A person can row a boat in still water at the rate of 5 km/hr. It takes him 4 times as long to row upstream of a river as to row downstream to cover same distance in the same river. The speed of flow of the stream is
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For problems where the ratio of upstream and downstream times is given for the same distance, you can use the direct formula: \( \frac{u}{v} = \frac{T_u + T_d}{T_u - T_d} \). Here, \( \frac{T_u}{T_d} = 4 \), so \( \frac{5}{v} = \frac{4+1}{4-1} = \frac{5}{3} \). This gives \(v = 3\) km/hr.
Step 1: Conceptual Foundation: This problem falls under the category of 'Boats and Streams' scenarios. The boat's velocity is influenced by the river's current. Downstream movement involves the summation of the boat's speed and the current's speed. Conversely, upstream movement requires subtracting the current's speed from the boat's speed.
Step 2: Governing Principles: Define \(u\) as the boat's speed in still water. Define \(v\) as the river's current speed. Downstream velocity (\(S_d\)) = \(u + v\). Upstream velocity (\(S_u\)) = \(u - v\). The relationship between time, distance, and speed is: Time = Distance / Speed. The problem specifies that the duration for upstream travel is four times the duration for downstream travel over an identical distance. \[ T_{\text{upstream}} = 4 \times T_{\text{downstream}} \]
Step 3: Solution Breakdown: Given parameters: Boat speed in still water, \(u = 5\) km/hr. River current speed = \(v\) km/hr. Distance = 'd' km.
Applying the time-distance-speed formula: \[ T_{\text{downstream}} = \frac{d}{S_d} = \frac{d}{5 + v} \] \[ T_{\text{upstream}} = \frac{d}{S_u} = \frac{d}{5 - v} \] Substitute these into the given condition \( T_{\text{upstream}} = 4 \times T_{\text{downstream}} \): \[ \frac{d}{5 - v} = 4 \times \left(\frac{d}{5 + v}\right) \] Since the distance 'd' is consistent for both journeys, it can be eliminated from the equation: \[ \frac{1}{5 - v} = \frac{4}{5 + v} \] Perform cross-multiplication to solve for \(v\): \[ 1 \times (5 + v) = 4 \times (5 - v) \] \[ 5 + v = 20 - 4v \] Rearrange the equation to group terms involving \(v\) and constants: \[ v + 4v = 20 - 5 \] \[ 5v = 15 \] \[ v = \frac{15}{5} \] \[ v = 3 \]
Step 4: Conclusion: The velocity of the river's flow is 3 km/hr.