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# Chain Rule

Formula for Chain Rule:
$$\dfrac{{{M_1}{E_1}{D_1}{H_1}}}{{{M_2}{E_2}{D_2}{H_2}}} = \dfrac{{{W_1}}}{{{W_2}}}$$
Here,
$M$ = Men or women or any person
$E$ = Efficiency of the person
$D$ = Days
$H$ = Hours
$W$ = Work
Note: In general $Inputs$ are on the left side, $Outputs$ are on the right side of the equation. Here, Men, efficiency etc are inputs. Work is output.

Solved Example:
If 12 carpenters working 6 hours a day can make 30 chairs in 24 days, how many chairs will 18 carpenters make in 36 days, each working 8 hours a day?
Solution:
Let us prepare small table to understand the problem.
$\begin{array}{*{20}{c}} {Men}&{Hours}&{Days}&{Chairs} \\ \hline {12}&6&{24}&{30} \\ {18}&8&{36}&? \\ \end{array}$
Formula: $\dfrac{{{M_1}{E_1}{D_1}{H_1}}}{{{M_2}{E_2}{D_2}{H_2}}} = \dfrac{{{W_1}}}{{{W_2}}}$
$\therefore \dfrac{{12 \times 6 \times 24}}{{18 \times 8 \times 36}} = \dfrac{{30}}{x}$
($\because$ E is omitted as the efficicney is same.)
$\Rightarrow \require{cancel}\dfrac{{{12} \times 6 \times \cancel{24}^2}}{{{18}\times 8 \times \cancel{36}^3}} = \dfrac{{30}}{x}$
$\Rightarrow \require{cancel}\dfrac{{{12} \times \cancel6 \times \cancel{24}^2}}{{\cancel{18}_3 \times 8 \times \cancel{36}^3}} = \dfrac{{30}}{x}$
$\Rightarrow \require{cancel}\dfrac{{\cancel{12}^4 \times \cancel6 \times \cancel{24}^2}}{{\cancel{18}_\cancel3 \times 8 \times \cancel{36}^3}} = \dfrac{{30}}{x}$
$\Rightarrow x = 90$

## Exercise

136 men can complete a piece of work in 18 days working 6 hours a day. In how many days will 27 men complete the same work working 8 hours a day?
A12
B18
C36
D48

23 pumps, working 6 hours a day, can empty a tank in 2 days. How many hours a day must 4 pumps work to empty the tank in 1 day?
A6
B9
C24
D32

3If five cats can kill five mice in five minutes, how long will it take 100 cats to kill 100 mice?
A5
B1
C100
D500

436 men can complete a piece of work in 18 days working 6 hours a day. In how many days will 27 men with half the efficiency complete "double" the work working 8 hours a day?
A18
B36
C72
D108

5If 18 pumps of 25 watts can raise 250 tonnes of water to a height of 15 mts in 10 days working 7 hours a day, how many pumps of 40 wtts will be required to raise 200 tonnes of water to a height of 12 mts in 14 days, working 9 hours a day?
A24
B8
C12
D4

6If 30 men working 7 hours a day can make 12 tables in 18 days, how many days will 45 women working 9 hours a day take to make 32 chairs? Given, 4 men can make 3 tables in the same time as 3 women can make 4 chairs.
A42
B21
C14
DCannot be determined

7If 9 engines consume 24 metric tonnes of coal, when each is working 8 hours per day, how much coal will be required for 8 engines, each running 13 hours a day, it is given that 3 engines of former type consume as much as 4 engines of latter type?
A13
B26
C32
DCannot be determined