A and B invest in a partnership business in the ratio of $3:4$. A receives Rs. 2700 as his share of the profit from a total profit of Rs. 5100. If the difference in the time periods for which A and B invested is 3 months, find the number of months A invested his money.

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**Solution:**

1. Let the investments of A and B be Rs. $3P$ and Rs. $4P$ respectively, according to the given ratio $3:4$.

2. Let the time periods for which A and B invested be $a$ months and $b$ months respectively.

3. The profit share is proportional to the product of investment and time, so:
   $$\frac{3P 	imes a}{4P 	imes b} = \frac{2700}{5100 - 2700} = \frac{2700}{2400} = \frac{9}{8}$$

4. Simplifying, we get:
   $$\frac{3a}{4b} = \frac{9}{8} \implies \frac{a}{b} = \frac{9}{8} 	imes \frac{4}{3} = \frac{3}{2}$$

5. Therefore, the ratio of the time periods is $a : b = 3m : 2m$ for some $m$.

6. Given the difference between the time periods is 3 months:
   $$3m - 2m = 3 \implies m = 3$$

7. Hence, the time period for which A invested is:
   $$a = 3m = 3 	imes 3 = 9 	ext{ months}$$

**Answer:** $9$ months

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