P and Q started a partnership business with the capital of Rs.15000 and Rs.12000 respectively. After certain months, R joined with the capital of Rs.45000. After 1 year, the ratio of the profit obtained by them is 2: 5: 2 respectively. After how many months did R invest his capital? P और Q ने क्रमशः 15000 रुपये और 12000 रुपये की पूंजी के साथ एक साझेदारी व्यवसाय शुरू किया। कुछ महीनों के बाद, R 45000 रुपये की पूंजी के साथ जुड़ गया। 1 वर्ष के बाद, उनके द्वारा प्राप्त लाभ का अनुपात क्रमशः 2: 5: 2 है। कितने महीनों के बाद R ने अपनी पूंजी का निवेश किया?

P invests 15000 for 12 months = 180000. Q invests 12000 for 12 months = 144000. R invests 45000 for x months. Ratio of investments: P:Q = 180000:144000 = 5:4. But profit ratio P:Q = 2:5. This is inconsistent - the investment ratio doesn't match profit ratio. R's investment for x months = 45000x. P's profit = 2 parts = 180000 (assuming), R's profit = 2 parts = 45000x. So 45000x = 180000, x = 4. But answer is 8. Let me reconsider: If profit ratio is 2:5:2 and investment ratio should match, then R invests 45000 for same effective period as P (180000/45000 = 4), so R invests for 4 months. But answer says 8. There might be additional context or calculation needed.