A man rows to a place 70 km in distance and back in 10 hours 30 min.  He found that he can row 5 km with the stream in the same time as he can row 4 km against the stream.  Find the rate of current.

3

Time is constant,

Let speed of man = x and speed of current = y                          

Speeds ⇒ \(\frac{Downstream}{Upstream}\) = \(\frac{x + y}{x - y}\)
⇒ \(\frac{x + y}{x - y}\) = \(\frac{5}{4}\) = \(\frac{5R}{4R}\)

ATQ,

⇒\(\frac{70}{x\;+\;y}\) + \(\frac{70}{x\;-\;y}\) = 10hrs 30 min = \(\frac{21}{2}\) hrs.

⇒ \(\frac{70}{5R}\) + \(\frac{70}{4R}\) = \(\frac{21}{2}\)

70 [\(\frac{1}{5R}\) + \(\frac{1}{4R}\)] = \(\frac{21}{2}\) 

⇒ \(\frac{9R}{20R^2}\) = \(\frac{21}{2\;×\;70}\)

⇒ \(\frac{3}{R}\) = 1

⇒ R = 3

Now,

⇒ \(\frac{x + y}{x - y}\) = \(\frac{5}{4}\)

by componendo and dividendo

\(\frac{x}{y}\) = \(\frac{5\;+\;4}{5\; -\;4 }\)  = \(\frac{9}{1}\)

y = 1R = 1 × 3 = 3 km/hr. (speed of current)