What are the compound amount and interest at the end of three years if bire 10,000 borrowed at annually interest rate of 8% A compounded monthly? B compounded quarterly C compounded semi annually ? D compounded annually E compounded continuously ?
To find the compound amount and interest at the end of three years, we'll use the formula:
A = P(1 + r/n)^(nt)
Where:
A is the amount (future value) at the end of the period,
P is the principal (the initial amount), which is $10,000,
r is the annual interest rate, which is 0.08 (8% as a decimal),
n is the number of times interest is compounded per year,
t is the number of years, which is 3 in this case.
For each case, we'll compute A and then the interest, which can be found by subtracting the initial amount (P) from A.
A) Compounded monthly (n = 12 times per year)
A = 10000(1 + 0.08/12)^(12*3)
A = 10000(1.0066667)^36
A ≈ $12,662.03
Interest = A - P = $12,662.03 - $10,000 = $2,662.03
B) Compounded quarterly (n = 4 times per year)
A = 10000(1 + 0.08/4)^(4*3)
A = 10000(1.02)^12
A ≈ $12,620.61
Interest = A - P = $12,620.61 - $10,000 = $2,620.61
C) Compounded semi-annually (n = 2 times per year)
A = 10000(1 + 0.08/2)^(2*3)
A = 10000(1.04)^6
A ≈ $12,596.90
Interest = A - P = $12,596.90 - $10,000 = $2,596.90
D) Compounded annually (n = 1 time per year)
A = 10000(1 + 0.08/1)^(1*3)
A = 10000(1.08)^3
A ≈ $12,544.32
Interest = A - P = $12,544.32 - $10,000 = $2,544.32
E) Compounded continuously
For this case, we'll use a different formula: A = Pe^(rt)
Where:
A is the amount (future value) at the end of the period,
P is the principal (the initial amount), which is $10,000,
r is the annual interest rate, which is 0.08 (8% as a decimal),
t is the number of years, which is 3,
e is the base of the natural logarithm, approximately 2.71828.
A = 10000 * e^(0.08*3)
A ≈ $12,680.09
Interest = A - P = $12,680.09 - $10,000 = $2,680.09
So the compound amount for each case is:
A) $12,662.03
B) $12,620.61
C) $12,596.90
D) $12,544.32
E) $12,680.09
And the interest at the end of three years for each case is:
A) $2,662.03
B) $2,620.61
C) $2,596.90
D) $2,544.32
E) $2,680.09