If a boat travels 18 km more downstream than upstream, in 3 hours and if the speed of the boat in still water is 20 km/hr, find the distance (in km) travelled by the boat downstream in 4 hours.

Options

A. 92
B. 68
C. 96
D. None of these

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Understand the Problem
The boat travels 18 km more downstream than upstream in 3 hours. The speed of the boat in still water is 20 km/hr. We need to find the distance traveled by the boat downstream in 4 hours.

### Step 2: Define Variables
Let:
- Speed of the boat in still water = 20 km/hr
- Speed of the stream = x km/hr
- Distance traveled upstream = d km
- Distance traveled downstream = d + 18 km

### Step 3: Calculate Speeds
- Speed upstream = Speed of boat - Speed of stream = 20 - x km/hr
- Speed downstream = Speed of boat + Speed of stream = 20 + x km/hr

### Step 4: Set Up Time Equations
Using the formula: Time = Distance / Speed

For upstream:
\[ \text{Time upstream} = \frac{d}{20 - x} \]
For downstream:
\[ \text{Time downstream} = \frac{d + 18}{20 + x} \]

### Step 5: Total Time
According to the problem, the total time for both upstream and downstream is 3 hours:
\[ \frac{d}{20 - x} + \frac{d + 18}{20 + x} = 3 \]

### Step 6: Express Distance in Terms of x
From the upstream distance, we can express d in terms of x:
\[ d = 3(20 - x) \quad \text{(from the upstream time)} \]
Substituting this into the downstream equation:
\[ \frac{3(20 - x)}{20 - x} + \frac{3(20 - x) + 18}{20 + x} = 3 \]
This simplifies to:
\[ 3 + \frac{60 - 3x + 18}{20 + x} = 3 \]
\[ \frac{78 - 3x}{20 + x} = 0 \]
This implies:
\[ 78 - 3x = 0 \implies 3x = 78 \implies x = 26 \text{ km/hr} \]

### Step 7: Calculate Downstream Speed
Now, we can find the downstream speed:
\[ \text{Speed downstream} = 20 + x = 20 + 26 = 46 \text{ km/hr} \]

### Step 8: Calculate Distance Downstream in 4 Hours
Using the formula for distance:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Substituting the values:
\[ \text{Distance} = 46 \text{ km/hr} \times 4 \text{ hours} = 184 \text{ km} \]

### Final Answer
The distance traveled by the boat downstream in 4 hours is **184 km**.

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