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 Alligation or Mixture :-
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Direct Formula / Rule 21 :

• Example :

• In an alloy, zinc and copper are in the ratio 1 : 2. In the second alloy the same elements are in the ratio 2 : 3. In what ratio should these two alloys be mixed to form a new alloy in which the two elements are in ratio 5 : 8 ?

• Detail Method :
Let them be mixed in the ratio x : y
Then, in 1st alloy, Zinc = x/3 and Copper = 2x/3
2nd alloy, Zinc = 2y/5 and Copper = 3y/5
Now, we have x/3 + 2y/5 : 2x/3 + 3y/5 = 5 : 8  or, 5x + 6y = 5 10x + 9y 8
or, 40x + 48y = 50x + 45y
or, 10x = 3y  ∴ x = 3 y 10
Thus, the required ratio = 3 : 10.

• Ailigation Method :
You must know that we can apply this rule over the fractional value of either zinc or copper.
Method I. Let us consider the fractional value of zinc.
 1st alloy 2nd alloy
 1/3 5/13 2/5 1/65 2/39
Therefore, they should be mixed in the ratio  = 1 : 2 or, 1 X 39 = 3 or, 3 : 10 65 39 65 2 10
Method II.Now, we try to solve it by taking fractional value of Copper.
 1st alloy 2nd alloy
 2/3 8/13 3/5 1/65 2/39
Therefore, they should be mixed in the ratio  = 1 : 2 or, 1 X 39 = 3 or, 3 : 10 65 39 65 2 10

Exercise :
1. Ajar full of whisky contains 40% of alcohol. A part of this whisky is replaced by another containing 19% alcohol and now the percentage of alcohol was found to be 26. The quantity of whisky replaced is:
2. In an alloy, zinc and copper are in the ratio 3 : 4. In the second alloy the same elements are in the ratio 4 : 5. In what ratio should these two alloys be mixed to form a new alloy in which the two elements are in ratio 7 : 3?
3. In an alloy, zinc and copper are in the ratio 2 : 3. In the second alloy the same elements are in the ratio 4 : 5. In what ratio should these two alloys be mixed to form a new alloy in which the two elements are in ratio 6 : 5?
4. Answers : 1 = 2/3,   2 = 161 : 171,   3 = 25 : 36

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