Question

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

• A. 5 km/hr
• B. 3 km/hr
• C. 6.5 km/hr
• D. 4 km/hr

Hint:

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.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a classic 'Boats and Streams' problem. The speed of the boat is affected by the speed of the river's current. When rowing downstream, the speeds add up. When rowing upstream, the river's speed is subtracted from the boat's speed.
Step 2: Key Formula or Approach:
Let \(u\) be the speed of the boat in still water.
Let \(v\) be the speed of the stream.
Speed downstream (\(S_d\)) = \(u + v\).
Speed upstream (\(S_u\)) = \(u - v\).
Time = Distance / Speed.
The problem states that the time to go upstream is 4 times the time to go downstream for the same distance. \[ T_{\text{upstream}} = 4 \times T_{\text{downstream}} \] Step 3: Detailed Explanation:
Given values:
Speed of the boat in still water, \(u = 5\) km/hr.
Let the speed of the stream be \(v\) km/hr.
Let the distance be 'd' km.
Using the time, distance, and speed relationship: \[ T_{\text{downstream}} = \frac{d}{S_d} = \frac{d}{5 + v} \] \[ T_{\text{upstream}} = \frac{d}{S_u} = \frac{d}{5 - v} \] Now, we use the given condition \( T_{\text{upstream}} = 4 \times T_{\text{downstream}} \): \[ \frac{d}{5 - v} = 4 \times \left(\frac{d}{5 + v}\right) \] The distance 'd' is the same on both sides, so it can be cancelled out: \[ \frac{1}{5 - v} = \frac{4}{5 + v} \] Now, we cross-multiply to solve for \(v\): \[ 1 \times (5 + v) = 4 \times (5 - v) \] \[ 5 + v = 20 - 4v \] Bring all the terms with \(v\) to one side and constants to the other: \[ v + 4v = 20 - 5 \] \[ 5v = 15 \] \[ v = \frac{15}{5} \] \[ v = 3 \] Step 4: Final Answer:
The speed of the flow of the stream is 3 km/hr.