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 Alligation or Mixture :-
 ::Important Formulas ::Examples

Direct Formula / Rule 7 :

• Theorem: A person has a liquid of Rs x per litre. The ratio in which water should be mixed in that liquid, so that after selling litre mixture at Rs y per litre he may get a profit of P%, is given by
= ( y )
 (x - y) + ( p ) x 100

• Example :

• A person has a chemical of Rs 25 per litre. In what ratio should water be mixed in that chemical so that after selling the mixture at Rs 20/litre he may get a profit of 25%.

• Detail Method :
Let the ratio of chemical to water in the mixture be a : b.
Cost price of the chemical is Rs 25 per litre
∴ cost price of a litre of the chemical = Rs 25
Assume that the cost price of water be Rs 0 per litre
Now, according to the question,
Selling price of the mixture = Rs 20 per litre
∴ Selling price of (a + b) litres of the mixture
= Rs (a + b) 20
Cost price of (a + b) litres of the mixture  = (a + b) X 20 X 100 125
(By the rule of fraction)
= Rs (a + b) 16
or,25 X a + 0 X b = (a + b) 16
or, 9a = 16b  or, a = 16 b 9
So a : b = 16 : 9
So Required ratio = 16 : 9.

• Ailigation Method :
In this question the alligation method is applicable on prices, so we should get the average price of mixture.
SP of mixture = Rs 20/litre, profit = 25%
 ∴ average price = 20 X 100 = Rs 16/litre 125
 Chemical Water
 25 16 0 16 9
∴ Chemical : Water = 16 : 9

• Quicker Method : Here you can use direct formula :
reqd ratio = ( 20 )
 (25 - 20) + ( 25 ) X 25 100
 = 80 = 16 = 16 : 9. 45 9

Exercise :
1. A milk seller pays Rs 500 per kilolitre for his milk. He adds water to it and sells the mixture at 56 P a litre, thereby making altogether 40% profit. Find the proportion of water to milk which his customers receive.
2. A person has a chemical of Rs 50 per litre. In what ratio should water be mixed in that chemical so that after selling the mixture at Rs 40 per litre he may get a profit of 50%.
3. A man buys milk at Rs 5 a litre and mixes it with water. By selling the mixture at Rs 4 a litre he gains 12 1/2 per cent on his outlay. How much water did each litre of the mixture contain?
4. Answers : 1 = 1 : 4,   2 = 8 : 7,   3 = 13/45 litre

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