A boat takes 5 hours more while going back in upstream than in downstream. If the distance between two places is 24 km and the speed of boat in still water be 5.5 km/h. What must be the speed of boat in still water so that it can row downstream, 24 km, in 4 hours?

Options

A. 1.5 km/h
B. 3.5 km/h
C. 4.5 km/h
D. 3 km/h

Answer

To solve the problem step by step, we will follow the logical reasoning outlined in the video transcript.

### Step 1: Understand the problem
We need to find the speed of the boat in still water (let's denote it as \( b \)) such that the boat can row downstream 24 km in 4 hours. We also know that the boat takes 5 hours longer to return upstream than it does downstream.

### Step 2: Define the variables
Let:
- \( b \) = speed of the boat in still water (in km/h)
- \( y \) = speed of the stream (in km/h)

### Step 3: Calculate downstream speed
The downstream speed of the boat can be expressed as:
\[ \text{Downstream Speed} = b + y \]
The time taken to travel downstream 24 km is given as 4 hours. Therefore, we can write:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \implies 4 = \frac{24}{b + y} \]
From this, we can rearrange to find:
\[ b + y = \frac{24}{4} = 6 \quad \text{(1)} \]

### Step 4: Calculate upstream speed
The upstream speed of the boat is:
\[ \text{Upstream Speed} = b - y \]
Let’s denote the time taken to travel upstream as \( t_u \). According to the problem, the time taken to go upstream is 5 hours more than the time taken to go downstream:
\[ t_u = 4 + 5 = 9 \quad \text{(2)} \]
Using the formula for time again, we have:
\[ 9 = \frac{24}{b - y} \]
Rearranging gives:
\[ b - y = \frac{24}{9} = \frac{8}{3} \quad \text{(3)} \]

### Step 5: Set up equations
Now we have two equations:
1. \( b + y = 6 \) (from equation (1))
2. \( b - y = \frac{8}{3} \) (from equation (3))

### Step 6: Solve the equations
We can add equations (1) and (3):
\[ (b + y) + (b - y) = 6 + \frac{8}{3} \]
This simplifies to:
\[ 2b = 6 + \frac{8}{3} \]
To combine the terms, convert 6 to a fraction:
\[ 6 = \frac{18}{3} \implies 2b = \frac{18}{3} + \frac{8}{3} = \frac{26}{3} \]
Thus, we find:
\[ b = \frac{26}{6} = \frac{13}{3} \approx 4.33 \text{ km/h} \]

### Step 7: Find the speed of the stream
Now substitute \( b \) back into equation (1):
\[ \frac{13}{3} + y = 6 \implies y = 6 - \frac{13}{3} = \frac{18}{3} - \frac{13}{3} = \frac{5}{3} \approx 1.67 \text{ km/h} \]

### Step 8: Conclusion
The speed of the boat in still water that allows it to row downstream 24 km in 4 hours is approximately \( 4.33 \) km/h.