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

1

36 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


2

3 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


3

If five cats can kill five mice in five minutes, how long will it take 100 cats to kill 100 mice?

A5
B1
C100
D500


4

36 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


5

If 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


6

If 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


7

If 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