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)
$$\begin{array}{c}\text{Pumps 4}:3\\ \text{Days 1}:2\end{array}\}::8:x$$
$$\therefore 4\times 1\times x=3\times 2\times 8$$
$$\Rightarrow x=\frac{\left(3\times 2\times 8\right)}{\left(4\right)}$$
$$\Rightarrow x=12$$

Let the required earning be ₹ x
5 men = 7 women
7 men = $$(\frac{7}{5}\times 7)$$   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)
$$\therefore 7:\frac{114}{5}::\text{ 5}250:x$$
$$\Rightarrow 7x=\left(\frac{114}{5}\times 5250\right)$$
$$\Rightarrow 7x=119700$$
$$\Rightarrow x=17100$$

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

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

Let the required number of labourers be x
Less working hour/day, More labours (Indirect proportion)
More days, Less labourers (Indirect proportion)
$$\begin{array}{c}\text{Working hours/day 6}:7\\ \text{Days 30}:18\end{array}\}::30:x$$
$$\therefore 6\times 30\times x=7\times 18\times 30$$
$$\Rightarrow 6x=126$$
$$\Rightarrow x=21$$

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

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

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,
$$\Rightarrow \frac{5}{8}:\frac{3}{8}::10:x$$
$$\Rightarrow \frac{5}{8}\times x=\frac{3}{8}\times 10$$
$$\Rightarrow x=\left(\frac{3}{8}\times 10\times \frac{8}{5}\right)$$
$$\Rightarrow x=6$$

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

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