Chain Rules

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

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)
\[\left. \begin{gathered} {\text{Maids & Mops 3}}:7 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Floors 7}}:3 \hfill \\ \end{gathered} \right\}::7:x\]
$$\therefore {\text{ }}3 \times 7 \times x = 7 \times 3 \times 7$$
$$ ⇒ x = \frac{{\left( {7 \times 3 \times 7} \right)}}{{\left( {3 \times 7} \right)}}$$
$$ ⇒ x = 7$$

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

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

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

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)
\[\left. \begin{gathered} \,\,\,\,{\text{Carpenters 12}}:18 \hfill \\ {\text{Hours per day 6}}:8 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 24}}:{\text{36}} \hfill \\ \end{gathered} \right\}::460:x\]
$$\therefore {\text{ }}12 \times 6 \times 24 \times x = 18 \times 8 \times 36 \times 460$$
$$ ⇒ x = \frac{{\left( {18 \times 8 \times 36 \times 460} \right)}}{{\left( {12 \times 6 \times 24} \right)}}$$
$$ ⇒ x = 1380$$
∴ 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)
\[\left. \begin{gathered} \,\,\,{\text{Spiders 5}}:100 \hfill \\ {\text{Minutes 5}}:100 \hfill \\ \end{gathered} \right\}::5:x\]
$$\therefore {\text{ }}5 \times 5 \times x = 100 \times 100 \times 5$$
$$ ⇒ x = \frac{{\left( {100 \times 100 \times 5} \right)}}{{\left( {5 \times 5} \right)}}$$
$$ ⇒ x = 2000$$

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

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

Let the required number of binders be x
Less books, Less binders (Direct proportion)
More days, Less binders (Indirect proportion)
\[\left. \begin{gathered} {\text{Books 1400}}:800 \hfill \\ \,\,\,\,\,\,\,\,\,{\text{Days 20}}:15 \hfill \\ \end{gathered} \right\}::21:x\]
$$\therefore {\text{ }}1400 \times 20 \times x = 800 \times 15 \times 21$$
$$ ⇒ x = \frac{{\left( {800 \times 15 \times 21} \right)}}{{\left( {1400 \times 20} \right)}}$$
$$ ⇒ x = 9$$