A started a business with an investment of Rs.1200. After some months, B join the business with an investment of Rs.3600 and after two more months C joins the business with Rs.2800. If the annual profit share of A and B is equal, then find for how many months C invested his money?

The correct option is 2 months.

Let's solve the problem step-by-step to find out how long C invested his money in the business.

Step 1: Understand the formula for profit sharing
The share of profit in a partnership is directly proportional to the product of the investment amount and the time period (in months) for which the money was invested.
$\text{Profit Share}=\text{Investment}\×\text{Time Period}$

Step 2: Calculate A's investment details
A started the business and remained for the entire year (12 months).
A's Investment = Rs. 1200
A's Time Period = 12 months
A's Profit Equivalent = $1200\×12=14400$

Step 3: Calculate B's investment details and time period
Let B join the business after $x$ months.
This means B invested money for $(12-x)$ months.
B's Investment = Rs. 3600
B's Profit Equivalent = $3600\×(12-x)$

According to the question, the annual profit share of A and B is equal:
$14400=3600\×(12-x)$

Dividing both sides by 3600:
$12-x=\frac{14400}{3600}$

$12-x=4$

$x=12-4=8$
Thus, B joined after 8 months, which means B invested his money for 4 months.

Step 4: Determine C's investment time period
The question states that C joined the business two more months after B joined.
Since B joined after 8 months, C joined after:
$8+2=10\text{months}$

Therefore, the number of months C invested his money in the 12-month period is:
$12-10=2\text{months}$

Conclusion:
C invested his money for 2 months.