Inverse Variation


 
 
Concept Explanation
 

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 quantities a and b vary inversely to each other and if  b_1,b_2 are the values of b corresponding to the values ,a_1,a_2 of a respectively ,then

a_1times b_1= constant ; (=k , say ); and; a_2 times b_2 =k

because; a_1b_1=a_2b_2

Rightarrow frac{a_1}{a_2}=frac{b_2}{b_1}   

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

Rightarrow ;;;;;58:28; =x; :35

Rightarrow ;;;;;;; frac{52}{28}=frac{x}{35}

Rightarrow ;;52times 35=28times x

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
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Sample Questions
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Question : 1

If a and b vary inversely, fill in the blanks:

a 16 32 8
b 4 ......... ....
Right Option : B
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Question : 2

The frequency of a vibrating guitar string varies inversely with its length. Suppose a guitar string 0.80 meters long vibrates 4 times per second. What frequency would a string 0.5 meters long have?

Right Option : A
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Question : 3

Amar takes 15 days to reduce 20 kilograms of his weight by doing 20 minutes of exercise per day. If he does exercise for 1 hour per day, how many days will he take to reduce the same weight?

Right Option : A
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Explanation
 
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