Compound Interest Calculation on Half-Yearly Rests

Given:
Principal amount (P) = Rs. 10,000
Rate of interest for the first and second years (r1) = 6% per annum
Rate of interest for the third year (r2) = 9% per annum

Formula for Compound Interest:
The formula to calculate compound interest is:
A = P(1 + r/n)^(nt) - P

Where:
A = the final amount including interest
P = the principal amount
r = the annual interest rate (as a decimal)
n = the number of times that interest is compounded per year
t = the number of years

In this case, the interest is compounded on a half-yearly basis, so n = 2.

Calculating interest for the first and second years:
- For the first and second years, the interest rate is 6% per annum.
- Since the interest is compounded half-yearly, the effective interest rate for each half-year period is 6%/2 = 3%.
- The total number of half-year periods for two years is 2*2 = 4.

Using the compound interest formula:
A1 = P(1 + r/n)^(nt) - P
A1 = 10000(1 + 0.03/2)^(2*2) - 10000
A1 = 10000(1.015)^4 - 10000
A1 = 10000(1.0616025) - 10000
A1 = 10616.03 - 10000
A1 = Rs. 616.03

Calculating interest for the third year:
- For the third year, the interest rate is 9% per annum.
- Since the interest is compounded half-yearly, the effective interest rate for each half-year period is 9%/2 = 4.5%.
- The total number of half-year periods for one year is 2*1 = 2.

Using the compound interest formula:
A2 = P(1 + r/n)^(nt) - P
A2 = 10000(1 + 0.045/2)^(2*1) - 10000
A2 = 10000(1.0225)^2 - 10000
A2 = 10000(1.04500625) - 10000
A2 = Rs. 450.06

Total Compound Interest:
The total compound interest is the sum of interest earned for the first and second years (A1) and interest earned for the third year (A2).

Total Compound Interest = A1 + A2
Total Compound Interest = Rs. 616.03 + Rs. 450.06
Total Compound Interest = Rs. 1066.09

Therefore, the compound interest on Rs. 10,000 at the given rates for the first and second years being 6% and for the third year 9% per annum, with half-yearly rests, is Rs. 1066.09.