P starts a business with an investment of Rs. 30 lakh. Two months later Q joined with Rs.90 lakh. Subsequently R joined with Rs. 180 lakh. The year-end profit of Rs. 1.2 crore was distributed in proportion of the investment and duration in the partnership. If the profit received by R was Rs. 60 lakh, how many months after Q. did R join?

P, Q, and R start a business with investments:
P=30 lakh,
Q=90 lakh, and
R=180 lakh.
The year-end profit is
1.2crore = 120 lakh.
Profits are distributed in proportion to investment × time.
R’s profit is 60 lakh.

Share of profit = Investment × Time (in months).

P = 12 months.
Q = 12−2=10 months.
R = X where X is the time R was in the business.

The shares of profit for P, Q, and R are proportional to:
P:30×12 = 360 
Q:90×10 = 900.
R:180×X = 180X.

Total profit share = 360+900+180t.

R’s profit is = $$\frac {180X} {360 + 900 + 180X} \times 120 = 60$$

​​$$\frac {180X} {360 + 900 + 180X}= \frac {1} {2}$$

360 + 900 +180X =360X.

1260 = 180X

X = 7

Thus, the correct answer is (D) 7