A train P with speed 64 km/hr crosses a pole in 27 second and an another train Q whose speed is `12(1)/(2)%` less than train P crosses a man in 36 second. If both train move in opposite direction to each other, then find in how much time will they cross each other?

Options

A. A)28.2 sec
B. B)36.2 sec
C. C)31.2 sec
D. D)38.2 sec

Answer

To solve the problem step by step, we will follow the outlined process to find the time taken for the two trains to cross each other.

### Step 1: Calculate the length of Train P
Train P crosses a pole in 27 seconds. The speed of Train P is given as 64 km/hr.

1. Convert the speed from km/hr to m/s:
\[ \text{Speed in m/s} = \text{Speed in km/hr} \times \frac{5}{18} \]
\[ \text{Speed of Train P} = 64 \times \frac{5}{18} = \frac{320}{18} = \frac{160}{9} \text{ m/s} \]

2. Calculate the length of Train P using the formula:
\[ \text{Length of Train P} = \text{Speed} \times \text{Time} \]
\[ \text{Length of Train P} = \frac{160}{9} \times 27 = \frac{160 \times 27}{9} = 480 \text{ meters} \]

### Step 2: Calculate the speed of Train Q
Train Q's speed is 12.5% less than Train P's speed.

1. Calculate the percentage decrease:
\[ \text{Speed of Train Q} = \text{Speed of Train P} - \left(12.5\% \text{ of Speed of Train P}\right) \]
\[ \text{Speed of Train Q} = 64 - \left(\frac{12.5}{100} \times 64\right) = 64 - 8 = 56 \text{ km/hr} \]

### Step 3: Calculate the length of Train Q
Train Q crosses a man in 36 seconds.

1. Convert the speed of Train Q to m/s:
\[ \text{Speed of Train Q} = 56 \times \frac{5}{18} = \frac{280}{18} = \frac{140}{9} \text{ m/s} \]

2. Calculate the length of Train Q:
\[ \text{Length of Train Q} = \text{Speed} \times \text{Time} \]
\[ \text{Length of Train Q} = \frac{140}{9} \times 36 = \frac{140 \times 36}{9} = 560 \text{ meters} \]

### Step 4: Calculate the time taken for both trains to cross each other
When two trains are moving in opposite directions, their speeds add up.

1. Total length to be crossed:
\[ \text{Total Length} = \text{Length of Train P} + \text{Length of Train Q} = 480 + 560 = 1040 \text{ meters} \]

2. Total speed when moving in opposite directions:
\[ \text{Total Speed} = \text{Speed of Train P} + \text{Speed of Train Q} = 64 + 56 = 120 \text{ km/hr} \]

3. Convert total speed to m/s:
\[ \text{Total Speed in m/s} = 120 \times \frac{5}{18} = \frac{600}{18} = \frac{100}{3} \text{ m/s} \]

4. Time taken to cross each other:
\[ \text{Time} = \frac{\text{Total Length}}{\text{Total Speed}} = \frac{1040}{\frac{100}{3}} = 1040 \times \frac{3}{100} = 31.2 \text{ seconds} \]

### Final Answer:
The time taken for the two trains to cross each other is **31.2 seconds**.

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