Question

The semi annual compound interest of a sum of money in 1 year and 2 years are Rs. 2100 and Rs. 4641 respectively, find the rate of interest and the sum.​

Answer 1

Rs. 20827.

Explanation:

Let's denote the principal amount as P and the annual interest rate as r. Since interest is compounded semi-annually, the interest rate per compounding period is r/2 and the number of compounding periods in 1 year and 2 years are 2 and 4, respectively.

Using the formula for compound interest, we can write:

P(1 + r/2)^2 - P = 2100 (1)

P(1 + r/2)^4 - P = 4641 (2)

Simplifying equation (1), we get:

P[(1 + r/2)^2 - 1] = 2100

P(r + 2)/(100*2) = 2100

Simplifying equation (2), we get:

P[(1 + r/2)^4 - 1] = 4641

P(r + 2)/(100*2) * [(1 + r/2)^2 + 1] = 4641

Dividing equation (2) by equation (1), we get:

[(1 + r/2)^2 + 1]/2 = 4641/2100

(1 + r/2)^2 + 1 = 4641/1050

(1 + r/2)^2 = (4641/1050) - 1

Taking the square root of both sides, we get:

1 + r/2 = sqrt[(4641/1050) - 1]

r/2 = sqrt[(4641/1050) - 1] - 1

r = 2 * [sqrt((4641/1050) - 1) - 1]

Using equation (1) to solve for P, we get:

P = 2100 / [(1 + r/2)^2 - 1]

Substituting the value of r into the above equation, we get:

P = 2100 / [(1 + 2 * [sqrt((4641/1050) - 1) - 1]/2)^2 - 1]

Simplifying, we get:

P = 42000 / (23 + 16 * sqrt((4641/1050) - 1))

Therefore, the rate of interest is approximately 10.14% and the principal amount is approximately Rs. 20827.