Chain Rules

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

Let the required earning be ₹ x
5 men = 7 women
7 men = $$ \left( \frac{7}{5} \times 7 \right) $$   women = $$\frac{49}{5}$$ women
∴ (7 men and 13 women)
$$ = \left( {\frac{{49}}{5} + 13} \right){\text{women}}$$
$$ = \frac{{114}}{5}{\text{ women}}$$
Now, More women, More earning (Direct proportion)
$$\therefore {\text{ }}7 : \frac{{114}}{5}::{\text{ 5}}250 : x $$
$$ ⇒ 7x = \left( {\frac{{114}}{5} \times 5250} \right)$$
$$ ⇒ 7x = 119700$$
$$ ⇒ x = 17100$$

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

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

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

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

Let the height of the building be x metres
Lees lengthy shadow, Less is the height (Direct proportion)
$$\therefore 40.25:28.75::17.5:x$$
$$ ⇒ 40.25 \times x = 28.75 \times 17.5$$
$$ ⇒ x = \frac{{\left( {28.75 \times 17.5} \right)}}{{40.25}}$$
$$ ⇒ x = 12.5$$

Work done = $$\frac{5}{8}$$
Balance work $$ = \left( {1 - \frac{5}{8}} \right)$$   $$ = \frac{3}{8}$$
Less work, Less days ( Direct proportion)
Let the required number of days be x
Then,
$$ ⇒ \frac{5}{8}:\frac{3}{8}::10:x$$
$$ ⇒ \frac{5}{8} \times x = \frac{3}{8} \times 10$$
$$ ⇒ x = \left( {\frac{3}{8} \times 10 \times \frac{8}{5}} \right)$$
$$ ⇒ x = 6$$

Let the required number of days be x
Less men, More days (Indirect proportion)
$$\therefore 42:56::24:x$$
$$ ⇒ 42 \times x = 56 \times 24$$
$$ ⇒ x = \left( {\frac{{56 \times 24}}{{42}}} \right)$$
$$ ⇒ x = 32$$

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