\]

**Semiannual Compounding: $28,147.72**

---

### (c) Monthly Compounding
For monthly compounding, $n = 12$.

\[
A = 19900 \left(1 + \frac{0.07}{12}\right)^{12 \times 5} = 19900 \left(1 + 0.00583333\right)^{60} = 19900 \left(1.00583333\right)^{60}
\]

Calculating:

\[
A \approx 19900 \times 1.432364 \approx 28534.24
\]

**Monthly Compounding: $28,534.24**

---

### (d) Daily Compounding
For daily compounding, $n = 365$.

\[
A = 19900 \left(1 + \frac{0.07}{365}\right)^{365 \times 5} = 19900 \left(1 + 0.00019178\right)^{1825} = 19900 \left(1.00019178\right)^{1825}
\]

Calculating:

\[
A \approx 19900 \times 1.438946 \approx 28623.16
\]

**Daily Compounding: $28,623.16**

---

### Summary of Values:
- (a) Annual: **$27,845.28**
- (b) Semiannual: **$28,147.72**
- (c) Monthly: **$28,534.24**
- (d) Daily: **$28,623.16**

Do you want more details on any specific calculation or have any questions? Here are some related questions to consider:

1. How does the frequency of compounding affect the total amount earned?
2. What would the values be if the interest rate were higher or lower?
3. How does inflation affect the real value of the investment?
4. What would be the total interest earned for each compounding method?
5. How would the calculations change if the investment period was longer or shorter?

**Tip:** Always compare the final amounts for different compounding frequencies to understand how your investment grows over time.

If 19900 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: Annual, Semiannual, Monthly, Daily.

Discover how to calculate the future value of a $19,900 investment at a 7% annual interest rate over 5 years using annual, semiannual, monthly, and daily compounding methods. Results show $27,845.28, $28,147.72, $28,534.24, and $28,623.16 respectively.

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