A man invested Rs $(P + 500)$ at an interest rate of 10\% per annum compounded annually for 2 years and received a compound interest of Rs 1785. If he had invested Rs $(P - 1000)$ at the same rate and for the same duration, what would be the compound interest earned by him (in Rs)?

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**Solution:**

1. Let the principal amount be $P + 500$.

2. The compound interest (CI) for 2 years at 10\% p.a. is given as Rs 1785.

3. The formula for compound interest is:
   $$	ext{CI} = P 	imes \left(1 + \frac{r}{100}ight)^n - P$$
   where $P$ is principal, $r$ is rate of interest, and $n$ is time in years.

4. Substitute the values:
   $$1785 = (P + 500) 	imes \left(1 + \frac{10}{100}ight)^2 - (P + 500)$$

5. Calculate the compound factor:
   $$\left(1 + \frac{10}{100}ight)^2 = (1.1)^2 = 1.21$$

6. So,
   $$1785 = (P + 500) 	imes 1.21 - (P + 500) = (P + 500)(1.21 - 1) = (P + 500) 	imes 0.21$$

7. Therefore,
   $$P + 500 = \frac{1785}{0.21} = 8500$$

8. Hence,
   $$P = 8500 - 500 = 8000$$

9. Now, the new principal is $P - 1000 = 8000 - 1000 = 7000$.

10. Calculate the compound interest for Rs 7000 at 10\% for 2 years:
    $$	ext{CI} = 7000 	imes 1.21 - 7000 = 7000 	imes 0.21 = 1470$$

**Answer:** Rs 1470

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