Solving the Boat and Stream Problem

This problem involves the concepts of speed, time, and distance, specifically applied to scenarios where a boat travels in water with a current (stream). We need to find the speed of the stream based on the given information about the man's rowing speed relative to the stream and the total time taken for a round trip.

Understanding Boat and Stream Concepts

• Speed of boat in still water (\(v_b\)): The speed at which the boat travels in the absence of any stream.
• Speed of the stream (\(v_s\)): The speed of the water current.
• Speed downstream (\(v_d\)): When the boat travels in the same direction as the stream, the speeds add up. \(v_d = v_b + v_s\)
• Speed upstream (\(v_u\)): When the boat travels against the direction of the stream, the stream speed is subtracted from the boat's speed. \(v_u = v_b - v_s\)

Remember that speed is distance divided by time (\(Speed = \frac{Distance}{Time}\)), which means time is distance divided by speed (\(Time = \frac{Distance}{Speed}\)).

Analyzing the Given Information

We are given two main pieces of information:

1. A man rows 100 km away and comes back in 10 hours (total round trip time).
2. He can row 12 km downstream or 8 km upstream in the same amount of time.

Step-by-Step Solution to find Speed of Stream

Step 1: Find the relationship between boat speed and stream speed

The second piece of information states that the time taken to cover 12 km downstream is the same as the time taken to cover 8 km upstream. Let this time be \(t\).

• Time downstream for 12 km = \(\frac{12}{v_d} = \frac{12}{v_b + v_s}\)
• Time upstream for 8 km = \(\frac{8}{v_u} = \frac{8}{v_b - v_s}\)

Since these times are equal:

\(\frac{12}{v_b + v_s} = \frac{8}{v_b - v_s}\)

Cross-multiplying:

\(12(v_b - v_s) = 8(v_b + v_s)\)

Expanding both sides:

\(12 v_b - 12 v_s = 8 v_b + 8 v_s\)

Collecting terms with \(v_b\) on one side and \(v_s\) on the other:

\(12 v_b - 8 v_b = 8 v_s + 12 v_s\)

\(4 v_b = 20 v_s\)

Dividing by 4, we get a relationship between the speed of the boat in still water and the speed of the stream:

\(v_b = 5 v_s\)

This means the speed of the man in still water is 5 times the speed of the stream.

Step 2: Use the total round trip time

The total distance for the round trip is 100 km downstream and 100 km upstream. The total time taken is 10 hours.

• Time taken to go downstream 100 km = \(\frac{100}{v_d} = \frac{100}{v_b + v_s}\)
• Time taken to come upstream 100 km = \(\frac{100}{v_u} = \frac{100}{v_b - v_s}\)

The sum of these times is the total round trip time:

\(\frac{100}{v_b + v_s} + \frac{100}{v_b - v_s} = 10\)

Step 3: Substitute the relationship from Step 1 into the equation from Step 2

We found that \(v_b = 5 v_s\). Substitute this into the equation:

\(\frac{100}{5 v_s + v_s} + \frac{100}{5 v_s - v_s} = 10\)

\(\frac{100}{6 v_s} + \frac{100}{4 v_s} = 10\)

Step 4: Solve for the speed of the stream (\(v_s\))

Find a common denominator for the fractions on the left side, which is \(12 v_s\).

\(\frac{100 \times 2}{6 v_s \times 2} + \frac{100 \times 3}{4 v_s \times 3} = 10\)

\(\frac{200}{12 v_s} + \frac{300}{12 v_s} = 10\)

\(\frac{200 + 300}{12 v_s} = 10\)

\(\frac{500}{12 v_s} = 10\)

Multiply both sides by \(12 v_s\):

\(500 = 10 \times 12 v_s\)

\(500 = 120 v_s\)

Divide by 120 to find \(v_s\):

\(v_s = \frac{500}{120}\)

Simplify the fraction by dividing both numerator and denominator by their greatest common divisor (which is 20):

\(v_s = \frac{500 \div 20}{120 \div 20} = \frac{25}{6}\)

So, the speed of the stream is \(\frac{25}{6}\) km/hr.

Checking the Answer Options

The calculated speed of the stream is \(\frac{25}{6}\) km/hr. Let's check the options provided:

Our calculated speed of stream, \(\frac{25}{6}\) km/hr, matches Option 3.

Summary of Calculations for Speed of Stream

We used the given information about relative distances covered in the same time to establish a relationship between boat speed and stream speed. Then, we used the total time for the 100 km round trip to set up an equation and solve for the speed of the stream.

\(v_b = 5 v_s\)

\(\frac{100}{v_b + v_s} + \frac{100}{v_b - v_s} = 10\)

Substituting \(v_b\):

\(\frac{100}{5v_s + v_s} + \frac{100}{5v_s - v_s} = 10\)

\(\frac{100}{6v_s} + \frac{100}{4v_s} = 10\)

\(\frac{200 + 300}{12v_s} = 10\)

\(\frac{500}{12v_s} = 10\)

\(v_s = \frac{500}{120} = \frac{25}{6}\) km/hr

Revision Table: Boat and Stream Formulas

Additional Information on Boat and Stream Problems

Boat and stream problems are common in quantitative aptitude tests. They are based on the principle of relative speed. When a boat moves with the stream, its speed increases. When it moves against the stream, its speed decreases.

Key strategies for solving these problems include:

• Defining variables for the speed of the boat in still water and the speed of the stream.
• Expressing downstream and upstream speeds in terms of these variables.
• Using the relationship between speed, distance, and time to set up equations based on the given information (total time, time for specific segments, etc.).
• Solving the system of equations to find the unknown variable(s), such as the speed of the stream or the speed of the boat.

Understanding the difference between speed in still water and speed relative to the ground (downstream or upstream speed) is crucial.