Alligation or Mixture problems
From a cask of wine containing 25 litres, 5 litres are withdrawn and the cask is refilled with water. The process is repeated a second and then a third time. Find the quantity of wine left in the cask and also the ratio of wine to water in the resulting mixture.
Here, quantity of wine left after third operation = [1 - (5 / 25)]3 x 25 = (4 / 5)3 x 25 = (64 / 125) x 25 = (64 / 5) = 12 4/5 liters. Final ratio of wine to water = (64 / 125) / (1- 64 /125) = (64 / 125) /(61 / 125) Wine : Water = (64 / 61)
Suppose the vessel initially contains 8 litres of liquid. Let x litres of this liquid be replaced with water. Quantity of water in new mixture = ❨ 3 - 3x + x ❩ litres 8 Quantity of syrup in new mixture = ❨ 5 - 5x ❩ litres 8 ∴ ❨ 3 - 3x + x ❩ = ❨ 5 - 5x ❩ 8 8 ⟹ 5x + 24 = 40 - 5x ⟹ 10x = 16 ⟹ x = 8 . 5 So, part of the mixture replaced = ❨ 8 x 1 ❩ = 1 . 5
w i n e ( l e f t ) w i n e ( a d d e d ) = 343 169 It means w i n e ( l e f t ) w i n e ( i n i t i a l a m o u n t ) = 343 512 (since 343 + 169 = 512) Thus, 343 x = 512 x 1 - 15 k 3 343 512 = 7 8 3 = 1 - 15 k 3 1 - 15 k = 7 8 = 1 - 1 8 Thus the initial amount of wine was 120 liters.
Ratio of milk and water = 2 : 1Quantity of milk = 60 X 2/3 = 40 litreQuantity of water = 20 litreTo make ratio, 1: 2, we have to double the water that of milkSo, water should be 80 litre.That means 80 ? 20 = 60 litre water to be added.