A SUM OF MONEY IS KEPT IN A BANK AT 8% INTEREST COMPOUNDED ANNUALLY.HAD THE INTEREST BEEN AT 10% COMPOUNDED SEMI ANNUALLY THE AMOUNT AFTER 1 YEAR WOULD HAVE BEEN 225 MORE.WHAT WAS THE PRINCIPAL AMOUNT KEPT IN THE BANK?
let the amount be x
x(1.05)^2 - x(1.08) = 225
x(1.05^2 - 1.08) = 225
x = 10000
Let's assume the principal amount kept in the bank is P.
At an annual interest rate of 8% compounded annually, the amount after 1 year would be given by the formula:
A1 = P(1 + r/100)^n
where A1 is the amount after 1 year, r is the interest rate, and n is the number of times the interest is compounded.
Therefore, in this case:
A1 = P(1 + 8/100)^1
A1 = P(1 + 0.08)
Now, at an interest rate of 10% compounded semi-annually, the amount after 1 year would be given by the formula:
A2 = P(1 + r/100)^n
where A2 is the amount after 1 year, r is the interest rate, and n is the number of times the interest is compounded.
In this case:
A2 = P(1 + 10/100/2)^2
A2 = P(1 + 0.05)^2
A2 = P(1.05)^2
Given that the amount would have been $225 more if the interest was 10% compounded semi-annually, we can set up the following equation:
A2 - A1 = $225
P(1.05)^2 - P(1 + 0.08) = $225
Now, let's solve for P:
P(1.05)^2 - P(1.08) = $225
P(1.1025) - P(1.08) = $225
1.1025P - 1.08P = $225
0.0225P = $225
Divide both sides by 0.0225:
P ≈ $10,000
So, the principal amount kept in the bank was approximately $10,000.
To find the principal amount kept in the bank, we need to use the concept of compound interest and solve the equation.
Let's assume the principal amount is P.
According to the given information, if the interest rate is 8% compounded annually, the amount after one year would be:
A = P(1 + r/100)^n
where A is the final amount, r is the interest rate, and n is the number of times the interest is compounded in a year.
For 8% compound interest compounded annually, n = 1, so the amount after one year is:
A1 = P(1 + 8/100)^1 = P(1.08)
Now, if the interest rate is 10% compounded semi-annually, the amount after one year would be:
A2 = P(1 + r/100/n)^(n*t)
where t is the time in years.
For 10% compound interest compounded semi-annually, n = 2, so the amount after one year is:
A2 = P(1 + 10/100/2)^(2*1) = P(1.05)^2 = P(1.1025)
The problem states that the amount after one year would have been 225 more if the interest rate was 10% compounded semi-annually. So, we can set up the equation:
A2 - A1 = 225
P(1.1025) - P(1.08) = 225
1.1025P - 1.08P = 225
0.0225P = 225
P = 225 / 0.0225
P = 10,000
Therefore, the principal amount kept in the bank was $10,000.