Boat and Stream Speed Calculation

This problem involves calculating the time taken to travel downstream given the speed of a boat upstream and its speed in still water. We need to find the speed of the stream first, then the speed of the boat downstream, and finally the time using the distance and downstream speed.

Understanding Key Concepts in Boat and Streams

• Speed of boat in still water ($V_b$): The speed at which the boat travels without the influence of the water current.
• Speed of stream ($V_s$): The speed at which the water current flows.
• Speed upstream ($V_u$): The effective speed of the boat when moving against the stream. It is the difference between the speed of the boat in still water and the speed of the stream: $V_u = V_b - V_s$.
• Speed downstream ($V_d$): The effective speed of the boat when moving with the stream. It is the sum of the speed of the boat in still water and the speed of the stream: $V_d = V_b + V_s$.

Step-by-Step Calculation

Given information:

• Speed of boat upstream ($V_u$) = 5 kmph
• Speed of boat in still water ($V_b$) = 10 kmph
• Distance to travel downstream ($D_d$) = 25 km

We need to find the time taken to travel 25 km downstream.

Step 1: Find the Speed of the Stream ($V_s$)

We know that the speed upstream is the speed of the boat in still water minus the speed of the stream. We can use the formula:

\(V_u = V_b - V_s\)

Substitute the given values:

\(5 \text{ kmph} = 10 \text{ kmph} - V_s\)

Rearrange the equation to find \(V_s\):

\(V_s = 10 \text{ kmph} - 5 \text{ kmph}\)

\(V_s = 5 \text{ kmph}\)

The speed of the stream is 5 kmph.

Step 2: Find the Speed Downstream ($V_d$)

The speed downstream is the sum of the speed of the boat in still water and the speed of the stream. We use the formula:

\(V_d = V_b + V_s\)

Substitute the values we know:

\(V_d = 10 \text{ kmph} + 5 \text{ kmph}\)

\(V_d = 15 \text{ kmph}\)

The speed of the boat downstream is 15 kmph.

Step 3: Calculate the Time Taken to Row 25 km Downstream

Time is calculated by dividing the distance by the speed. The distance is 25 km and the downstream speed is 15 kmph.

We use the formula:

\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)

\(T_d = \frac{D_d}{V_d}\)

\(T_d = \frac{25 \text{ km}}{15 \text{ kmph}}\)

\(T_d = \frac{25}{15} \text{ hours}\)

\(T_d = \frac{5}{3} \text{ hours}\)

Step 4: Convert Time from Hours to Minutes

The question asks for the time in minutes. To convert hours to minutes, we multiply by 60.

\(T_d \text{ in minutes} = \frac{5}{3} \times 60 \text{ minutes}\)

\(T_d \text{ in minutes} = 5 \times 20 \text{ minutes}\)

\(T_d \text{ in minutes} = 100 \text{ minutes}\)

It will take 100 minutes to row 25 km downstream.

The final answer is 100 minutes.

Revision Table: Boat and Stream Formulas

Additional Information: Solving Boat and Stream Problems

Boat and stream problems are common in quantitative aptitude tests. They test your understanding of relative speed. The key is to correctly identify whether the boat is moving with or against the stream and to use the appropriate formula for the effective speed.

• When the boat moves downstream, the speed of the stream adds to the speed of the boat, making it faster.
• When the boat moves upstream, the speed of the stream opposes the boat, making it slower.
• The speed in still water is the boat's inherent speed without any current influence.
• Remember to pay attention to the units (kmph, m/s, hours, minutes) and convert them if necessary to ensure consistency in calculations. In this problem, we converted hours to minutes as required by the question.