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.
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