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

Direct Formula / Rule 10 :

• Theorem: A trader has N kg of a certain item, a part of which he sells at x% profit and the rest of y% loss. He gains P% on the whole  Then the quantity sold at x% profit is = ( P + y ) N Kg x + y
 and the quantity sold at y% loss is = ( x - P ) N Kg x + y

• Example :

• A trader has 50 kg of rice, a part ofwhich he sells at 10% profit and the rest at 5% loss. He gains 7% on the whole. What is the quantity sold at 10% gain and 5% loss?

• Detail Method :
Let the quantity sold at 10% profit be x kg.
Then the quantity sold at 5% loss will be (50 - x) kg.
For a matter of convenience suppose that the price of rice is 1 rupee per kg.
Then price of x kg rice = Rs x and price of (50 - x) kg rice = Rs (50 - x)
Now we get an equation,
10% profit of x + 5% loss of (50 - x) = 7% gain of 50
or, 10% of x - 5% of (50 - x) = 7% of 50
or, 10x-250 +5x =350
∴ x = 40 kg and (50 - x)= 50 - 40 = 10 kg.
Therefore, the quantity sold at 10% profit = 40 kg and
the quantity sold at 5% loss = 50 - 40 = 10 kg.

• Ailigation Method :

 I Part II Part
 10 7 (-)5 12 3
Ratio of quantities sold at 10% profit and 5% loss
= 12 : 3 = 4 : 1
Therefore, the quantity sold at 10% profit  = 50 X 4 = 40 kg, 4 + 1
and the quantity sold at 5% loss = 50 - 40 = 10 kg.
Note: Whenever there is loss, take the negative value. Here, difference between 7 and (-5) = 7 - (-5) = 7 + 5 = 12. Never take the difference that counts negative value.

• Quicker Method : Here you can use direct formula :
 Quantity sold at 10% profit = ( 7 + 5 ) X 50 10 + 5
 = 12 X 50 = 40 kg. 15
 Quantity sold at 5% loss = ( 10 - 7 ) X 50 = 10 kg. 10 + 5

Exercise :
1. A trader has 90 kg of pulse, a part of which he sells at 20% profit and the rest at 10% loss. He gains 14% on the whole. What is the quantity sold at 20% gain and 10% loss?
2. A trader has 45 kg of wheat, a part of which he sells at 30% profit and the rest at 15% loss. He gains 21% on the whole. What is the quantity sold at 30% gain and 15% loss?
3. A trader has 40 kg of tea, a part of which he sells at 12% profit and the rest at 8% loss. He gains 9% on the whole. What is the quantity sold at 12% gain and 8% loss?
4. Answers : 1 = 72 kg, 18 kg,   2 = 36 kg, 9 kg,   3 = 34 kg, 6 kg

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