A tran crosses a pole in 15 seconds and a platform of length 100 m in 30 seconds. Find the speed and length of the train.

Options

A. 100 m, `6 (2)/(3)` m/sec
B. 100 m, 6 m/sec
C. 120 m, 8 m/sec
D. 100 m, 8 m/sec

Answer

To solve the problem, we need to find the speed and length of the train based on the information given. Let's break down the solution step by step.

### Step 1: Define Variables
Let:
- \( L \) = Length of the train (in meters)
- \( S \) = Speed of the train (in meters per second)

### Step 2: Train Crossing a Pole
When the train crosses a pole, it travels a distance equal to its own length. The time taken to cross the pole is given as 15 seconds. Therefore, we can express the relationship as:
\[ L = S \times 15 \quad \text{(Equation 1)} \]

### Step 3: Train Crossing a Platform
The train crosses a platform of length 100 meters in 30 seconds. When crossing the platform, the distance covered by the train is the sum of its own length and the length of the platform. Thus, we have:
\[ L + 100 = S \times 30 \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
From Equation 1, we can substitute \( L \) in Equation 2:
\[ 15S + 100 = 30S \]

### Step 5: Rearranging the Equation
Rearranging the equation gives us:
\[ 100 = 30S - 15S \]
\[ 100 = 15S \]

### Step 6: Solve for Speed \( S \)
Now, we can solve for \( S \):
\[ S = \frac{100}{15} = \frac{20}{3} \text{ m/s} \]
Converting this to a mixed number:
\[ S = 6 \frac{2}{3} \text{ m/s} \]

### Step 7: Find Length \( L \)
Now that we have the speed, we can find the length of the train using Equation 1:
\[ L = 15S = 15 \times \frac{20}{3} = 100 \text{ m} \]

### Conclusion
Thus, the speed of the train is \( 6 \frac{2}{3} \) m/s and the length of the train is 100 m.

### Final Answer
- Length of the train: 100 m
- Speed of the train: \( 6 \frac{2}{3} \) m/s

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