Practice MCQ Questions and Answer on Chain Rule

Solution:

 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

Solution:

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

Solution:

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

Solution:

 Let the required number of days be x Less working hours, More days (Indirect Proportion) More men, Less days (Indirect Proportion) \[\left. \begin{gathered} {\text{Working hours 4}}:8 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Men 18}}:12 \hfill \\ \end{gathered} \right\}::30:x\] $$\eqalign{ & \therefore {\text{ }}4 \times 18 \times x = 8 \times 12 \times 30 \cr & \Leftrightarrow x = \frac{{\left( {8 \times 12 \times 30} \right)}}{{\left( {4 \times 18} \right)}} \cr & \Leftrightarrow x = 40 \cr} $$

Solution:

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

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:

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

Solution:

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

Solution:

 Let, L = large ships, M = medium ships, S = small ships Now from the question, 4L ≡ 7S $$\eqalign{ & {\text{15L}} = \left( {\frac{7}{4} \times 15} \right){\text{S}} = \frac{{105}}{4}{\text{S}} \cr & {\text{Also, 2L}} = \left( {\frac{7}{4} \times 2} \right){\text{S}} = \frac{7}{2}{\text{S}} \cr & {\text{3M}} \equiv 2{\text{L}} + {\text{1S }} \equiv \left( {\frac{7}{2} + 1} \right){\text{S}} = \frac{9}{2}{\text{S}} \cr & \Leftrightarrow {\text{ 7M}} = \left( {\frac{9}{2} \times \frac{1}{3} \times 7} \right)\,\,{\text{S}} = \frac{{21}}{2}{\text{S}} \cr & \therefore \,\left( {{\text{15L}} + {\text{7M}} + {\text{14S}}} \right){\text{ ships}} \cr & \equiv \left( {\frac{{105}}{4} + \frac{{21}}{2} + 14} \right){\text{S}} \cr & = \frac{{203}}{4}{\text{S}} \cr & \left( {{\text{12 large}} + 14{\text{ medium}} + 2{\text{1 small}}} \right){\text{ ships}} \cr & \equiv \left[ {\left( {\frac{7}{4} \times 12} \right) + \left( {\frac{{21}}{2} \times 2} \right) + 21} \right]{\text{S}} \cr & = \left( {21 + 21 + 21} \right){\text{S}} \cr & = 63\,{\text{S}} \cr} $$ Let the required number of journeys be x More ships, Less journeys (Indirect proportion) $$\eqalign{ & \therefore \,63:\frac{{203}}{4}::36:x \cr & \Leftrightarrow 63x = \frac{{203}}{4} \times 36 = 1827 \cr & \Leftrightarrow x = \frac{{1827}}{{63}} \cr & \Leftrightarrow x = 29 \cr} $$ Alternate: Now from the question, $$\eqalign{ & 4L = 7S \cr & \Rightarrow \frac{L}{S} = \frac{7}{4} \cr & \Rightarrow L = 7x;S = 4x \cr & 3M = 2L + S \cr & \Rightarrow M = \frac{{2 \times 7x + 4x}}{3} \cr & \Rightarrow M = 6x \cr} $$ Thus, ratio of large, medium and small = 7 : 6 : 4 So, numbers of journeys: $$\eqalign{ & = \frac{{(15 \times 7 + 7 \times 6 + 14 \times 4)36}}{{12 \times 7 + 14 \times 6 + 21 \times 4}} \cr & = \frac{{7308}}{{252}} \cr & = 29 \cr} $$

Solution:

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