The simple interest on a certain sum of money for 4 years at 10% p.a. is half the compound interest on Rs. 1000 for 2 years at 20% per annum. What is the sum placed on simple interest?

First, let's calculate the compound interest on Rs. 1000 for 2 years at 20% per annum.

The formula for compound interest is:

A = P (1 + r/n)^(nt)

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

In this case, P = 1000, r = 20/100 = 0.2, n = 1 (since it's compounded annually), and t = 2.

So,

A = 1000 (1 + 0.2/1)^(1*2) A = 1000 (1 + 0.2)^2 A = 1000 * 1.2^2 A = 1000 * 1.44 A = 1440

The compound interest is then A - P = 1440 - 1000 = Rs. 440.

The problem states that the simple interest is half of this, so the simple interest is 440/2 = Rs. 220.

The formula for simple interest is I = PRT, where I is the interest, P is the principal, R is the rate, and T is the time.

We know I = 220, R = 10/100 = 0.1, and T = 4. We want to find P.

So,

220 = P * 0.1 * 4 220 = 0.4P P = 220 / 0.4 P = Rs. 550

So, the sum placed on simple interest is Rs. 550.