At the beginning of the 1974/74 school year,a man in vested 2000.00 in a bank a 12 and half% compound interest.if he withdrew 650.00 at the end of each school year after payment of interest, to pay his son school fees, how much did he have left at the beginning of the 1978/79 school year
To determine how much the man had left at the beginning of the 1978/79 school year, we need to calculate the interest for each school year, subtract the amount he withdrew each year, and then determine the remaining balance.
First, let's calculate the interest for each school year. The interest is compounded annually at a rate of 12.5%. We can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment (including interest)
P = the principal amount (the initial investment)
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years
For each school year, the principal amount (P) is the remaining balance from the previous year. The interest rate (r) is 12.5%, which can be written as 0.125. The number of times interest is compounded per year (n) is 1 because the interest is compounded annually. And the number of years (t) is 1 for each school year.
Let's calculate the balance for each school year:
1. Calculate the balance at the end of the 1974/75 school year:
P₁ = 2000.00
r = 0.125
n = 1
t = 1
A₁ = P₁(1 + r/n)^(nt)
A₁ = 2000.00(1 + 0.125/1)^(1*1)
A₁ = 2000.00(1.125)^1
A₁ = 2000.00(1.125)
A₁ = 2250.00
After paying the school fees:
Balance₁ = A₁ - 650.00
Balance₁ = 2250.00 - 650.00
Balance₁ = 1600.00
2. Calculate the balance at the end of the 1975/76 school year:
P₂ = 1600.00
r = 0.125
n = 1
t = 1
A₂ = P₂(1 + r/n)^(nt)
A₂ = 1600.00(1 + 0.125/1)^(1*1)
A₂ = 1600.00(1.125)^1
A₂ = 1600.00(1.125)
A₂ = 1800.00
After paying the school fees:
Balance₂ = A₂ - 650.00
Balance₂ = 1800.00 - 650.00
Balance₂ = 1150.00
3. Calculate the balance at the end of the 1976/77 school year:
P₃ = 1150.00
r = 0.125
n = 1
t = 1
A₃ = P₃(1 + r/n)^(nt)
A₃ = 1150.00(1 + 0.125/1)^(1*1)
A₃ = 1150.00(1.125)^1
A₃ = 1150.00(1.125)
A₃ = 1293.75
After paying the school fees:
Balance₃ = A₃ - 650.00
Balance₃ = 1293.75 - 650.00
Balance₃ = 643.75
4. Calculate the balance at the end of the 1977/78 school year:
P₄ = 643.75
r = 0.125
n = 1
t = 1
A₄ = P₄(1 + r/n)^(nt)
A₄ = 643.75(1 + 0.125/1)^(1*1)
A₄ = 643.75(1.125)^1
A₄ = 643.75(1.125)
A₄ = 725.78
After paying the school fees:
Balance₄ = A₄ - 650.00
Balance₄ = 725.78 - 650.00
Balance₄ = 75.78
Therefore, at the beginning of the 1978/79 school year, the man had $75.78 left.