If 8000 dollars is invested in a bank account at an interest rate of 10 per cent per year, find the amount in the bank after 10 years if interest is compounded annually (a)
b) Find the amount in the bank after 10 years if interest is compounded quaterly
c) Find the amount in the bank after 10 years if interest is compounded monthly
d) Finally, find the amount in the bank after 10 years if interest is compounded continuously
To answer these questions, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = The amount of money in the bank after t years
P = The principal amount (initial investment)
r = The annual interest rate (as a decimal)
n = The number of times interest is compounded per year
t = The number of years
Now, let's calculate the amount in the bank for each case:
a) Compounded Annually
For this case, n = 1 (compounded once per year).
So, A = 8000(1 + 0.10/1)^(1*10)
A = 8000(1.10)^10
A ≈ $21,589.89
b) Compounded Quarterly
For this case, n = 4 (compounded four times per year).
So, A = 8000(1 + 0.10/4)^(4*10)
A = 8000(1.025)^40
A ≈ $22,093.54
c) Compounded Monthly
For this case, n = 12 (compounded twelve times per year).
So, A = 8000(1 + 0.10/12)^(12*10)
A = 8000(1.0083)^120
A ≈ $22,184.82
d) Compounded Continuously
For this case, n approaches infinity, but we can approximate it using the formula:
A = Pe^(rt)
Where e is the base of the natural logarithm (approximately 2.71828).
So, A = 8000e^(0.10*10)
A = 8000e^1.0
A ≈ $22,163.53
Therefore, the amount in the bank after 10 years would be approximately:
a) $21,589.89 (compounded annually)
b) $22,093.54 (compounded quarterly)
c) $22,184.82 (compounded monthly)
d) $22,163.53 (compounded continuously)