Chain Rule - Aptitude MCQ Questions and Solutions with Explanations

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)
\[\left. \begin{gathered} {\text{Pumps 4}}:3 \hfill \\ {\text{Days}}\,\,\,\,\,\,\,\,{\text{1}}:2 \hfill \\ \end{gathered} \right\}::8:x\]
$$\eqalign{ & \therefore 4 \times 1 \times x = 3 \times 2 \times 8 \cr & \Rightarrow x = \frac{{ {3 \times 2 \times 8} }}{{ 4 }} \cr & \Rightarrow x = 12 \cr} $$

$$\eqalign{ & {\text{Cost of }}x{\text{ metres}} \cr & = {\text{Rs}}{\text{. }}d \cr & {\text{Coast of 1 metre}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{d}{x}} \right) \cr & {\text{Cost of }}y{\text{ metres}} \cr & = {\text{Rs}}{\text{.}}\left( {\frac{d}{x} \times y} \right) \cr & = {\text{Rs}}{\text{.}} {\frac{{yd}}{x}} \cr} $$

Let the required number of bottles be x.
More machines, More bottles (Direct Proportion)
More minutes, More bottles (Direct Proportion)
\[\left. \begin{gathered} {\text{Machines}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{6}}:10 \hfill \\ {\text{Time(in min}}{\text{.)}}\,\,{\text{1}}:4 \hfill \\ \end{gathered} \right\}::270:x\]
$$\eqalign{ & \therefore 6 \times 1 \times x = 10 \times 4 \times 270 \cr & \Rightarrow x = \frac{{ {10 \times 4 \times 270} }}{{ 6 }} \cr & \Rightarrow x = 1800 \cr} $$

After 10 days : 150 men had food for 35 days.
Suppose 125 men had food for x days.
Now, Less men, More days (Indirect Proportion)
$$\eqalign{ & \therefore 125:150::35:x \cr & \Rightarrow 125 \times x = 150 \times 35 \cr & \Rightarrow x = \frac{{150 \times 35}}{{125}} \cr & \Rightarrow x = 42 \cr} $$

Let the required number of days bex.
Less persons, More days (Indirect Proportion)
More working hours per day, Less days (Indirect Proportion)
\[\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Persons }}\,30:39 \hfill \\ {\text{Working hour/day }}\,{\text{6}}:{\text{5}} \hfill \\ \end{gathered} \right\}::12:x\]
$$\eqalign{ & \therefore 30 \times 6 \times x = 39 \times 5 \times 12 \cr & \Rightarrow x = \frac{{ {39 \times 5 \times 12} }}{{ {30 \times 6} }} \cr & \Rightarrow x = 13 \cr} $$

$$\eqalign{ & {\text{Work}}\,{\text{done}} = \frac{5}{8} \cr & {\text{Balance}}\,{\text{work}} = {1 - \frac{5}{8}} = \frac{3}{8} \cr & {\text{Let}}\,{\text{the}}\,{\text{required}}\,{\text{number}}\,{\text{of}}\,{\text{days}}\,{\text{be}}\,x \cr & {\text{Then}}, \cr &\frac{5}{8}:\frac{3}{8} :: 10:x \cr & \Rightarrow \frac{5}{8} \times x = \frac{3}{8} \times 10 \cr & \Rightarrow x = {\frac{3}{8} \times 10 \times \frac{8}{5}} \cr & \Rightarrow x = 6 \cr} $$

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{required}}\,{\text{weight}}\,{\text{be}}\,x\,{\text{kg}}. \cr & {\text{Less}}\,{\text{weight,}}\,{\text{less}}\,{\text{cost}}\,\left( {{\text{Direct}}\,{\text{Proportion}}} \right) \cr & \therefore 250:200::60:x \cr & \Rightarrow 250 \times x = {200 \times 60} \cr & \Rightarrow x = \frac{{ {200 \times 60} }}{{250}} \cr & \Rightarrow x = 48 \cr} $$

Let the required number of days be x.
Less cows, More days (Indirect Proportion)
Less bags, Less days (Direct Proportion)
\[\left. \begin{gathered} {\text{Cows}}\,\,\,\,\,1:40 \hfill \\ {\text{Bags}}\,\,\,\,\,40:1 \hfill \\ \end{gathered} \right\}::40:x\]
$$\eqalign{ & \therefore 1 \times 40 \times x = 40 \times 1 \times 40 \cr & \Rightarrow x = 40 \cr} $$

Let the required number of revolutions made by larger wheel be x.
Then, More cogs, Less revolutions (Indirect Proportion)
$$\eqalign{ & \therefore 14:6::21:x \cr & \Rightarrow 14 \times x = 6 \times 21 \cr & \Rightarrow x = \frac{{6 \times 21}}{{14}} \cr & \Rightarrow x = 9 \cr} $$

Let the required number days be x.
Less spiders, More days (Indirect Proportion)
Less webs, Less days (Direct Proportion)
\[\left. \begin{gathered} {\text{Spiders}}\,\,\,\,\,1:7 \hfill \\ \,\,\,{\text{Webs}}\,\,\,\,\,7:1 \hfill \\ \end{gathered} \right\}::7:x\]
$$\eqalign{ & \therefore 1 \times 7 \times x = 7 \times 1 \times 7 \cr & \Rightarrow x = 7 \cr} $$