Calculation of Compound Interest

Given: Principal amount (P) = 10000

Rate of interest (r) for the first and second year = 6%

Rate of interest (r) for the third year = 9%

Time period (t) = 3 years

Half-yearly compound interest formula:

A = P(1 + r/2)^(2t)

Where, A is the amount after 3 years

Calculation of Amount after 2 years

For the first two years, the rate of interest is 6%

Time period (t) = 2 years

A = P(1 + r/2)^(2t)

A = 10000(1 + 0.06/2)^(2*2)

A = 10000(1 + 0.03)^4

A = 10000(1.03)^4

A = 11255.52

Calculation of Amount after 3 years

For the third year, the rate of interest is 9%

Time period (t) = 1 year

Amount after 3 years = Amount after 2 years * (1 + r/2)

Amount after 3 years = 11255.52 * (1 + 0.09/2)

Amount after 3 years = 12247.34

Calculation of Compound Interest

Compound Interest = Amount after 3 years - Principal amount

Compound Interest = 12247.34 - 10000

Compound Interest = 2247.34

Therefore, the compound interest on 10000 at half-yearly rests for three years at the rate of 6% for the first two years and 9% for the third year is 2247.34.

Explanation

Compound interest is the interest earned on the principal amount as well as the accumulated interest. In this problem, we need to calculate the compound interest for three years at half-yearly rests. We first calculate the amount after two years using the given rate of interest of 6% for the first two years. Then, we calculate the amount after three years using the rate of interest of 9% for the third year. Finally, we subtract the principal amount from the amount after three years to obtain the compound interest.