Chain Rules

Let the total number of men to be engaged be $$x$$
More length, More labourers (Direct proportion)
Less days, More labourers (Indirect proportion)
\[\left. \begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,{\text{Lentgh 26}}:39 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{Days 6}}:18 \hfill \\ {\text{Hours per day }}9:8 \hfill \\ \end{gathered} \right\}::17:x\]
$$\therefore {\text{ }}26 \times 6 \times 9 \times×= 39 \times 18 \times 8 \times 17$$
$$ ⇒ x = \frac{{\left( {39 \times 18 \times 8 \times 17} \right)}}{{\left( {26 \times 6 \times 9} \right)}}$$
$$ ⇒ x = 68$$
∴ Number of more labourers = (68 -17) = 51

Let the total number of men be x
Work done = $$\frac{1}{3}$$
Remaining work
= $$\left( {1 - \frac{1}{3}} \right) = \frac{2}{3}$$
More work, More men (Direct proportion)
More days, Less men (Indirect proportion)
\[\left. \begin{gathered} \,{\text{Work }}\frac{1}{3}:\frac{2}{3} \hfill \\ {\text{Days 25}}:20 \hfill \\ \end{gathered} \right\}::20:x\]
$$\therefore {\text{ }}\left( {\frac{1}{3} \times 25 \times x} \right) = \left( {\frac{2}{3} \times 20 \times 20} \right)$$
$$ ⇒ x = \frac{{800}}{{25}}$$
$$ ⇒ x = 32$$
∴ More men to be employed = (32 - 20) = 12

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

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)
\[\left. \begin{gathered} \,\,\,\,\,{\text{Work }}\frac{4}{7}:\frac{3}{7} \hfill \\ \,\,\,\,\,\,\,{\text{Men 13}}:33 \hfill \\ {\text{Hour/day }}9:8 \hfill \\ \end{gathered} \right\}::117:x\]
$$\therefore {\text{ }}\frac{4}{7} \times 13 \times 9 \times×= \frac{3}{7} \times 33 \times 8 \times 117$$
$$ ⇒ x = \frac{{\left( {3 \times 33 \times 8 \times 117} \right)}}{{\left( {4 \times 13 \times 9} \right)}}$$
$$ ⇒ x = 198$$
∴ Addition men to be employed = (198 - 117) = 81

Let the required number of persons be x
Less days , More persons (Indirect proportion)
More work , More persons (Direct proportion)
\[\left. \begin{gathered} \,{\text{Days 2}}:4 \hfill \\ {\text{Work 1}}:8 \hfill \\ \end{gathered} \right\}::12:x\]
$$\therefore \,2 \times 1 \times×= 4 \times 8 \times 12$$
$$ ⇒ x = \frac{{\left( {4 \times 8 \times 12} \right)}}{{\left( 2 \right)}}$$
$$ ⇒ x = 192$$

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

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
$$ = \left( {\frac{{120}}{{200}} \times 50} \right){\text{men}}$$
$$ = {\text{30 men}}$$

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

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

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