A man travels 404 kilometre partly by train and steamer. He spends 10 hours more time on steamer. If the speed of the steamer is 30km/hr and the velocity of train is 50km/hr, how much distance does he cover by steamer?

Options

A. 407km
B. 509km
C. 610km
D. 339km

Answer

To solve the problem step-by-step, we will use the information given about the total distance traveled, the speeds of the train and steamer, and the time difference between the two modes of transport.

### Step 1: Define Variables
Let:
- \( d_s \) = distance covered by the steamer (in km)
- \( d_t \) = distance covered by the train (in km)

From the problem, we know that:
\[ d_s + d_t = 404 \quad \text{(1)} \]

### Step 2: Express Time for Each Mode of Transport
The time taken by the steamer can be expressed as:
\[ \text{Time by steamer} = \frac{d_s}{30} \quad \text{(2)} \]
The time taken by the train can be expressed as:
\[ \text{Time by train} = \frac{d_t}{50} \quad \text{(3)} \]

### Step 3: Set Up the Time Difference Equation
According to the problem, the man spends 10 hours more on the steamer than on the train:
\[ \frac{d_s}{30} = \frac{d_t}{50} + 10 \quad \text{(4)} \]

### Step 4: Substitute \( d_t \) from Equation (1) into Equation (4)
From Equation (1), we can express \( d_t \) in terms of \( d_s \):
\[ d_t = 404 - d_s \]
Substituting this into Equation (4):
\[ \frac{d_s}{30} = \frac{404 - d_s}{50} + 10 \]

### Step 5: Clear the Fractions
To eliminate the fractions, we can multiply the entire equation by 150 (the least common multiple of 30 and 50):
\[ 5d_s = 3(404 - d_s) + 1500 \]

### Step 6: Simplify the Equation
Expanding and simplifying:
\[ 5d_s = 1212 - 3d_s + 1500 \]
\[ 5d_s + 3d_s = 1212 + 1500 \]
\[ 8d_s = 2712 \]

### Step 7: Solve for \( d_s \)
Now, divide both sides by 8:
\[ d_s = \frac{2712}{8} = 339 \]

### Step 8: Find the Distance Covered by the Train
Using Equation (1):
\[ d_t = 404 - d_s = 404 - 339 = 65 \]

### Final Answer
The distance covered by the steamer is:
\[ \boxed{339 \text{ km}} \]