Chain Rule - Important Formulas, Questions and Study Material

Chain Rule Important Questions - Page 3
Chain Rule Video Lecture - Page 4

When quantities of different kinds are connected to one another so that we know how much of one quantity is equivalent to a given quantity of a second, etc. We can determine how much of the last kind is equivalent to a given of first kind by the chain rule.

Proportion and Indirect Proportion

If increase or decrease of a quantity $Q_{1}$ causes increase or decrease of another quantity $Q_{2}$ in the same extent then, $Q_{1}$ is directly proportional to $Q_{2}\Rightarrow Q_{1}\propto Q_{2}$

1) Number of persons $\propto$ Amount of work done, i.e. more persons, more work

2) Number of days $\propto$ Amount of work, i.e. more days, more work

3) Working rate $\propto$ Amount of work, i.e. more working rate, more work

4) Efficiency of man $\propto$ Amount of work, i.e. more efficiency of man, more work
Combining I,II,III,and IV, $(Man\times Days\times Work \,rate\times Efficiency) \propto$ Amount of work. If increase of a quantity $Q_{1} is\, indirectly \,proportional\, to \,Q_{2}\Rightarrow Q_{1}\propto \frac{1}{Q_{2}}$

5) Number of men $\propto\frac{1} {No.\,of \,days}$ , i.e. more the men, less the no. of days required

Chain Rule Important Formulas

1) $\frac{Man_{1}\times Days_{1}\times Work\, rate_{1}}{Amount \,of\, work\,done_{1}}=\frac{Man_{2}\times Days_{2}\times Work\,rate_{2}}{Amount \,of \,work\,done_{2}}$

Remember, “Man days” required per unit work is always same. In fact, $Man \times Days$ specify the volume of job or work.

2) If in place of men there are engines burning coal for certain number of hours, then, the above equation changes to

$\frac{Number\, of \,engine_{1}\times Hours_{1}\times Consumption \,Rate_{1}}{Amount\, of\, coal \,burnt_{1}}=\frac{Engine_{2}\times Hours_{2}\times Consumption \,Rate_{2}}{Amount\, of\, coal\, burnt_{2}}$

Because, here the job of engine is to burn the coal.

3) If number of examiners examining a number of answer books in number of days by working a number of hours or day, since the job of examiner is to check the answer books,

then, $\frac{Number\, of\, examiner_{1}\times Days_{1}\times Work\, Rate_{1}}{No.\,of \,answer\, books checked_{1}}=\frac{Examiner_{2}\times Days_{2}\times Work\, Rate_{2}}{No.\,of\, answer\, books\, checked_{2}}$