A man invested Rs. $X$ at 15\% simple interest and Rs. $2X$ at 8\% compound interest (compounded annually) for 2 years. If the difference between the compound interest and simple interest received after 2 years is Rs. 820, find the value of $X$.

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**Solution:**

1. Let the principal amounts be Rs. $X$ at 15\% simple interest and Rs. $2X$ at 8\% compound interest.

2. Calculate the compound amount for Rs. $2X$ after 2 years at 8\% compounded annually:
   $$A = 2X 	imes (1 + \frac{8}{100})^2 = 2X 	imes (1.08)^2 = 2X 	imes 1.1664 = 2.3328X$$

3. The compound interest (CI) on Rs. $2X$ is:
   $$CI = A - P = 2.3328X - 2X = 0.3328X$$

4. Calculate the simple interest (SI) on Rs. $X$ for 2 years at 15\%:
   $$SI = \frac{X 	imes 15 	imes 2}{100} = 0.3X$$

5. According to the question, the difference between compound interest and simple interest is Rs. 820:
   $$CI - SI = 820$$
   $$0.3328X - 0.3X = 820$$
   $$0.0328X = 820$$

6. Solve for $X$:
   $$X = \frac{820}{0.0328} = 25000$$

**Answer:** Rs. 25000

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