☼ Variation  :   If the value of  one variable changes according to the changes in the value of  some other variable, then the relation between them is called variation.

Symbol    is  used  to  denote  variation.

   x    y       x   =   k y   ;  for some non–zero integer k  (  k  0 ).

■ Some special type of  Variation :

 (i) Direct variation  :  If  the value of  one variable (x) increases  then its dependent other variable (y)  also increases and if   the value of  first variable (x) decreases then its dependent other variable (y)  also decreases, then their relationship is called direct variation.

i.e.

In this case,

one can write  x    y      x   =   k y  :  Where,  k  0

  e.g. (i) The circumference of  a circle is in direct variation with its diameter.

   (ii) The area and the altitude of  a triangle having the same base are in direct proportion

 (ii) Indirect ( or inverse ) variation  :  If  the value of  one variable (x) increases  then its dependent other variable (y)  decrease and vice versa, then their relationship is called indirect variation.

In this case,

one can write  x    y      x   =   k y  :  Where,  k  0

e.g. (i) If   price  increases,  then  demand  decreases.

(ii) The length and breadth of  a rectangle having fixed area are in inverse variation.

(iii) If  pressure increase, then volume decreases.

( i.e.  PV = k   or    )

 (iii) Compound variation  :  If  a variable z varies as the product of  two other variables x and y, then z is said to be in compound variation with x and y.

It is also said that z varies jointly as  x  and  y.

It is denoted by  z    xy    x   =   kxy  :  Where,  k  0

z  xy, z    & z    are different types of compound variation of  z.

  e.g. (i) The area of  the rectangle varies jointly as its length and breadth.

   (ii) The curved surface area of  a cylinder varies jointly as its radius and height.

 (iv) Partial variation  :   If  a variable y is divided into two parts such that one part  ‘a’  remains constant and the other  part varies ( directly or inversely ) as x, then it is said that y varies partially as x. The relation between y and x is called partial variation.

If  y varies partially directly as x, then one can write y  =  a  +  kx  ;  k  0

  e.g. (i) The total expense of  a canteen is partly constant and partly varies directly as the number of  customers.

   (ii) The cost of  any article is constant and partly depends on raw materials.

■ Some important results  :

 (i) If   x  y  and  y  z,  then  x  z.

 (ii) If   x  y  and  x  z,  then  x  ( y  z )

 (iii) If   x  z  and  y  z,  then  ( x  y )  z.

 (iv) If   x  yz,  then  y  and   z  

 (v) If   a  b  and  c  d,  then  ac  bd.

 (vi) If   x y,  then 

 (vii) If   x  y,  then    xz  yz  ( Where, z is any quantity, variable or constant. )

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