Understanding Boat and Stream Speed Concepts

This problem involves the concepts of boat speed in still water and the speed of the stream. When a boat moves downstream, its speed is increased by the speed of the stream. When it moves upstream, its speed is decreased by the speed of the stream.

Let's define the variables:

• Let \(B\) be the speed of the boat in still water (in km/h).
• Let \(S\) be the speed of the stream (in km/h).

Based on these definitions, we can determine the speeds in different directions:

• Speed downstream = Speed of boat in still water + Speed of stream = \(B + S\) km/h.
• Speed upstream = Speed of boat in still water - Speed of stream = \(B - S\) km/h.

We are given information about the distance covered upstream and downstream and the total time taken for two different scenarios. The relationship between distance, speed, and time is:
\(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\)

Setting Up Equations for Boat and Stream Travel

From the problem statement, we can formulate two equations based on the total time taken for each journey.

Scenario 1: 25 km upstream and 39 km downstream in 8 hours.

• Time taken for 25 km upstream = \(\frac{25}{B - S}\) hours.
• Time taken for 39 km downstream = \(\frac{39}{B + S}\) hours.

Total time for Scenario 1:

\[ \frac{25}{B - S} + \frac{39}{B + S} = 8 \quad \text{(Equation 1)} \]

Scenario 2: 35 km upstream and 52 km downstream in 11 hours.

• Time taken for 35 km upstream = \(\frac{35}{B - S}\) hours.
• Time taken for 52 km downstream = \(\frac{52}{B + S}\) hours.

Total time for Scenario 2:

\[ \frac{35}{B - S} + \frac{52}{B + S} = 11 \quad \text{(Equation 2)} \]

Solving the System of Equations

We have a system of two linear equations with two variables, \(\frac{1}{B-S}\) and \(\frac{1}{B+S}\). To make solving easier, let's use substitution.

Let \(u = \frac{1}{B - S}\) and \(v = \frac{1}{B + S}\).

Substituting these into Equation 1 and Equation 2, we get:

\[ 25u + 39v = 8 \quad \text{(Equation 3)} \]

\[ 35u + 52v = 11 \quad \text{(Equation 4)} \]

Now we can solve this system of linear equations using methods like elimination or substitution. Let's use elimination. We can multiply Equation 3 by the coefficient of \(u\) in Equation 4 (35) and Equation 4 by the coefficient of \(u\) in Equation 3 (25), or find a common multiple for coefficients of \(u\) or \(v\). Let's try to eliminate \(u\). The least common multiple of 25 and 35 is 175.

• Multiply Equation 3 by 7: \(7 \times (25u + 39v) = 7 \times 8\)
• This gives: \(175u + 273v = 56 \quad \text{(Equation 5)}\)

• Multiply Equation 4 by 5: \(5 \times (35u + 52v) = 5 \times 11\)
• This gives: \(175u + 260v = 55 \quad \text{(Equation 6)}\)

Now subtract Equation 6 from Equation 5:

\[ (175u + 273v) - (175u + 260v) = 56 - 55 \]

\[ 175u - 175u + 273v - 260v = 1 \]

\[ 13v = 1 \]

\[ v = \frac{1}{13} \]

Now substitute the value of \(v\) back into either Equation 3 or Equation 4 to find \(u\). Let's use Equation 3:

\[ 25u + 39\left(\frac{1}{13}\right) = 8 \]

\[ 25u + 3 = 8 \]

\[ 25u = 8 - 3 \]

\[ 25u = 5 \]

\[ u = \frac{5}{25} \]

\[ u = \frac{1}{5} \]

Finding Boat Speed and Stream Speed

We found that \(u = \frac{1}{5}\) and \(v = \frac{1}{13}\). Now substitute back the original expressions for \(u\) and \(v\):

• Since \(u = \frac{1}{B - S}\), we have \(\frac{1}{B - S} = \frac{1}{5}\). This implies \(B - S = 5\) (Upstream speed).
• Since \(v = \frac{1}{B + S}\), we have \(\frac{1}{B + S} = \frac{1}{13}\). This implies \(B + S = 13\) (Downstream speed).

Now we have a simpler system of two linear equations:

\[ B - S = 5 \quad \text{(Equation 7)} \]

\[ B + S = 13 \quad \text{(Equation 8)} \]

To find \(B\) and \(S\), we can add Equation 7 and Equation 8:

\[ (B - S) + (B + S) = 5 + 13 \]

\[ B + B - S + S = 18 \]

\[ 2B = 18 \]

\[ B = \frac{18}{2} \]

\[ B = 9 \]

The speed of the boat in still water is 9 km/h.

Now substitute the value of \(B\) into either Equation 7 or Equation 8 to find \(S\). Using Equation 7:

\[ 9 - S = 5 \]

\[ S = 9 - 5 \]

\[ S = 4 \]

Using Equation 8 as a check:

\[ 9 + S = 13 \]

\[ S = 13 - 9 \]

\[ S = 4 \]

The speed of the stream is 4 km/h.

Verifying the Solution

Let's check if \(B=9\) km/h and \(S=4\) km/h satisfy the original conditions.

• Upstream speed = \(B - S = 9 - 4 = 5\) km/h.
• Downstream speed = \(B + S = 9 + 4 = 13\) km/h.

Check Scenario 1: 25 km upstream and 39 km downstream.

• Time upstream = \(\frac{25 \text{ km}}{5 \text{ km/h}} = 5\) hours.
• Time downstream = \(\frac{39 \text{ km}}{13 \text{ km/h}} = 3\) hours.
• Total time = \(5 + 3 = 8\) hours. (Matches the given information)

Check Scenario 2: 35 km upstream and 52 km downstream.

• Time upstream = \(\frac{35 \text{ km}}{5 \text{ km/h}} = 7\) hours.
• Time downstream = \(\frac{52 \text{ km}}{13 \text{ km/h}} = 4\) hours.
• Total time = \(7 + 4 = 11\) hours. (Matches the given information)

Both scenarios match the given conditions, confirming our calculated speeds are correct.

Final Answer

The speed of the stream is 4 km/h.

Revision Table: Boat and Stream Problem Steps

Additional Information on Boat and Stream Problems

Boat and stream problems are common in competitive exams. They test your understanding of relative speed. Here are some key points:

• The speed of the boat in still water is its speed without any influence from the current.
• The speed of the stream is the speed of the water flow.
• Upstream movement is against the current, reducing the effective speed.
• Downstream movement is with the current, increasing the effective speed.
• If the boat speed is less than the stream speed (B < S), the boat cannot move upstream. In these problems, B is usually assumed to be greater than S.
• The speeds \(B\) and \(S\) are typically assumed to be uniform (constant) throughout the journey unless otherwise specified.
• These problems often lead to systems of linear equations that can be solved using substitution or elimination methods.

Understanding the basic formulas and how to set up the equations is crucial for solving these types of questions accurately.