Alligation or Mixture -> Formula/Rule 6

Direct Formula / Rule 6 :
• Theorem: There are 'N' students in a class. Rs X are distributed among them so that each boy gets Rs x and each girl gets Rs y. Then the ratio of boys to the girls is given by

  =	(	X - Ny	)	and
  		Nx - X

  the no. of boys =	(	X - Ny	)	and
  		x - y

  the no. of girls are =	(	Nx - X	)	and
  		x - y
• Example :
There are 65 students in a class, 39 rupees are distributed among them so that each boy gets 80 P and each girl gets 30 P. Find the number of boys and girls in that class.
• Detail Method : 
  Let the ratio of boys to the girls in the

  and the no. of girls =	65 X a
  	a + b

  No. of boys =	65 X b
  	a + b

  or,	65 X a	X 80 +	65 X b	X 30 = 3900
  	a + b		a + b

  or, (5200 - 3900)a = (3900 - 1950)b

  or,	a	=	1950	=	3
  	b		1300		2

  ∴ a : b = 3 : 2

  ∴ the no. of boys =	65 X 3	= 39 and
  	5

  the no. of girls =	65 X 2	= 26
  	5

• Ailigation Method : 
  Here alligation is applicable for "money per boy or girl."

  Mean value of money per student =	3900	= 60 P
  	65

  Boys

  Girls

  80		60		30
  30				20

  ∴ Boys : Girls = 3 : 2
  

  ∴ Number of boys =	65	X 3 = 39
  	3 + 2

  and number of girls = 65 - 39 = 26.
• Quicker Method : Here you can use direct formula : 
  Applying the above theorem, we have
  N = 65 
  X = Rs 39 =3900 P
  x= 80 P 
  y = 30 P

  No. of boys =	3900 - 65 X 30	=	1950	= 39
  	80 - 30		50

  No. of girls =	65 X 80 - 3900	=	1300	= 26
  	80 - 30		50

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Exercise :
1. There are 60 students in a class, 120 rupees are distributed among them so that each boy gets Rs 2.50 and each girl gets 50 P. Find the ratio of boys to the girls.
2. There are 75 students in a class, 48 rupees are distributed among them so that each boy gets Re 1 and each girl gets 40 P. Find the number of boys and girls in that class.
3. There are 50 students in a class, 32 rupees are distributed among them so that each boy gets Re 1 and each girl gets 50 P. Find the number of girls -and boys in that class. 
Answers : 1 = 3 : 5,   2 = 30, 45,   3 = 36 girls, 14 boys