Train A takes 45 minutes more than train B to travel 450 km. Due to engine trouble, speed of train B falls by a quarter. So it takes 30 minutes more than Train A to complete the same journey. Find the speed of Train A.

Let us assume the speeds of trains A and B to be a km/hr and b km/hr, respectively.

The total distance that has to be travelled = 450 km

It is given that Train A takes 45 minutes more than Train B to travel 450 km.

Time taken by train A to travel 450 km = $$\dfrac{450}{a}$$

Time taken by train B to travel 450 km = $$\dfrac{450}{b}$$

We know that Train A takes 45 minutes more than Train B

$$\dfrac{450}{a}=\dfrac{45}{60}+\dfrac{450}{b}$$  ------equation (1)  (We have divided 45 by 60 to convert in hour)

Also, due to engine trouble, the speed of train B falls by a quarter. So it takes 30 minutes more than Train A to complete the same journey.

The time taken by A to complete the journey will remain the same.

Time taken by B to complete the journey will become = $$\dfrac{450}{\frac{3b}{4}}$$ (As it's speed falls by a quarter, new speed = 3b/4)

Thus, $$\dfrac{450}{\frac{3b}{4}}=\dfrac{450}{a}+\dfrac{30}{60}$$   -----equation (2)

=> Substituting the value of $$\dfrac{450}{a}$$ from equation (1) into (2)

=> $$\dfrac{450}{\frac{3b}{4}}=\dfrac{45}{60}+\dfrac{450}{b}+\dfrac{30}{60}$$

=>$$\dfrac{600}{b}=\dfrac{75}{60}+\dfrac{450}{b}$$

=> $$\dfrac{150}{b}=\dfrac{75}{60}$$

=> b = 120 km/hr

=> $$\dfrac{450}{a}=\dfrac{45}{60}+\dfrac{450}{120}$$

=> $$\dfrac{450}{a}=\dfrac{3}{4}+\dfrac{15}{4}=\dfrac{18}{4}$$

Thus, a = 100 km/hr