Chain Rule - Aptitude MCQ Questions and Solutions with Explanations

5 cm on the map represents 1450 km
∴ 1 cm on the map represents $$\left( {\frac{{1450}}{5}} \right)$$  km = 290 km
Hence, the scale is 1 cm : 290 km i.e., $$1:29 \times {10^6}{\text{ }}$$
$$\left[ {\because 290{\text{km}} = \left( {29 \times {{10}^6}} \right){\text{cm}}} \right]$$

Let the height of the building be x metres
Lees lengthy shadow, Less is the height (Direct proportion)
$$\eqalign{ & \therefore 40.25:28.75::17.5:x \cr & \Leftrightarrow 40.25 \times x = 28.75 \times 17.5 \cr & \Leftrightarrow x = \frac{{\left( {28.75 \times 17.5} \right)}}{{40.25}} \cr & \Leftrightarrow x = 12.5 \cr} $$

Work done = $$\frac{5}{8}$$
Balance work $$ = \left( {1 - \frac{5}{8}} \right)$$   $$ = \frac{3}{8}$$
Less work, Less days ( Direct proportion)
Let the required number of days be x
Then,
$$\eqalign{ & \Leftrightarrow \frac{5}{8}:\frac{3}{8}::10:x \cr & \Leftrightarrow \frac{5}{8} \times x = \frac{3}{8} \times 10 \cr & \Leftrightarrow x = \left( {\frac{3}{8} \times 10 \times \frac{8}{5}} \right) \cr & \Leftrightarrow x = 6 \cr} $$

Let the required number of days be x
Less men, More days (Indirect proportion)
$$\eqalign{ & \therefore 42:56::24:x \cr & \Leftrightarrow 42 \times x = 56 \times 24 \cr & \Leftrightarrow x = \left( {\frac{{56 \times 24}}{{42}}} \right) \cr & \Leftrightarrow x = 32 \cr} $$

Let the required number of men be x
More days, Less men (Indirect proportion)
$$\eqalign{ & \therefore {\text{ }}20:16::30:x \cr & \Leftrightarrow 20x = 16 \times 30 \cr & \Leftrightarrow x = \left( {\frac{{16 \times 30}}{{20}}} \right) \cr & \Leftrightarrow x = 24 \cr} $$

Let the required number of mats be x
More weavers , More mates ( Direct proportion )
More days, More mats ( Direct proportion )
\[\left. \begin{gathered} {\text{Weavers }}4:8 \hfill \\ \,\,\,\,\,\,\,\,{\text{Days }}4:8 \hfill \\ \end{gathered} \right\}::4:x\]
$$\eqalign{ & \therefore 4 \times 4 \times x = 8 \times 8 \times 4 \cr & \Leftrightarrow x = \frac{{\left( {8 \times 8 \times 4} \right)}}{{\left( {4 \times 4} \right)}} \cr & \Leftrightarrow x = 16 \cr} $$

Since each maid would work with one mop,
So,we shall consider 1 maid and 1 mop as 1 unit
Let the required time be x hours
Less maids and mops, More time (Indirect proportion)
Less floor, Less time (Direct proportion)
\[\left. \begin{gathered} {\text{Maids & Mops 3}}:7 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Floors 7}}:3 \hfill \\ \end{gathered} \right\}::7:x\]
$$\eqalign{ & \therefore {\text{ }}3 \times 7 \times x = 7 \times 3 \times 7 \cr & \Leftrightarrow x = \frac{{\left( {7 \times 3 \times 7} \right)}}{{\left( {3 \times 7} \right)}} \cr & \Leftrightarrow x = 7 \cr} $$

Since each gardener would work with one grass mower,
So, we shall consider 1 gardener and 1 grass mower as one unit.
Let the required time be x hours
More gardeners and grass mowers, Less time (Indirect proportion)
More area, More time (Direct proportion)
\[\left. \begin{gathered} {\text{Gardeners & grass mowers 8}}:4 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Area 400}}:800 \hfill \\ \end{gathered} \right\}::4:x\]
$$\eqalign{ & \therefore {\text{ }}8 \times 400 \times x = 4 \times 800 \times 4 \cr & \Leftrightarrow x = \frac{{\left( {4 \times 800 \times 4} \right)}}{{\left( {8 \times 400} \right)}} \cr & \Leftrightarrow x = 4 \cr} $$

Let the required number of bottles be x
More machines, More bottles produced (Direct proportion)
Less time, Less bottles produce (Direct proportion)
\[\left. \begin{gathered} {\text{Machines 6}}:15 \hfill \\ \,\,\,\,\,\,{\text{Times 60}}:30 \hfill \\ \end{gathered} \right\}::180:x\]
$$\eqalign{ & \therefore {\text{ }}6 \times 60 \times x = 15 \times 30 \times 180 \cr & \Leftrightarrow x = \frac{{\left( {15 \times 30 \times 180} \right)}}{{\left( {6 \times 60} \right)}} \cr & \Leftrightarrow x = 225 \cr} $$

Let the weekly earning be Rs. x
More persons , More earning (Direct proportion)
Less working hours, Less earning (Direct proportion)
\[\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Persons 6}}:9 \hfill \\ {\text{Working hours 8}}:6 \hfill \\ \end{gathered} \right\}::8400:x\]
$$\eqalign{ & \therefore {\text{ }}6 \times 8 \times x = 9 \times 6 \times 8400 \cr & \Leftrightarrow x = \frac{{\left( {9 \times 6 \times 8400} \right)}}{{\left( {6 \times 8} \right)}} \cr & \Leftrightarrow x = 9450 \cr} $$