Let the initial speed of Train A be $v_A = 90$ km/hr and the initial speed of Train B be $v_B = 108$ km/hr. They would have crossed each other at 2:30 p.m., meaning that the time taken to cross without any speed reduction is 1.5 hours (from 1 p.m. to 2:30 p.m.).

The total distance between P and Q is calculated as: $\text{Distance} = (\text{Relative Speed}) \times (\text{Time})$ Relative speed initially = $90 + 108 = 198$ km/hr. Therefore: $\text{Distance} = 198 \times 1.5 = 297 \text{ km}$

When the speed reduction occurs at 1 p.m., the trains instead cross each other at 5:30 p.m., which is 4.5 hours after 1 p.m. This implies the effective reduced speed allows them to cover 297 km in 4.5 hours.

The effective reduced relative speed is: $\text{Reduced Relative Speed} = \frac{297}{4.5} = 66 \text{ km/hr}$

Since the reduction in speed is the same for both trains, let $x$ be the reduction in speed for each train.

The new speeds of the trains are: $\text{Reduced Speed of Train A} = 90 - x$ $\text{Reduced Speed of Train B} = 108 - x$

Since their combined speed now equals the reduced relative speed, we have: $(90 - x) + (108 - x) = 66$ $198 - 2x = 66$ $2x = 198 - 66$ $2x = 132$ $x = 66$

Therefore, the reduced speed of Train B is: $108 - 66 = 42 \text{ km/hr}$

Answer:

The reduced speed of Train B is 42 km/hr.

Would you like a detailed breakdown of each step or have further questions?

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Here are some related questions:

1. How would the solution change if the trains had different initial speeds?
2. What if the trains did not start simultaneously but at different times?
3. How can the concept of relative speed be applied to trains moving in the same direction?
4. What are other practical applications of speed reduction analysis in real life?
5. How does signal timing impact travel time and scheduling in real-world train systems?

Tip: In problems involving reduced speeds, relative speed concepts can simplify the calculations by focusing on combined effects rather than individual train speeds.