Step 1: Understanding the Question:
The problem requires us to determine the exact speed of the water stream.
A man completes a round trip by rowing to a destination that is 48 km away and then returning back to the starting point.
The total time taken for this entire round trip is 14 hours.
We are given an additional condition: the time it takes him to row 4 km downstream is identical to the time it takes him to row 3 km upstream.
Step 2: Key Formula or Approach:
Let the downstream speed be denoted as $D$ and the upstream speed as $U$.
The speed of the stream can be calculated using the formula $\text{Stream Speed} = \frac{D - U}{2}$.
We will use the "same time" condition to establish a direct ratio between $D$ and $U$.
Then, we will set up an equation for the total time of the 48 km round trip to find the actual values of $D$ and $U$.
Step 3: Detailed Explanation:
Let $D$ be the downstream speed (rowing with the stream) in km/hr.
Let $U$ be the upstream speed (rowing against the stream) in km/hr.
According to the problem, the time taken to travel 4 km downstream equals the time taken to travel 3 km upstream.
Since time equals distance divided by speed, we can write: \[ \frac{4}{D} = \frac{3}{U} \]
By rearranging this equation, we can find the ratio of downstream speed to upstream speed.
\[ \frac{D}{U} = \frac{4}{3} \]
Because this is a ratio, we can introduce a common variable $x$.
Let $D = 4x$ km/hr and $U = 3x$ km/hr.
The total time for the 48 km round trip is the sum of the downstream time and the upstream time, which equals 14 hours.
\[ \text{Time downstream} + \text{Time upstream} = 14 \]
\[ \frac{48}{D} + \frac{48}{U} = 14 \]
We substitute our expressions for $D$ and $U$ into this equation.
\[ \frac{48}{4x} + \frac{48}{3x} = 14 \]
Simplifying the fractions gives us a very clean equation.
\[ \frac{12}{x} + \frac{16}{x} = 14 \]
Since the denominators are the same, we can add the numerators.
\[ \frac{28}{x} = 14 \]
To solve for $x$, we rearrange the equation.
\[ x = \frac{28}{14} = 2 \]
Now we can find the actual downstream and upstream speeds.
Downstream speed $D = 4x = 4 \times 2 = 8$ km/hr.
Upstream speed $U = 3x = 3 \times 2 = 6$ km/hr.
Finally, we use the formula to calculate the rate of the stream.
Rate of the stream = $\frac{D - U}{2}$.
Rate of the stream = $\frac{8 - 6}{2} = \frac{2}{2} = 1$ km/hr.
Step 4: Final Answer:
The rate of the stream is 1 km/hr.