A train leaves New Delhi for Ludhiana, 324 km away, at 9 a.m. One hour later, another train leaves Ludhiana for New Delhi. They meet at noon. If the second train had started at 9 a.m. and the first train at 10.30 am., they both would still have met at noon. Find the speed of each train.

Let the speed of the first train be u km/hr and that of the second train be v km/hr.

The total distance between New Delhi and Ludhiana is 324 km.

For the first case:

The first train starts at 9:00 a.m. and meets the second train at 12:00 p.m.

Time taken by 1st train = (12 - 9) = 3 hrs

The second train starts at 10:00 a.m. and meets the first train at 12:00 p.m.

Time taken by 2nd train = (12 - 10) = 2 hrs

We know that, distance = speed x time

⇒ 3u + 2v = 324 ………………..(1)

For the second case:

The first train starts at 10:30 a.m. and meets the second train at 12:00 p.m.

Time taken by 1st train = (12 - 10:30) = 1.5 hrs

The second train starts at 9:00 a.m. and meets the first train at 12:00 p.m.

Time taken by 2nd train = (12 - 9) = 3 hrs

⇒ 1.5u + 3v = 324 ………………..(2)

Multiply equation (1) by 3 and equation (2) by 2, we get:

⇒ (3u + 2v = 324) x 3

⇒ 9u + 6v = 972 ………………..(3)

And, (1.5u + 3v = 324) x 2

⇒ 3u + 6v = 648 ………………..(4)

Now, subtract equation (4) from equation (3):

$\begin{matrix} & 9u & + & 6v & = & 972 \ & 3u & + & 6v & = & 648 \ & - & &- & & - \ \hline & 6u & & & = & 324 \ & u & & & = & \dfrac{324}{6} \ \end{matrix}$

⇒ u = 54

Substituting the value of u in equation (1), we get:

⇒ 3 $\times$ 54 + 2v = 324

⇒ 162 + 2v = 324

⇒ 2v = 324 - 162

⇒ 2v = 162

⇒ v = $\dfrac{162}{2}$ = 81

Hence, the speed of the first train is 54km/hr and the speed of the second train is 81km/hr.