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Maths / Direct and Inverse Variation / Inverse Variation
(A Brief Glimpse of ABHYAS Content - Have aLook !!!!)

# Inverse Variation

If two quantities x and y vary with each other in such a manner that the product ab remains constant and is positive ,then we say that a and b vary inversely as each other or a varies inversely as b and b varies inversely as a.  Thus, if two quantities a and b vary inversely as each other, then the product ab always remains constant .The product ab is called the constant of variation .

If  two quanities a and b vary inversely to each other and if  $\dpi{120} b_1,b_2$ are the values of b corresponding to the values ,$\dpi{120} a_1,a_2$ of a respectively ,then

$\dpi{120} a_1\times b_1= constant \; (=k , say )\; and\; a_2 \times b_2 =k$

$\dpi{120} \because\; a_1b_1=a_2b_2$

$\dpi{120} \Rightarrow \frac{a_1}{a_2}=\frac{b_2}{b_1}$

$\dpi{120} \Rightarrow \; a_1:a_2::b_2:b_1$

Rule: if  two quantities a and b vary inversely as each other , then the ratio of any two values of  a is equal to the inverse ratio of the corresponding values of b.

Illustration:  If 52 men can do a piece of work in 35 days , in how many days 28 men will do it ?

Solution: Suppose 28 men will do the piece of the work in x days . The given information can be exhibited in the following tabular form.

 Number of men 52 28 Number of days 35 x

Clearly, we can say that less is the number of men ,more will be the number of days required to finish the work.

It is therefore , the case of inverse variation .

Ratio of Number of men = Inverse ratio of number of days

$\dpi{120} \Rightarrow \;\;\;\;\;58:28\; =x\; :35$

$\dpi{120} \Rightarrow \;\;\;\;\;\;\; \frac{52}{28}=\frac{x}{35}$

$\dpi{120} \Rightarrow \;\;52\times 35=28\times x$

$\dpi{120} \Rightarrow \frac{52 \times 35}{28}=65$

Hence, 28 men will do the work in 65 days.

Illustration: Raghu has enough money to buy 75 machines worth Rs. 200 each. How many machines he can buy if he gets a discount of Rs. 50 on each machine?

Solution: Let x be the number of machines he can buy if he gets a discount of Rs. 50 on each machine.

This information can be formulated in the following table as follows

 Number of machines 75 x Cost of machine(in Rupees) 200 150

As Raghu is getting a discount of Rs. 50 on each machine. So price of machine ig getting reduced. If price of machine is less, then he can buy more number of machines.

Hence the number of machines and cost of machine vary inversely.

Hence by the inverse variation, we have

$\frac{75}{x}=\frac{150}{200}$

By the cross multiplication, we have

$x=\frac{75\times 200}{150}$

$x=100$

Hence he can buy 100 machines.

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