Train X having length 130 m and train Y having length 145 m moving in opposite direction. They enter into a tunnel which have length equal to the sum of length of both trains, Trains meet after 10 second of entering in the tunnel. What percent of train X part is left out the tunnel when it meet train Y if they have there speed in the ratio of 5:6.

Options

A. `2(11)/(13)%`
B. `3(11)/(13)%`
C. `4(1)/(3)%`
D. `5%`

Answer

To solve the problem step by step, we will follow the given information and perform the necessary calculations.

### Step 1: Understand the problem
We have two trains, Train X and Train Y, moving in opposite directions. The lengths of the trains are:
- Train X = 130 m
- Train Y = 145 m

The length of the tunnel is equal to the sum of the lengths of both trains.

### Step 2: Calculate the length of the tunnel
The length of the tunnel (L_tunnel) is given by:
\[ L_{tunnel} = L_{trainX} + L_{trainY} = 130 \, m + 145 \, m = 275 \, m \]

### Step 3: Determine the time taken to meet
The trains meet after 10 seconds of entering the tunnel.

### Step 4: Set up the speed ratio
The speeds of Train X and Train Y are in the ratio of 5:6. Let the speed of Train X be \( 5x \) and the speed of Train Y be \( 6x \).

### Step 5: Calculate the total distance covered
The total distance covered by both trains when they meet is the length of the tunnel:
\[ \text{Total Distance} = L_{tunnel} = 275 \, m \]

### Step 6: Calculate the combined speed
The combined speed of both trains when moving towards each other is:
\[ \text{Combined Speed} = 5x + 6x = 11x \]

### Step 7: Use the time to find the speed
Using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
we can write:
\[ 275 = 11x \times 10 \]
From this, we can solve for \( x \):
\[ 275 = 110x \]
\[ x = \frac{275}{110} = 2.5 \, m/s \]

### Step 8: Calculate the speeds of Train X and Train Y
Now we can find the individual speeds:
- Speed of Train X:
\[ S_{trainX} = 5x = 5 \times 2.5 = 12.5 \, m/s \]
- Speed of Train Y:
\[ S_{trainY} = 6x = 6 \times 2.5 = 15 \, m/s \]

### Step 9: Calculate the distance covered by Train X in 10 seconds
The distance covered by Train X when they meet is:
\[ D_{trainX} = S_{trainX} \times \text{Time} = 12.5 \, m/s \times 10 \, s = 125 \, m \]

### Step 10: Calculate the part of Train X left in the tunnel
The length of Train X is 130 m, so the part of Train X that is left in the tunnel when it meets Train Y is:
\[ \text{Part left} = L_{trainX} - D_{trainX} = 130 \, m - 125 \, m = 5 \, m \]

### Step 11: Calculate the percentage of Train X left in the tunnel
To find the percentage of Train X that is left in the tunnel:
\[ \text{Percentage left} = \left( \frac{\text{Part left}}{L_{trainX}} \right) \times 100 = \left( \frac{5}{130} \right) \times 100 \]
\[ = \frac{500}{130} \approx 3.846 \, \text{or} \, 3 \frac{11}{13} \% \]

### Final Answer
The percentage of Train X that is left out of the tunnel when it meets Train Y is approximately \( 3 \frac{11}{13} \% \).

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