Three friends A, B, and C enter into a partnership investing Rs 2800, Rs 3600, and Rs 4000 respectively. After $x$ months, C leaves the business. If the ratio of C's annual profit share to the combined profit share of A and B is $5:12$, find the number of months C was involved in the business.

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**Solution:**

1. Let the number of months C stayed in the business be $x$.

2. The profit share is proportional to the product of investment and time period. So, the shares of A, B, and C are:
   - A's share = $2800 	imes 12$
   - B's share = $3600 	imes 12$
   - C's share = $4000 	imes x$

3. According to the problem, the ratio of C's profit to the combined profit of A and B is $5:12$. Therefore,
   $$\frac{4000 	imes x}{2800 	imes 12 + 3600 	imes 12} = \frac{5}{12}$$

4. Calculate the denominator:
   $$2800 	imes 12 + 3600 	imes 12 = (2800 + 3600) 	imes 12 = 6400 	imes 12 = 76800$$

5. Substitute and solve for $x$:
   $$\frac{4000x}{76800} = \frac{5}{12}$$

Multiply both sides by 76800:
   $$4000x = \frac{5}{12} 	imes 76800$$

Calculate right side:
   $$\frac{5}{12} 	imes 76800 = 5 	imes 6400 = 32000$$

So,
   $$4000x = 32000$$

$$x = \frac{32000}{4000} = 8$$

**Answer:** $8$ months

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