Find the ratio of the times taken for a train 300 m long and travelling at a speed of 63 km/h to cross a person travelling at a speed of 9 km/h in the opposite direction and to cross a bridge 400 m long.

Answer

To solve the problem, we need to find the ratio of the times taken by a train to cross a person and to cross a bridge. Let's break down the solution step by step.

### Step 1: Convert the speeds from km/h to m/s
1. **Speed of the train (Vt)**:
\[ Vt = 63 \text{ km/h} = 63 \times \frac{5}{18} = 17.5 \text{ m/s} \]

2. **Speed of the person (Vp)**:
\[ Vp = 9 \text{ km/h} = 9 \times \frac{5}{18} = 2.5 \text{ m/s} \]

### Step 2: Calculate the time taken to cross the person
1. **Length of the train (Lt)**: 300 m
2. **Relative speed when crossing the person** (since they are moving towards each other):
\[ \text{Relative speed} = Vt + Vp = 17.5 + 2.5 = 20 \text{ m/s} \]

3. **Time taken to cross the person**:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{Lt}{\text{Relative speed}} = \frac{300}{20} = 15 \text{ seconds} \]

### Step 3: Calculate the time taken to cross the bridge
1. **Length of the bridge (Lb)**: 400 m
2. **Total distance to cross the bridge**:
\[ \text{Total distance} = \text{Length of train} + \text{Length of bridge} = 300 + 400 = 700 \text{ m} \]

3. **Time taken to cross the bridge**:
\[ \text{Time} = \frac{\text{Total distance}}{\text{Speed}} = \frac{700}{17.5} = 40 \text{ seconds} \]

### Step 4: Find the ratio of the times
1. **Ratio of time taken to cross the person to the time taken to cross the bridge**:
\[ \text{Ratio} = \frac{\text{Time to cross person}}{\text{Time to cross bridge}} = \frac{15}{40} = \frac{3}{8} \]

### Final Answer
The ratio of the times taken for the train to cross a person and to cross a bridge is \( \frac{3}{8} \).

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