What is the ratio of their speeds?

Two, trains, one from Howrah to Patna and the other from Patna to Howrah, start simultaneously. After they meet, the trains reach their destinations after 9 hours and 16 hours respectively.
What is the ratio of their speeds?

Solution -

Let the distance between Howrah and Patna be D kilometer and let the trains cross at a distance of x from Howrah. Let us call the train from Howrah to Patna as T1 and other T2 and their corresponding speeds s1 and s2. The remaining distance in case of first train is D-x and in the other x. Since the first train covers its remaining distance in 9 hours and the other in 16 hours s1=(D-x)/9 similarly s2=x/16............(1)
There fore s1/s2=16*(D-x)/(9*x)
Since the two train cross after equal time from the start, their distances covered are in proportion to their speeds.
This gives s1/s2=x/(D-x).................(2)
From equation (1) and (2)
x/(D-x)=s1/s2=16s2/9s1
Cross multiplying gives s1/s2 = 4/3