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Maths / Direct and Inverse Variation / Direct Variation
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Direct Variation

If two quantities are linked in such a way that an increase in one quantity leads to a corresponding increase in the other and vice-versa,then such a variation is called a direct variation. If two quantities are in direct variation ,then we also say that they are proportional to each other .

Illustration: If the cost of 93 m of a certain kind of plastic sheet Rs 1395, then what would it cost to buy 105 m of such plastic sheet?

Solution: Let the cost of plastic sheet 105m is x

Plastic Sheet (m) 93 105
Cost of Sheet (Rs) 1395 x

According to Direct Variation:

frac{93}{1395}=frac{105}{x}

93times x= 1395times105

On solving as these numbers we see that 1395 is divisible by 93.

x= frac{15times 93times105}{93}= 1575

x = 1575 Rs

Illustration: In which of the following tables,a and b vary directly. Also ,find the constant of variation if a and b are in direct variation .

(i)

a 4 7 21 28
b 12 21 63 84

(ii)

a 2.5 5 7.5 10 15
b 10 20 30 40 60

(iii)

a 1 2 3 4 5
b 2 1 6 3 2/5

Solution:

(i) We have,

 frac{4}{12}=frac{7}{21}=frac{21}{63}=frac{28}{84}= frac{1}{3}   

Thus, the ratio of the corresponding values of a and b is constant

frac{a}{b}=frac{1}{3}

Hence, a and b are in direct variation with the constant of variation equal to 1/3

(ii) We have, 

frac{2.5}{10}=frac{5}{20}=frac{7.5}{30}=frac{10}{40}=frac{15}{60}=frac{1}{4}.  

This shows that the ratio of the corresponding values of a and b is constant and is equal to frac{1}{4} .Thus, a and b vary directly. The constant of variation is frac{1}{4}

(iii) It is evident from the table that the ratio of the corresponding values of a and b is not constant. So, a and b are not in direct variation i.e they do not vary directly .


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