Ratio And Proportion

Initial State: Jar 1: 8L Brandy (B), 0L Water(W) Jar2: 0L B, 8L W Step1: Transfer 1L B from J1 to J2. Jar1: 7L B, 0L W (Total 7L) Jar2: 1L B, 8L W (Total 9L) Step2: Transfer 1L mixture from J2 to J1. Mixture in J2: 1/9 B, 8/9 W concentration. Amount transferred: (1/9)L B, (8/9)L W. Final State: Jar 1: B = 7 - 1/9 = 62/9 L. W = 0 + 8/9 = 8/9 L. Jar 2: B=1 - 1/9=8/9L.W=8 - 8/9=64/9L. RatioRequired: (WaterinJ1): (Brandy in J2) = (8/9) : (8/9) =1:1 This matches option (3).

Let longer side = L, shorter side = S. Distance along sides = L + S. Distance along diagonal = √(L^2 + S^2) .Distance saved = (L + S) - √(L^2 + S^2) = L/2. Rearrange: √(L^2 + S^2) = L + S - L/2 = L/2 + S. Square both sides: L^2 + S^2= (L/2 + S)^2 = (L^2)/4 + LS + S^2. Simplify: L^2 = (L^2)/4 + LS (Subtract S^2 from both sides). Multiply by 4: 4L^2 = L^2 + 4LSSubtract L^2: 3L^2 = 4LS. DividebyL(since L>0): 3L = 4S → S/L = 3/4 .The ratio is 3 : 4 . This matches option (1).

CP of Spirit = 60/L. CP of Water = 0/L. SP of Mixture = 75/L. Profit % = 37.5% SP = CP * (1+Profit/100) CP of Mixture = SP / (1 + Profit%/100) = 75 / (1+ 37.5/100) = 75/ 1.375 = 54.54/L. Let the ratio of Spirit to Water = S:W. Using Alligation method, CP of Spirit CP of Water (0) CP of Mixture / / (60-54.54) = 5.45 (54.54 - 0) = 54.54 60 / / 54.54 5.45 = 10:1

Let B’s share = x. A’s share = x - 1000. C’s share = x - 1200. Total share = A + B + C=5600. (x - 1000) + x + (x - 1200) = 5600 3x - 2200=5600 3x =5600 + 2200=7800 x = 7800/3 = 2600. B’s share is 2,600. This matches option (2).