Answer :
To calculate the compound amount and compound interest, we'll use different compounding frequencies as described in the question. First, let's establish the key variables involved in the calculations:
- Principal (`P`): The initial amount invested, which is Birr 25,000.
- Annual interest rate (`r`): 8%, or 0.08 as a decimal for calculation purposes.
- Time (`t`): The investment period, which is 17 years.
The formula for calculating the compound amount with different compounding frequencies is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex] where: - `A` is the future value of the investment (compound amount), - `P` is the principal amount, - `r` is the annual interest rate (in decimal form), - `n` is the number of times the interest is compounded per year, - `t` is the time in years. **Compounded annually (`n=1`):** - We plug the above values into our formula: [tex]\[ A = 25000 \left(1 + \frac{0.08}{1}\right)^{1 \times 17} \][/tex] [tex]\[ A = 25000 \times (1.08)^{17} \][/tex] - Calculate the compound amount using a calculator or a piece of software that can handle exponentiation. The compound interest is then calculated as the difference between the compound amount and the principal: [tex]\[ \text{Compound Interest} = A - P \][/tex] **Compounded semiannually (`n=2`):** - Using the same formula with `n=2`: [tex]\[ A = 25000 \left(1 + \frac{0.08}{2}\right)^{2 \times 17} \][/tex] [tex]\[ A = 25000 \times \left(1.04\right)^{34} \][/tex] - Calculate the amount and then the interest. **Compounded quarterly (`n=4`):** - Similarly, for quarterly compounding we have `n=4`: [tex]\[ A = 25000 \left(1 + \frac{0.08}{4}\right)^{4 \times 17} \][/tex] [tex]\[ A = 25000 \times \left(1.02\right)^{68} \][/tex] - Calculate the amount and then the interest. **Compounded monthly (`n=12`):** [tex]\[ A = 25000 \left(1 + \frac{0.08}{12}\right)^{12 \times 17} \][/tex] [tex]\[ A = 25000 \times \left(1 + \frac{0.08}{12}\right)^{204} \][/tex] - Calculate the amount and then the interest. **Compounded weekly (`n=52`):** [tex]\[ A = 25000 \left(1 + \frac{0.08}{52}\right)^{52 \times 17} \][/tex] [tex]\[ A = 25000 \times \left(1 + \frac{0.08}{52}\right)^{884} \][/tex] - Calculate the amount and then the interest. **Compounded daily (`n=365`):** [tex]\[ A = 25000 \left(1 + \frac{0.08}{365}\right)^{365 \times 17} \][/tex] [tex]\[ A = 25000 \times \left(1 + \frac{0.08}{365}\right)^{6205} \][/tex] - Calculate the amount and then the interest. **Compounded hourly (`n=8760`, assuming 24 hours in a day):** [tex]\[ A = 25000 \left(1 + \frac{0.08}{8760}\right)^{8760 \times 17} \][/tex] [tex]\[ A = 25000 \times \left(1 + \frac{0.08}{8760}\right)^{148920} \][/tex] - Calculate the amount and then the interest. **Compounded continuously:** The formula for continuous compounding is a bit different: [tex]\[ A = Pe^{rt} \][/tex] where `e` is Euler's number, an irrational and transcendental number approximately equal to 2.71828. Using the values given: [tex]\[ A = 25000 \times e^{0.08 \times 17} \][/tex] [tex]\[ A = 25000 \times e^{1.36} \][/tex]
- Calculate the amount and then the interest.
For all the calculations above, it is essential to use a scientific calculator or appropriate software to accurately compute the powers and exponentials to find the compound amount for each compounding frequency. Once the amount (`A`) is determined for each case, subtract the principal (`P`) to find the compound interest accumulated over the period.
- Principal (`P`): The initial amount invested, which is Birr 25,000.
- Annual interest rate (`r`): 8%, or 0.08 as a decimal for calculation purposes.
- Time (`t`): The investment period, which is 17 years.
The formula for calculating the compound amount with different compounding frequencies is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex] where: - `A` is the future value of the investment (compound amount), - `P` is the principal amount, - `r` is the annual interest rate (in decimal form), - `n` is the number of times the interest is compounded per year, - `t` is the time in years. **Compounded annually (`n=1`):** - We plug the above values into our formula: [tex]\[ A = 25000 \left(1 + \frac{0.08}{1}\right)^{1 \times 17} \][/tex] [tex]\[ A = 25000 \times (1.08)^{17} \][/tex] - Calculate the compound amount using a calculator or a piece of software that can handle exponentiation. The compound interest is then calculated as the difference between the compound amount and the principal: [tex]\[ \text{Compound Interest} = A - P \][/tex] **Compounded semiannually (`n=2`):** - Using the same formula with `n=2`: [tex]\[ A = 25000 \left(1 + \frac{0.08}{2}\right)^{2 \times 17} \][/tex] [tex]\[ A = 25000 \times \left(1.04\right)^{34} \][/tex] - Calculate the amount and then the interest. **Compounded quarterly (`n=4`):** - Similarly, for quarterly compounding we have `n=4`: [tex]\[ A = 25000 \left(1 + \frac{0.08}{4}\right)^{4 \times 17} \][/tex] [tex]\[ A = 25000 \times \left(1.02\right)^{68} \][/tex] - Calculate the amount and then the interest. **Compounded monthly (`n=12`):** [tex]\[ A = 25000 \left(1 + \frac{0.08}{12}\right)^{12 \times 17} \][/tex] [tex]\[ A = 25000 \times \left(1 + \frac{0.08}{12}\right)^{204} \][/tex] - Calculate the amount and then the interest. **Compounded weekly (`n=52`):** [tex]\[ A = 25000 \left(1 + \frac{0.08}{52}\right)^{52 \times 17} \][/tex] [tex]\[ A = 25000 \times \left(1 + \frac{0.08}{52}\right)^{884} \][/tex] - Calculate the amount and then the interest. **Compounded daily (`n=365`):** [tex]\[ A = 25000 \left(1 + \frac{0.08}{365}\right)^{365 \times 17} \][/tex] [tex]\[ A = 25000 \times \left(1 + \frac{0.08}{365}\right)^{6205} \][/tex] - Calculate the amount and then the interest. **Compounded hourly (`n=8760`, assuming 24 hours in a day):** [tex]\[ A = 25000 \left(1 + \frac{0.08}{8760}\right)^{8760 \times 17} \][/tex] [tex]\[ A = 25000 \times \left(1 + \frac{0.08}{8760}\right)^{148920} \][/tex] - Calculate the amount and then the interest. **Compounded continuously:** The formula for continuous compounding is a bit different: [tex]\[ A = Pe^{rt} \][/tex] where `e` is Euler's number, an irrational and transcendental number approximately equal to 2.71828. Using the values given: [tex]\[ A = 25000 \times e^{0.08 \times 17} \][/tex] [tex]\[ A = 25000 \times e^{1.36} \][/tex]
- Calculate the amount and then the interest.
For all the calculations above, it is essential to use a scientific calculator or appropriate software to accurately compute the powers and exponentials to find the compound amount for each compounding frequency. Once the amount (`A`) is determined for each case, subtract the principal (`P`) to find the compound interest accumulated over the period.
