If the rate of interest is 20% p.a. and Rs. 16,000 is lent at the compound interest, half-yearly, what is the equivalent simple rate of interest for the first year?

To calculate the equivalent simple rate of interest for the first year, we need to understand how compound interest works when it is compounded half-yearly.

1. The rate of interest is 20% per annum. But since the interest is compounded half-yearly, the rate of interest for each half year would be 20%/2 = 10%.

2. Now, let's calculate the compound interest for the first year. The formula for compound interest is A = P(1 + r/n)^(nt), where:

   • A is the amount of money accumulated after n years, including interest.
   • P is the principal amount (the initial amount of money).
   • r is the annual interest rate (in decimal).
   • n is the number of times that interest is compounded per year.
   • t is the time the money is invested for in years.

3. Substituting the given values into the formula, we get:

   • A = 16000(1 + 0.10/2)^(2*1)
   • A = 16000(1 + 0.05)^2
   • A = 16000 * 1.05^2
   • A = 16000 * 1.1025
   • A = Rs. 17640

4. The interest earned in the first year is A - P = 17640 - 16000 = Rs. 1640.

5. The equivalent simple interest rate is calculated as (Interest/Principal) * 100. So, the equivalent simple interest rate for the first year is (1640/16000) * 100 = 10.25%.

So, the equivalent simple rate of interest for the first year is 10.25%.