Practice MCQ Questions and Answer on Chain Rule

Solution:

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

Solution:

 Cost of 21 pencils and 9 clippers = Rs. 819 Cost of 7 pencils and 3 clippers = $$\frac{{819}}{3}$$ = Rs. 273

Solution:

 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

Solution:

 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) \[\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Persons 2}}:6 \hfill \\ \,\,\,\, {\text{Working hours 2}}:6 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 2}}:{\text{6}} \hfill \\ \end{gathered} \right\}::2:x\] $$\eqalign{ & \therefore {\text{ }}2 \times 2 \times 2 \times x = 6 \times 6 \times 6 \times 2 \cr & \Leftrightarrow x = \frac{{\left( {6 \times 6 \times 6 \times 2} \right)}}{{\left( {2 \times 2 \times 2} \right)}} \cr & \Leftrightarrow x = 54 \cr} $$

Solution:

 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) \[\left. \begin{gathered} {\text{Gardeners & grass mowers 8}}:4 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Area 400}}:800 \hfill \\ \end{gathered} \right\}::4:x\] $$\eqalign{ & \therefore {\text{ }}8 \times 400 \times x = 4 \times 800 \times 4 \cr & \Leftrightarrow x = \frac{{\left( {4 \times 800 \times 4} \right)}}{{\left( {8 \times 400} \right)}} \cr & \Leftrightarrow x = 4 \cr} $$

Solution:

 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

Solution:

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

Solution:

 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., $$1:29 \times {10^6}{\text{ }}$$ $$\left[ {\because 290{\text{km}} = \left( {29 \times {{10}^6}} \right){\text{cm}}} \right]$$

Solution:

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

Solution:

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