A boy standing by the side of a railway track finds that an Up train crosses him in 8 seconds and a down train of twice the length of the Up train crosses him in 20 seconds. How long (in seconds) will the two trains take to cross each other?

Options

A. `13 1/3`
B. `15`
C. `20`
D. `12 1/3`

Answer

To solve the problem step by step, we will use the information given about the two trains and the time they take to cross the boy standing by the railway track.

### Step 1: Define Variables
Let:
- \( L_u \) = Length of the Up train
- \( L_d \) = Length of the Down train = \( 2L_u \) (since it is twice the length of the Up train)
- \( S_u \) = Speed of the Up train
- \( S_d \) = Speed of the Down train

### Step 2: Calculate the Length of the Up Train
The Up train crosses the boy in 8 seconds. Using the formula for distance:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
We can express the length of the Up train as:
\[ L_u = S_u \times 8 \]

### Step 3: Calculate the Length of the Down Train
The Down train crosses the boy in 20 seconds. Using the same formula, we have:
\[ L_d = S_d \times 20 \]
Since \( L_d = 2L_u \), we can substitute \( L_u \) into this equation:
\[ 2L_u = S_d \times 20 \]

### Step 4: Relate Speeds of the Trains
From Step 2, we have \( L_u = S_u \times 8 \). From Step 3, substituting \( L_u \):
\[ 2(S_u \times 8) = S_d \times 20 \]
This simplifies to:
\[ 16S_u = S_d \times 20 \]
Thus, we can express \( S_d \) in terms of \( S_u \):
\[ S_d = \frac{16}{20} S_u = \frac{4}{5} S_u \]

### Step 5: Calculate the Time to Cross Each Other
When the two trains cross each other, the time taken is given by:
\[ \text{Time} = \frac{\text{Length of Up train} + \text{Length of Down train}}{\text{Relative Speed}} \]
The relative speed \( S_u + S_d \) is:
\[ S_u + S_d = S_u + \frac{4}{5} S_u = \frac{9}{5} S_u \]
The total length when they cross each other is:
\[ L_u + L_d = L_u + 2L_u = 3L_u \]
So, the time taken to cross each other is:
\[ \text{Time} = \frac{3L_u}{\frac{9}{5} S_u} \]
This simplifies to:
\[ \text{Time} = \frac{3L_u \times 5}{9S_u} = \frac{15L_u}{9S_u} = \frac{5L_u}{3S_u} \]

### Step 6: Substitute for \( L_u \)
From Step 2, we know \( L_u = S_u \times 8 \). Substituting this into the time equation:
\[ \text{Time} = \frac{5(S_u \times 8)}{3S_u} = \frac{40}{3} \]

### Final Answer
The time taken for the two trains to cross each other is:
\[ \frac{40}{3} \text{ seconds} \approx 13.33 \text{ seconds} \]