Let's denote the variables as follows:

• $u_A$ and $d_A$ are the upstream and downstream speeds of boat A, respectively.
• $u_B$ and $d_B$ are the upstream and downstream speeds of boat B, respectively.
• $v$ is the speed of the stream (current).
• $M$ is the distance covered by both boats upstream and downstream.
• $t$ is the time taken by boat A to cover $M$ km upstream.
• $t - 4$ is the time taken by boat B to cover $M$ km downstream.

Given:

• The speed of boat A in still water is $14$ km/hr.
• The speed of boat B in still water is $22$ km/hr.
• The downstream speed of boat A is equal to the upstream speed of boat B, i.e., $d_A = u_B$.

Step 1: Express upstream and downstream speeds

• For boat A:

  • Upstream speed: $u_A = 14 - v$
  • Downstream speed: $d_A = 14 + v$

• For boat B:

  • Upstream speed: $u_B = 22 - v$
  • Downstream speed: $d_B = 22 + v$

Step 2: Use the given condition $d_A = u_B$

Given $d_A = u_B$: $14 + v = 22 - v$ Solving for $v$: $2v = 22 - 14$ $2v = 8 \quad \Rightarrow \quad v = 4 \text{ km/hr}$

Step 3: Use the time and distance relationships

Now that we know $v = 4$ km/hr, we can calculate the upstream and downstream speeds:

• For boat A:

  • $u_A = 14 - 4 = 10$ km/hr
  • $d_A = 14 + 4 = 18$ km/hr

• For boat B:

  • $u_B = 22 - 4 = 18$ km/hr
  • $d_B = 22 + 4 = 26$ km/hr

Step 4: Set up the distance equations

For boat A (upstream): $M = u_A \times t = 10t$

For boat B (downstream): $M = d_B \times (t - 4) = 26(t - 4)$

Step 5: Equate the two expressions for $M$

Since both equations represent $M$: $10t = 26(t - 4)$

Step 6: Solve for $t$

Expanding and solving for $t$: $10t = 26t - 104$ $104 = 26t - 10t$ $104 = 16t$ $t = \frac{104}{16} = 6.5 \text{ hours}$

Step 7: Find the value of $M$

Substitute $t = 6.5$ into the equation $M = 10t$: $M = 10 \times 6.5 = 65 \text{ km}$

Thus, the value of $M$ is $\boxed{65}$.

Would you like more details or have any questions? Here are 5 related questions for you to consider:

1. What would be the distance $M$ if the speed of the stream was different?
2. How does the time taken by boat B downstream change with different values of $v$?
3. Can we determine the time difference if the distances were different for both boats?
4. What would happen if the speed of one boat in still water was reduced?
5. How can we use these equations to solve for other related boat speed problems?

Tip: Always check if there's a direct relationship between upstream and downstream speeds that can simplify the problem-solving process.