Summer money lant a sum of money land 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 annually, the compound interest for 2 years at a rate of 20% per annum would be:
CI_annual = P(1 + 0.20)^2 = P(1.20)^2
If the interest is payable half-yearly, the compound interest for 2 years would be calculated as:
CI_half_yearly = P(1 + 0.10)^4 = P(1.10)^4
According to the given information, the interest payable half-yearly would fetch $482 more than if it was payable annually:
CI_half_yearly - CI_annual = 482
P(1.10)^4 - P(1.20)^2 = 482
P[(1.10)^4 - (1.20)^2] = 482
Using a calculator, we can find that (1.10)^4 = 1.4641 and (1.20)^2 = 1.44.
P(1.4641 - 1.44) = 482
P(0.0241) = 482
P = 482 / 0.0241
P ≈ $20,033.19
Therefore, the sum of money is approximately $20,033.19.
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount after interest
P = principal amount
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
Let's consider the scenario where interest is payable annually first. We'll assume that the principal amount is 'x'.
When interest is payable annually,
A = x(1 + 0.2)^2
A = x(1.2)^2
A = 1.44x
Now, let's consider the scenario where interest is payable half-yearly. In this case, the interest rate should be halved to 20%/2 = 10%.
When interest is payable half-yearly,
A' = x(1 + 0.1)^(2*2)
A' = x(1.1)^4
A' = 1.4641x
According to the given information, the difference between the two scenarios is $482.
A' - A = $482
1.4641x - 1.44x = $482
0.0241x = $482
x = $482 / 0.0241
x ≈ $20,000
Therefore, the sum of money is approximately $20,000.