A sum of money lent at compound interest for 2 years at 20% per annum would fetch 482 more, if the interest was payable half yearly then it was payable annually. The sum is:
Let's assume the principal amount is P.
If the interest is payable half-yearly, then the interest rate would be 20%/2 = 10% per half-year.
So, after 2 years, the amount will be P(1 + 0.1)^4 = P(1.1)^4.
If the interest is payable annually, then the interest rate would be 20% per annum.
So, after 2 years, the amount will be P(1 + 0.2)^2 = P(1.2)^2.
According to the given information, the difference between the two amounts is $482.
So, P(1.2)^2 - P(1.1)^4 = 482.
Simplifying the above equation, we get:
1.44P - 1.4641P = 482,
-0.0241P = 482,
P = 482 / -0.0241,
P ≈ -20041.49.
Since the principal amount cannot be negative, we ignore the negative sign.
Therefore, the sum of money lent is approximately $20,041.49.
To find the sum of money, we can set up two equations based on the given information.
Let's assume the principal amount (sum of money) is P.
First, let's calculate the amount with half-yearly compounding:
Amount after 2 years with half-yearly compounding:
A1 = P(1 + (20%/2))^4 (compounded half-yearly, for 2 years)
Now, let's calculate the amount with annual compounding:
Amount after 2 years with annual compounding:
A2 = P(1 + 20%)^2 (compounded annually, for 2 years)
According to the given information, the amount with half-yearly compounding is 482 more than the amount with annual compounding. So we can set up the following equation:
A1 = A2 + 482
Substituting the values we calculated above, we get:
P(1 + (20%/2))^4 = P(1 + 20%)^2 + 482
Simplifying this equation, we have:
P(1 + 0.1)^4 = P(1 + 0.2)^2 + 482
P(1.1)^4 = P(1.2)^2 + 482
1.4641P = 1.44P + 482
1.4641P - 1.44P = 482
0.0241P = 482
P = 482 / 0.0241
P ≈ 20000
Therefore, the sum of money (principal amount) is approximately 20000.