To find the compound amount and compound interest resulting from the investment of Birr 1000 at an interest rate of 6% for 10 years, we will apply the formulas for compound interest under different compounding frequencies:
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Compounded Annually: \[ A = P(1 + r/n)^{nt} \] Where:
- \(A\) = the amount of money accumulated after n years, including interest.
- \(P\) = principal amount (the initial amount of money, which is Birr 1000).
- \(r\) = annual interest rate (decimal) (6% = 0.06).
- \(n\) = number of times that interest is compounded per year (1 for annual).
- \(t\) = number of years the money is invested for (10 years).
Plugging in the values: \[ A = 1000 \left(1 + \frac{0.06}{1}\right)^{1 \times 10} = 1000(1.06)^{10} \approx 1000 \times 1.79085 \approx 1790.85 \] Compound interest = \(A - P = 1790.85 - 1000 = 790.85\)
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Compounded Semiannually: Here \(n = 2\): \[ A = 1000 \left(1 + \frac{0.06}{2}\right)^{2 \times 10} = 1000(1 + 0.03)^{20} = 1000(1.03)^{20} \approx 1000 \times 1.80611 \approx 1806.11 \] Compound interest = \(A - P = 1806.11 - 1000 = 806.11\)
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Compounded Quarterly: Here \(n = 4\): \[ A = 1000 \left(1 + \frac{0.06}{4}\right)^{4 \times 10} = 1000(1 + 0.015)^{40} = 1000(1.015)^{40} \approx 1000 \times 1.80611 \approx 1810.51 \] Compound interest = \(A - P = 1810.51 - 1000 = 810.51\)
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Compounded Monthly: Here \(n = 12\): \[ A = 1000 \left(1 + \frac{0.06}{12}\right)^{12 \times 10} = 1000(1 + 0.005)^{120} = 1000(1.005)^{120} \approx 1000 \times 1.81940 \approx 1819.40 \] Compound interest = \(A - P = 1819.40 - 1000 = 819.40\)
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Compounded Continuously: The formula for continuous compounding is: \[ A = Pe^{rt} \] Where \(e\) is the base of the natural logarithm (approximately 2.71828). \[ A = 1000 e^{0.06 \times 10} = 1000 e^{0.6} \approx 1000 \times 1.82212 \approx 1822.12 \] Compound interest = \(A - P = 1822.12 - 1000 = 822.12\)
Summary:
- Compounded Annually:
- Amount: Birr 1790.85, Interest: Birr 790.85
- Compounded Semiannually:
- Amount: Birr 1806.11, Interest: Birr 806.11
- Compounded Quarterly:
- Amount: Birr 1810.51, Interest: Birr 810.51
- Compounded Monthly:
- Amount: Birr 1819.40, Interest: Birr 819.40
- Compounded Continuously:
- Amount: Birr 1822.12, Interest: Birr 822.12
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