Direct and Inverse Variation


What is Direct Variation ?
Two quantities x and y are said to be in direct variation, if they increase or decrease together such that the ratio of their corresponding values remains the same.
If x and y vary directly and x1, x2 are the values of x and y1, y2 are the corresponding values of y, then 
x1/y1 = x/y2 = k   where k (a positive number) is a constant called the constant of variation.
       


What is Inverse Variation ?
Two quantities x and y are said to be in inverse variation, if an increase in x results in a decrease in y and a decrease in x results in an increase in y such that their product is a constant.
If x and y vary inversely and x1 and x2 are values of x and y1, y2 are the corresponding values of y, then
x1y1 = x2y2 = k where k is the constant of variation.

Q.1 The cost of 4 metres of a particular quality of cloth is Rs. 168. How much will be the cost of 2, 6, 10 and 12 meters of the same type ?
Solution : Suppose the length of cloth is x metres and its cost, in Rs., is y.
 
x 2 4 10 12
y y2 168 y3 y4

As the length of cloth increases, cost of the cloth also increases in the same ratio. It is a case of direct proportion.

We make use of the relation of type  x1/y1 = x2/y2  

(i) Here x1 =4, y1 = 168 and x2 = 2 
Therefore x1/y1 = x2/y gives 4/168 = 2/y2 or 4y2 = (2x168) / 4 = 84

(ii) If x3 =10, then 4/168 = 10/y or 4y3 = (168 x 10)/4 = 420

(iii) If x4 = 12, then 4/168 = 12/y or 4y4 = (168 x 12)/4 = 504.

Therefore, the missing entries are 84, 420 and 504.
 
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