The difference between the compound interest and the simple interest on a certain sumof money at 15% p.a for three years is Rs.283.50. Find the sum.

To find the sum, we need to calculate the compound interest and the simple interest separately.

Let's start with the compound interest. The formula to calculate compound interest is:

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

Where: A = the final amount P = the principal amount (the initial sum of money) r = the annual interest rate (in decimal form) n = the number of times that interest is compounded per year t = the number of years

In this case, the interest rate is 15% per annum, which is equivalent to 0.15 in decimal form. The time period is 3 years.

Now, let's calculate the compound interest. We know that the difference between the compound interest and the simple interest is Rs. 283.50. Therefore, we can set up the equation:

A - P = 283.50

Substituting the values into the compound interest formula, we have:

P(1 + 0.15/n)^(n*3) - P = 283.50

Simplifying the equation, we get:

P[(1 + 0.15/n)^(3n) - 1] = 283.50

Next, let's calculate the simple interest. The formula to calculate simple interest is:

SI = P * r * t

Substituting the values into the simple interest formula, we have:

SI = P * 0.15 * 3

Simplifying the equation, we get:

SI = 0.45P

Now, we can set up another equation using the given information:

0.45P = 283.50

Solving this equation, we find:

P = 283.50 / 0.45

P = 630

Therefore, the sum of money is Rs. 630.