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 Alligation or Mixture :-
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Direct Formula / Rule 19 :

• Theorem: If x glasses of different sizes, say S1, S2, S3...Sx, are filled with a mixture of spirit and water. The ratio of spirit and water in each glass are as follows, a1 : b1, a2 : b2, a3 : b3,...., ax : bx. If the contents of all the glasses are emptied into a single vessel, then proportion of spirit and water in it is given by  = ( a1S1 + a2S2 + ... + axSx ) : ( b1S1 + b2S2 + ... + bxSx ) a1 + b1 a2 + b2 ax + bx a1 + b1 a2 + b2 ax + bx

• Example :

• Three glasses of sizes 3 litres, 4 litres and 5 litres contain mixture of spirit and water in the ratio 2 : 3, 3 : 7 and 4 : 11 respectively. The contents of all the three glasses are poured into single vessel. Find the ratio of spirit to water in the resultant mixture

• Quicker Method : Here you can use direct formula :
 = ( 2 X 3 + 3 X 4 + 4 X 5 ) : ( 3 X 3 + 7 X 4 + 11 X 5 ) 2 + 3 3 + 7 4 + 11 2 + 3 3 + 7 4 + 11
 = ( 6 + 12 + 20 ) : ( 9 + 28 + 55 ) 5 10 15 5 10 15
 = 56 : 124 = 56 : 124 15 15
or, Spirit : Water = 14 : 31.

Exercise :
1. Three glasses of capacity 2 litres, 5 litres and 9 litres contain mixture of milk and water with milk concentrations 90%, 80% and 70% respectively. The contents of three glasses are emptied into a large vessel. Find the milk concentration and ratio of milk to water in the resultant mixture.
2. Four glasses of sizes 3 litres, 4 litres, 6 litres and 7 litres contain mixture of milk and water in the ratio 2 : 1, 5 : 3, 6 : 3 and 9 : 5 respectively. Find the ratio milk to water if the contents of all the four glasses are poured into one large vessel.
3. Two casks of 48 and 42 litres are filled with mixtures of wine and water, the proportions in the two casks being respectively 13 : 7 and 18 : 17. If the contents of the two casks be mixed, and 20 litres of water added to the whole what will be the proportion of wine to water in the result?
4. Answers : 1 = 7 : 3,   2 = 13 : 7,   3 = 12 : 13

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