Chain Rule - Aptitude MCQ Questions and Solutions with Explanations

Let the required number of chairs be x
Then, More carpenters, More chairs (Direct proportion)
More hours per day, More chairs (Direct proportion)
More days, More chairs (Direct proportion)
\[\left. \begin{gathered} \,\,\,\,{\text{Carpenters 12}}:18 \hfill \\ {\text{Hours per day 6}}:8 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 24}}:{\text{36}} \hfill \\ \end{gathered} \right\}::460:x\]
$$\eqalign{ & \therefore {\text{ }}12 \times 6 \times 24 \times x = 18 \times 8 \times 36 \times 460 \cr & \Leftrightarrow x = \frac{{\left( {18 \times 8 \times 36 \times 460} \right)}}{{\left( {12 \times 6 \times 24} \right)}} \cr & \Leftrightarrow x = 1380 \cr} $$
∴ Required number of chairs = 1380

Let the required number of chairs be x.
Then, More spiders, More flies (Direct proportion)
More time, More flies (Direct proportion)
\[\left. \begin{gathered} \,\,\,{\text{Spiders 5}}:100 \hfill \\ {\text{Minutes 5}}:100 \hfill \\ \end{gathered} \right\}::5:x\]
$$\eqalign{ & \therefore {\text{ }}5 \times 5 \times x = 100 \times 100 \times 5 \cr & \Leftrightarrow x = \frac{{\left( {100 \times 100 \times 5} \right)}}{{\left( {5 \times 5} \right)}} \cr & \Leftrightarrow x = 2000 \cr} $$

Let the required number of machine be x
More persons, More machines (Direct proportion)
More working working hours, More machines (Direct proportion)
More days, More machines (Direct proportion)
\[\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Persons 2}}:6 \hfill \\ \,\,\,\, {\text{Working hours 2}}:6 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 2}}:{\text{6}} \hfill \\ \end{gathered} \right\}::2:x\]
$$\eqalign{ & \therefore {\text{ }}2 \times 2 \times 2 \times x = 6 \times 6 \times 6 \times 2 \cr & \Leftrightarrow x = \frac{{\left( {6 \times 6 \times 6 \times 2} \right)}}{{\left( {2 \times 2 \times 2} \right)}} \cr & \Leftrightarrow x = 54 \cr} $$

Let the required number of days be x
7 men = 10 women
(14 men and 20 women) = (20 + 20) women = 40 women
More length, More days (Direct proportion)
More women, Less days (Indirect proportion)
\[\left. \begin{gathered} {\text{Length 100}}:600 \hfill \\ \,\,{\text{Women 40}}:10 \hfill \\ \end{gathered} \right\}::10:x\]
$$\eqalign{ & \therefore {\text{ }}100 \times 40 \times x = 600 \times 10 \times 10 \cr & \Leftrightarrow x = \frac{{\left( {600 \times 10 \times 10} \right)}}{{\left( {100 \times 40} \right)}} \cr & \Leftrightarrow x = 15 \cr} $$

Let the required number of binders be x
Less books, Less binders (Direct proportion)
More days, Less binders (Indirect proportion)
\[\left. \begin{gathered} {\text{Books 1400}}:800 \hfill \\ \,\,\,\,\,\,\,\,\,{\text{Days 20}}:15 \hfill \\ \end{gathered} \right\}::21:x\]
$$\eqalign{ & \therefore {\text{ }}1400 \times 20 \times x = 800 \times 15 \times 21 \cr & \Leftrightarrow x = \frac{{\left( {800 \times 15 \times 21} \right)}}{{\left( {1400 \times 20} \right)}} \cr & \Leftrightarrow x = 9 \cr} $$

Let the total number of men to be engaged be $$x$$
More length, More labourers (Direct proportion)
Less days, More labourers (Indirect proportion)
\[\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,{\text{Lentgh 26}}:39 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 6}}:18 \hfill \\ {\text{Hours per day }}9:8 \hfill \\ \end{gathered} \right\}::17:x\]
$$\eqalign{ & \therefore {\text{ }}26 \times 6 \times 9 \times x = 39 \times 18 \times 8 \times 17 \cr & \Leftrightarrow x = \frac{{\left( {39 \times 18 \times 8 \times 17} \right)}}{{\left( {26 \times 6 \times 9} \right)}} \cr & \Leftrightarrow x = 68 \cr} $$
∴ Number of more labourers = (68 -17) = 51

Let the total number of men be x
Work done = $$\frac{1}{3}$$
Remaining work
= $$\left( {1 - \frac{1}{3}} \right) = \frac{2}{3}$$
More work, More men (Direct proportion)
More days, Less men (Indirect proportion)
\[\left. \begin{gathered} \,{\text{Work }}\frac{1}{3}:\frac{2}{3} \hfill \\ {\text{Days 25}}:20 \hfill \\ \end{gathered} \right\}::20:x\]
$$\eqalign{ & \therefore {\text{ }}\left( {\frac{1}{3} \times 25 \times x} \right) = \left( {\frac{2}{3} \times 20 \times 20} \right) \cr & \Leftrightarrow x = \frac{{800}}{{25}} \cr & \Leftrightarrow x = 32 \cr} $$
∴ More men to be employed = (32 - 20) = 12

Let the required number of rounds be x
More radius, Less rounds (Indirect proportion)
$$\eqalign{ & \therefore {\text{ }}20:14::70:x \cr & \Leftrightarrow \left( {20 \times x} \right) = \left( {14 \times 70} \right) \cr & \Leftrightarrow x = \frac{{\left( {14 \times 70} \right)}}{{20}} \cr & \Leftrightarrow x = 49 \cr} $$

Remaining work
= $$\left( {1 - \frac{4}{7}} \right){\text{ = }}\frac{3}{7}$$
Remaining period = (46 - 33) = 13 days
Let the total men working at it be x
Less work, Less men (Direct proportion)
Less days, More men (Indirect proportion)
More hour/day, Less men (Indirect proportion)
\[\left. \begin{gathered} \,\,\,\,\,{\text{Work }}\frac{4}{7}:\frac{3}{7} \hfill \\ \,\,\,\,\,\,\,{\text{Men 13}}:33 \hfill \\ {\text{Hour/day }}9:8 \hfill \\ \end{gathered} \right\}::117:x\]
$$\eqalign{ & \therefore {\text{ }}\frac{4}{7} \times 13 \times 9 \times x = \frac{3}{7} \times 33 \times 8 \times 117 \cr & \Leftrightarrow x = \frac{{\left( {3 \times 33 \times 8 \times 117} \right)}}{{\left( {4 \times 13 \times 9} \right)}} \cr & \Leftrightarrow x = 198 \cr} $$
∴ Addition men to be employed = (198 - 117) = 81

Let the required number of persons be x
Less days , More persons (Indirect proportion)
More work , More persons (Direct proportion)
\[\left. \begin{gathered} \,{\text{Days 2}}:4 \hfill \\ {\text{Work 1}}:8 \hfill \\ \end{gathered} \right\}::12:x\]
$$\eqalign{ & \therefore \,2 \times 1 \times x = 4 \times 8 \times 12 \cr & \Leftrightarrow x = \frac{{\left( {4 \times 8 \times 12} \right)}}{{\left( 2 \right)}} \cr & \Leftrightarrow x = 192 \cr} $$