Question : A boat goes 2 km upstream and 3 km downstream in 20 minutes. It goes 7 km upstream and 2 km downstream in 53 minutes. What is the speed (in km/hr) of the boat in still water?

Option 1: $\frac{75}{7}$

Option 2: $\frac{120}{7}$

Option 3: $\frac{135}{7}$

Option 4: $\frac{150}{7}$

Correct Answer: $\frac{135}{7}$

Solution : Let the upstream speed be $x$ km/hr and the downstream speed be $y$ km/hr.
A boat goes 2 km upstream and 3 km downstream in 20 minutes.
According to the question,
$\frac{2}{x} + \frac{3}{y} = \frac{20}{60}$
⇒ $\frac{120}{x} + \frac{180}{y} =20$
⇒ $\frac{240}{x} + \frac{360}{y} = 40$ ............................ (1) [Multiplying by 2 on both sides]
The boat goes 7 km upstream and 2 km downstream in 53 minutes.
According to the question,
$\frac{7}x + \frac{2}y = \frac{53}{60}$
⇒ $\frac{420}{x} + \frac{120}{y} = 53$
⇒ $\frac{1260}{x} + \frac{360}{y} = 159$ ........ (2) [Multiplying by 3 on both sides]
Applying (2) – (1) we get,
$\frac{1260}{x} – \frac{240}{x} = 159 – 40$
⇒ $\frac{1020}{x} = 119$
⇒ $x = \frac{1020}{119}$
$\therefore x = \frac{60}{7}$
Putting the value of $x$ in (1), we get,
$240 \times (\frac{7}{60}) + \frac{360}{y} = 40$
⇒ $\frac{360}{y} = 40 – 28$
⇒ $\frac{360}{y} = 12$
⇒ $y = \frac{360}{12}$
$\therefore y = 30$
$\therefore$ Upstream speed = $\frac{60}{7}$ km/h and downstream speed = 30 km/hr
$\therefore$ Speed of the boat in still water =$\frac{ [\frac{60}{7}+ 30]}{2}$ = $\frac{270}{14}$ km/h = $\frac{135}{7}$ km/hr
Hence, the correct answer is $\frac{135}{7}$ km/hr.