Chain-Rule - 02

Explanation:-

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

Explanation:-

Let the required number of labourers be x
Less working hour/day, More labours (Indirect proportion)
More days, Less labourers (Indirect proportion)
$$\left. \begin{gathered} {\text{Working hours/day 6}}:7 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 30}}:18 \hfill \\ \end{gathered} \right\}::30:x$$
$$\eqalign{ & \therefore 6 \times 30 \times x = 7 \times 18 \times 30 \cr & \Leftrightarrow 6x = 126 \cr & \Leftrightarrow x = 21 \cr}$$

Explanation:-

Let the required number of days be x
Then,
More pumps, Less days (Indirect proportion)
Less working hour/day, More days (Indirect proportion)
$$\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Pumps 196}}:49 \hfill \\ {\text{Working hour/day 5}}:8 \hfill \\ \end{gathered} \right\}::\frac{{13}}{2}:x$$
$$\eqalign{ & \therefore {\text{ }}196 \times 5 \times x = 49 \times 8 \times \frac{{13}}{2} \cr & \Leftrightarrow x = \left( {49 \times 8 \times \frac{{13}}{2} \times \frac{1}{{196 \times 5}}} \right) \cr & \Leftrightarrow x = \frac{{13}}{5} \cr & \Leftrightarrow x = 2\frac{3}{5} \cr}$$

Explanation:-

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)
$$\eqalign{ & \therefore {\text{ }}32:6::20:x \cr & \Leftrightarrow x = \left( {\frac{{6 \times 20}}{{32}}} \right) \cr & \Leftrightarrow x = \frac{{15}}{4} \cr & \Leftrightarrow x = 3\frac{3}{4} \cr}$$

Explanation:-

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

Explanation:-

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

Explanation:-

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

Explanation:-

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

Explanation:-

Let the required quantity be x kg.
More workers, More quantity (Direct proportion)
More days, More quantity (Direct proportion)
$$\left. \begin{gathered} {\text{Workers 5}}:8 \hfill \\ \,\,\,\,\,\,\,\,{\text{Days 3}}:5 \hfill \\ \end{gathered} \right\}::60:x$$
$$\eqalign{ & \therefore {\text{ }}5 \times 3 \times x = 8 \times 5 \times 60 \cr & \Leftrightarrow x = \frac{{\left( {8 \times 5 \times 60} \right)}}{{\left( {5 \times 3} \right)}} \cr & \Leftrightarrow x = 160 \cr}$$

Explanation:-

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