Chain Rule - Aptitude MCQ Questions and Solutions with Explanations

Let the required quantity be x kg.
More workers, More quantity (Direct proportion)
More days, More quantity (Direct proportion)
\[\left. \begin{gathered} {\text{Workers 5}}:8 \hfill \\ \,\,\,\,\,\,\,\,{\text{Days 3}}:5 \hfill \\ \end{gathered} \right\}::60:x\]
$$\eqalign{ & \therefore {\text{ }}5 \times 3 \times x = 8 \times 5 \times 60 \cr & \Leftrightarrow x = \frac{{\left( {8 \times 5 \times 60} \right)}}{{\left( {5 \times 3} \right)}} \cr & \Leftrightarrow x = 160 \cr} $$

Let the required number of days be x
Less people, More days (Indirect proportion)
Less quantity, Less days ( Direct proportion)
\[\left. \begin{gathered} \,\,\,\,\,\,\,{\text{People 35}}:50 \hfill \\ {\text{Quantity 350}}:50 \hfill \\ \end{gathered} \right\}::30:x\]
$$\eqalign{ & \therefore {\text{ }}35 \times 350 \times x = 50 \times 50 \times 30 \cr & \Leftrightarrow x = \frac{{\left( {50 \times 50 \times 30} \right)}}{{\left( {35 \times 350} \right)}} \cr & \Leftrightarrow x = \frac{{300}}{{49}} \cr & \Leftrightarrow x = 6\frac{6}{{49}} \cr} $$

Let the required number of days be x
Less working hours, More days (Indirect Proportion)
More men, Less days (Indirect Proportion)
\[\left. \begin{gathered} {\text{Working hours 4}}:8 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Men 18}}:12 \hfill \\ \end{gathered} \right\}::30:x\]
$$\eqalign{ & \therefore {\text{ }}4 \times 18 \times x = 8 \times 12 \times 30 \cr & \Leftrightarrow x = \frac{{\left( {8 \times 12 \times 30} \right)}}{{\left( {4 \times 18} \right)}} \cr & \Leftrightarrow x = 40 \cr} $$

Let the number of working hours per day be x
More persons, Less working hours ( Indirect proportion)
Less days, More working hours (Indirect proportion)
\[\left. \begin{gathered} {\text{Persons 7}}:5 \hfill \\ {\text{Quantity 4}}:8 \hfill \\ \end{gathered} \right\}::7:x\]
$$\eqalign{ & \therefore {\text{ }}7 \times 4 \times x = 5 \times 8 \times 7 \cr & \Leftrightarrow x = \frac{{\left( {5 \times 8 \times 7} \right)}}{{\left( {7 \times 4} \right)}} \cr & \Leftrightarrow x = 10 \cr} $$

Let the number of working hours per day be x
More persons, Less working hours ( Indirect proportion)
Less days, More working hours (Indirect proportion)
\[\left. \begin{gathered} {\text{Persons 7}}:4 \hfill \\ {\text{Quantity 4}}:8 \hfill \\ \end{gathered} \right\}::7:x\]
$$\eqalign{ & \therefore {\text{ }}7 \times 4 \times x = 4 \times 8 \times 7 \cr & \Leftrightarrow x = \frac{{\left( {4 \times 8 \times 7} \right)}}{{\left( {7 \times 4} \right)}} \cr & \Leftrightarrow x = 8 \cr} $$

Let the required number of working hours per day be x
More pumps, Less working hours per day (Indirect proportion)
Less days, More working hours per day (Indirect proportion)
\[\left. \begin{gathered} {\text{Pumps 4}}:3 \hfill \\ \,\,\,\,\,\,{\text{Days 1}}:2 \hfill \\ \end{gathered} \right\}::8:x\]
$$\eqalign{ & \therefore {\text{ }}4 \times 1 \times x = 3 \times 2 \times 8 \cr & \Leftrightarrow x = \frac{{\left( {3 \times 2 \times 8} \right)}}{{\left( 4 \right)}} \cr & \Leftrightarrow x = 12 \cr} $$

Let the required earning be Rs. x
5 men = 7 women
7 men = $$ \left( \frac{7}{5} \times 7 \right) $$   women = $$\frac{49}{5}$$ women
∴ (7 men and 13 women)
$$\eqalign{ & = \left( {\frac{{49}}{5} + 13} \right){\text{women}} \cr & = \frac{{114}}{5}{\text{ women}} \cr} $$
Now, More women, More earning (Direct proportion)
$$\eqalign{ & \therefore {\text{ }}7 : \frac{{114}}{5}::{\text{ 5}}250 : x \cr & \Leftrightarrow 7x = \left( {\frac{{114}}{5} \times 5250} \right) \cr & \Leftrightarrow 7x = 119700 \cr & \Leftrightarrow x = 17100 \cr} $$

Let the required number of days be x
3 men = 6 women
⇒12 men = (2 × 12) women = 24 women
∴ 12 men and 8 women = (24 + 8) women = 32 women
Now, More women, Less days (Indirect Proportion)
$$\eqalign{ & \therefore {\text{ }}32:6::20:x \cr & \Leftrightarrow x = \left( {\frac{{6 \times 20}}{{32}}} \right) \cr & \Leftrightarrow x = \frac{{15}}{4} \cr & \Leftrightarrow x = 3\frac{3}{4} \cr} $$

Let the required number of days be x
Then,
More pumps, Less days (Indirect proportion)
Less working hour/day, More days (Indirect proportion)
\[\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Pumps 196}}:49 \hfill \\ {\text{Working hour/day 5}}:8 \hfill \\ \end{gathered} \right\}::\frac{{13}}{2}:x\]
$$\eqalign{ & \therefore {\text{ }}196 \times 5 \times x = 49 \times 8 \times \frac{{13}}{2} \cr & \Leftrightarrow x = \left( {49 \times 8 \times \frac{{13}}{2} \times \frac{1}{{196 \times 5}}} \right) \cr & \Leftrightarrow x = \frac{{13}}{5} \cr & \Leftrightarrow x = 2\frac{3}{5} \cr} $$

Let the required number of labourers be x
Less working hour/day, More labours (Indirect proportion)
More days, Less labourers (Indirect proportion)
\[\left. \begin{gathered} {\text{Working hours/day 6}}:7 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 30}}:18 \hfill \\ \end{gathered} \right\}::30:x\]
$$\eqalign{ & \therefore 6 \times 30 \times x = 7 \times 18 \times 30 \cr & \Leftrightarrow 6x = 126 \cr & \Leftrightarrow x = 21 \cr} $$