‘P’ and ‘Q’ started a business together such that investment of ‘Q’ was 25% more than that of ‘P’. After six months, ‘P’ left the business and ‘R’ joined the business whose investment was Rs. 6,000 more than that of ‘Q’. The annual profit share of ‘Q’ is (5/11)th of the total profit. Which of the following statement(s) is/are true?
I. Ratio of annual profit share of ‘R’ and ‘P’ is 5 : 4, respectively.
II. Average investment of ‘P’ and ‘Q’ is Rs. 5,400.
III. Investment of ‘R’ is Rs. 12,000.

Let the investment of ‘P’ be Rs. ‘4m’ Investment of ‘Q’ = 1.25 × 4m = Rs. ‘5m’ Investment of ‘R’ = Rs. (5m + 6000) Ratio of annual profit share of ‘P’, ‘Q’ and ‘R’ respectively: = (4m × 6) : (5m × 12) : [(5m + 6000) × 6] = 24m : 60m : (30m + 36000) Given, Annual profit share of ‘Q’ = {5/11} of total profit {60m}/{24m + 60m + 30m + 36000} = {5/11} 11 × 60m = 5 × (114m + 36000)
660m = 570m + 180000
90m = 180000
m = 2000 Statement I Required ratio (R : P) = (30m + 36000) : 24m = (60000 + 36000) : 48000 = 96000 : 48000 = 2 : 1 But given ratio is 5 : 4 So, statement I is false. Statement II Required average = (4m + 5m) ÷ 2 = 4.5m = 4.5 × 2000 = Rs. 9,000 But given average is Rs. 5,400 So, statement II is false. Statement III Investment of ‘R’ = 5m + 6000 = 10000 + 6000 = Rs. 16,000 But given value is Rs. 12,000 So, statement III is false. None of the statements is true.