A train passes a standing man in 6 s and a 210 m long platform in 16 s. Find the length and the speed of the train.

Answer

To solve the problem step by step, we will use the information given in the question about the train passing a standing man and a platform.

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

### Step 2: Create the First Equation
When the train passes a standing man, it travels its own length in 6 seconds. Therefore, we can write the equation:
\[ \frac{X}{Y} = 6 \]
From this, we can express \( X \) in terms of \( Y \):
\[ X = 6Y \quad \text{(Equation 1)} \]

### Step 3: Create the Second Equation
When the train passes a 210-meter long platform, it travels the length of the train plus the length of the platform in 16 seconds. Therefore, we can write the equation:
\[ \frac{X + 210}{Y} = 16 \]
From this, we can express \( X + 210 \) in terms of \( Y \):
\[ X + 210 = 16Y \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
Now, we can substitute \( X \) from Equation 1 into Equation 2:
\[ 6Y + 210 = 16Y \]

### Step 5: Solve for \( Y \)
Rearranging the equation gives:
\[ 210 = 16Y - 6Y \]
\[ 210 = 10Y \]
\[ Y = \frac{210}{10} = 21 \text{ m/s} \]

### Step 6: Find \( X \)
Now that we have \( Y \), we can substitute it back into Equation 1 to find \( X \):
\[ X = 6Y = 6 \times 21 = 126 \text{ m} \]

### Final Answer
- Length of the train \( X = 126 \) meters
- Speed of the train \( Y = 21 \) meters per second