Certainly! Let's solve the problem step by step for each compounding method to find the value of the investment at the end of 5 years.
We have the following parameters:
- Principal amount ([tex]\(P\)[/tex]): [tex]$10,800
- Annual interest rate (\(r\)): 7% or 0.07
- Time period (\(t\)): 5 years
The general formula for compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \(A\) = the future value of the investment
- \(P\) = the principal investment amount
- \(r\) = the annual interest rate (decimal)
- \(n\) = number of times the interest is compounded per year
- \(t\) = the number of years the money is invested
Let's go through each compounding method.
### 1) Annual Compounding (n = 1)
For annual compounding, the interest is compounded once per year:
\[ n = 1 \]
Plugging in the values, we get:
\[ A_{\text{annual}} = 10800 \left(1 + \frac{0.07}{1}\right)^{1 \times 5} \]
This simplifies to:
\[ A_{\text{annual}} = 10800 (1.07)^5 \]
The value of the investment at the end of 5 years with annual compounding is:
\[
\boxed{15147.56}
\]
### 2) Semiannual Compounding (n = 2)
For semiannual compounding, the interest is compounded twice per year:
\[ n = 2 \]
Plugging in the values, we get:
\[ A_{\text{semiannual}} = 10800 \left(1 + \frac{0.07}{2}\right)^{2 \times 5} \]
This simplifies to:
\[ A_{\text{semiannual}} = 10800 \left(1.035\right)^{10} \]
The value of the investment at the end of 5 years with semiannual compounding is:
\[
\boxed{15234.47}
\]
### 3) Monthly Compounding (n = 12)
For monthly compounding, the interest is compounded twelve times per year:
\[ n = 12 \]
Plugging in the values, we get:
\[ A_{\text{monthly}} = 10800 \left(1 + \frac{0.07}{12}\right)^{12 \times 5} \]
This simplifies to:
\[ A_{\text{monthly}} = 10800 \left(1.0058333\right)^{60} \]
The value of the investment at the end of 5 years with monthly compounding is:
\[
\boxed{15310.35}
\]
### 4) Daily Compounding (n = 365)
For daily compounding, the interest is compounded 365 times per year:
\[ n = 365 \]
Plugging in the values, we get:
\[ A_{\text{daily}} = 10800 \left(1 + \frac{0.07}{365}\right)^{365 \times 5} \]
This simplifies to:
\[ A_{\text{daily}} = 10800 \left(1.00019178\right)^{1825} \]
The value of the investment at the end of 5 years with daily compounding is:
\[
\boxed{15325.42}
\]
In summary, the value of the investment at the end of 5 years with different compounding methods is:
1. Annual: $[/tex]15,147.56
2. Semiannual: [tex]$15,234.47
3. Monthly: $[/tex]15,310.35
4. Daily: $15,325.42
