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 Alligation or Mixture :-
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Direct Formula / Rule 14 :

• Theorem: In a group, there are some 4-legged creatures and some 2-legged creatures. If heads are counted, there are x and if leggs are counted there are y,
then the no. of 4-legged creatures are given by  = ( y - 2x ) 2
 or = ( Total legs - 2 X Total heads ) and 2
the no. of 2-legged creatures are given by  = ( 4x - y ) 2
 or = ( 4 X Total heads - Total legs ) and 2

• Example :

• In a zoo, there are rabbits and pigeons. If heads are counted, there are 200 and if legs are counted, there are 580. How many pigeons are there?

• Detail Method :
Let the no. of rabbits be R and the pigeons be P.
According to the question
R + P = 200 ....(i) and
4R + 2P = 580 .... (ii)
[so Rabbits are 4-legged creatures and pigeons are 2-legged creatures.]
From solving eqn (i) and (ii) we get
R = 90, and P = 110
No. of rabbits = 90 and
No. of pigeons = 110.

• Ailigation Method :
Rule of Alligation is applicable on number of legs per head.
 Average number of legs per head = 580 = 29 200 10
 Rabbit Pigeons
 4 29/10 2 9/10 11/10
= Rabbit : Pigeons = 9 : 11  So number of pigeons = 200 X 11 = 110 9 + 11

• Quicker Method : Here you can use direct formula :
 the number of pigeons (2-legged) = 4 X 200 - 580 = 110 2

Exercise :
1. In a courtyard there are many chickens and goats. If heads are counted, it comes to 100 but when legs are counted, it comes to 320. Find the number of chickens and goats in the courtyard.
2. In a zoo, there are rabbits and pigeons. lf heads are counted, there are 100 and if legs are counted, there are 290. How many rabbits are there?
3. In a zoo, there are rabbits and pigeons. If heads are counted, there are 50 and if legs are counted, there are 140. How many pigeons are there?
4. Answers : 1 = 40, 60,   2 = 45,   3 = 30

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