Kartik, Prakash and Vimal enter into a Partnership with investment in the ratio of (\frac{3}{2}:\frac{8}{5}:\frac{5}{3}). After six months, Kartik increases his share by 40%. If the total profit at the end of the year was Rs.136800 then find the share of Prakash in the profit.

First convert the investment ratio to common form: (3/2):(8/5):(5/3) = 45:48:50 (using LCM 30 as denominator). Kartik:Prakash:Vimal = 45:48:50. Kartik increases by 40% after 6 months, so his investment for the year is effectively 45 × 6 + 45 × 1.4 × 6 = 270 + 378 = 648. Prakash: 48 × 12 = 576. Vimal: 50 × 12 = 600. Profit ratio = 648:576:600 = 27:24:25. Total profit Rs.136800 split in 76 parts (27+24+25). Prakash's share = (24/76) × 136800 = Rs.43200, matching option C.