Amit, Bittu, and Chinky initiated a business venture with investments in the proportions of 8:10:9, respectively. Their shares of the profit were distributed in the ratio of 6:5:9. Calculate the ratio of the time periods during which Amit, Bittu, and Chinky invested in the business, respectively.

ATQ, Let, the time for which 'Amit', 'Bittu', and 'Chinky' invested in the business be 'a' months, 'b' months, and 'c' months, respectively. Let, the investment of 'Amit', 'Bittu', and 'Chinky' in the business be, Rs. '8p', Rs. '10p', and Rs. '9p', respectively. Profit share of 'Amit' to 'Bittu' to that of 'Chinky' = (8p × a) :(10p × b) :(9p × c) = 6:5:9 We can say that, (8p × a) :(10p × b) = 6:5 Or, (4/5) × (a/b) = (6/5) Or, (a/b) = (6/5) × (5/4) So, a:b = 3:2 (10p × b) :(9p × c) = 5:9 Or, (10/9) × (b/c) = (5/9) Or, (b/c) = (5/9) × (9/10) So, b:c = 1:2 Let, b = 2q a = (3/2) × b Or, a = (3/2) × 2q So, a = '3q' c = 2b Or, c = 2 × 2q So, c = '4q Ratio of time period of 'Amit' to 'Bittu' to that of 'Chinky' = 3q:2q:4q = 3:2:4