Chain Rule — Part 1 MCQs 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) \[\left. \begin{gathered} {\text{Spiders}}\,\,\,\,\,1:7 \hfill \\ \,\,\,{\text{Webs}}\,\,\,\,\,7:1 \hfill \\ \end{gathered} \right\}::7:x\] $$\eqalign{ & \therefore 1 \times 7 \times x = 7 \times 1 \times 7 \cr & \Rightarrow x = 7 \cr} $$

Let the height of the building x metres. Less lengthy shadow, Less in the height (Direct Proportion) $$\eqalign{ & \therefore 40.25:28.75::17.5:x \cr & \Rightarrow 40.25 \times x = 28.75 \times 17.5 \cr & \Rightarrow x = \frac{{28.75 \times 17.5}}{{40.25}} \cr & \Rightarrow x = 12.5 \cr} $$

There is a meal for 200 children 150 children have taken the meal Remaining meal is to be catered to 50 children Now, 200 children = 120 men ∴ 50 children $$\eqalign{ & = \left( {\frac{{120}}{{200}} \times 50} \right){\text{men}} \cr & = {\text{30 men}} \cr} $$

Le the required time be x seconds. More metres, More time (Direct Proportion) $$\eqalign{ & \therefore 0.128:25::1:x \cr & \Rightarrow 0.128x = 25 \times 1 \cr & x = \frac{{25}}{{0.128}} = \frac{{25 \times 1000}}{{128}} \cr & \Rightarrow x = 195.31 \cr} $$ ∴ Required time = 195 sec (approximately)

Let the required number of days be x . Less men, More days (Indirect Proportion) $$\eqalign{ & \therefore 27:36::18:x \cr & \Rightarrow 27 \times x = 36 \times 18 \cr & \Rightarrow x = \frac{{36 \times 18}}{{27}} \cr & \Rightarrow x = 24 \cr} $$

Let the required price be Rs. x More mangoes, More price (Direct proportion) $$\eqalign{ & \therefore 357:\left( {49 \times 12} \right)::1517.25:x \cr & \Leftrightarrow 357x = \left( {49 \times 12 \times 1517.25} \right) \cr & \Leftrightarrow x = \frac{{\left( {49 \times 12 \times 1517.25} \right)}}{{357}} \cr & \Leftrightarrow x = 2499 \cr} $$ Hence, the approximate price is Rs. 2500

Let the dimension of the shorter side be x More is the longer side, More is the shorter side (Direct proportion) $$\eqalign{ & \therefore 2\frac{1}{2}:4::1\frac{7}{8}:x \cr & \Leftrightarrow \frac{5}{2}x = 4 \times \frac{{15}}{8} \cr & \Leftrightarrow x = \left( {\frac{{15}}{2} \times \frac{2}{5}} \right) \cr & \Leftrightarrow x = 3 \cr} $$

Total number of days = (30 + 31 + 30) = 91 Let the number of bananas be x More days, More bananas (Direct proportion) $$\eqalign{ & \therefore 7:91::651:x \cr & \Rightarrow 7x = 91 \times 651 \cr & \Rightarrow x = \left( {\frac{{91 \times 651}}{7}} \right) \cr & \Rightarrow x = 8463 \cr} $$

Remaining part $$\eqalign{ & = \left( {1 - \frac{4}{9}} \right) \cr & = \frac{5}{9}{\text{ }} \cr} $$ Let the required time be x minutes More volume to be filled , More time taken (Direct proportion) $$\eqalign{ & \therefore {\text{ }}\frac{4}{9}:\frac{5}{9}::1:x \cr & \Rightarrow \frac{4}{9}x = \frac{5}{9} \cr & \Rightarrow x = \left( {\frac{5}{9} \times \frac{9}{4}} \right) \cr & \Rightarrow x = \frac{5}{4}{\text{ }} \cr} $$

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) $$\eqalign{ & \therefore {\text{ }}20:16::30:x \cr & \Leftrightarrow 20x = 16 \times 30 \cr & \Leftrightarrow x = \left( {\frac{{16 \times 30}}{{20}}} \right) \cr & \Leftrightarrow x = 24 \cr} $$

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) \[\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Persons 6}}:9 \hfill \\ {\text{Working hours 8}}:6 \hfill \\ \end{gathered} \right\}::8400:x\] $$\eqalign{ & \therefore {\text{ }}6 \times 8 \times x = 9 \times 6 \times 8400 \cr & \Leftrightarrow x = \frac{{\left( {9 \times 6 \times 8400} \right)}}{{\left( {6 \times 8} \right)}} \cr & \Leftrightarrow x = 9450 \cr} $$

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 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) \[\left. \begin{gathered} {\text{Working hours/day 6}}:7 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 30}}:18 \hfill \\ \end{gathered} \right\}::30:x\] $$\eqalign{ & \therefore 6 \times 30 \times x = 7 \times 18 \times 30 \cr & \Leftrightarrow 6x = 126 \cr & \Leftrightarrow x = 21 \cr} $$

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) \[\left. \begin{gathered} {\text{Length 100}}:600 \hfill \\ \,\,{\text{Women 40}}:10 \hfill \\ \end{gathered} \right\}::10:x\] $$\eqalign{ & \therefore {\text{ }}100 \times 40 \times x = 600 \times 10 \times 10 \cr & \Leftrightarrow x = \frac{{\left( {600 \times 10 \times 10} \right)}}{{\left( {100 \times 40} \right)}} \cr & \Leftrightarrow x = 15 \cr} $$