A train x running at 84 km/h crosses another train y running at 52 km/h in opposite direction in 12 seconds. If the length of y is two-third that of x, then what is the length of x?

It is given that,
Speed of train X$$=84Km/hour$$
Speed of train Y$$=52Km/hour$$
Both trains are crossing each other in the time $$t=12 Sec$$
Both the trains are running in the opposite direction, so the net relative speed $$=84+52=136 Km/hour=\dfrac{136\times1000}{60\times60}=\dfrac{340}{9}$$ m/s

Let the length of the train X $$=l$$

So, length of the train Y $$=\dfrac{2l}{3}$$

Both train X and Y will cross each other if total distance = total length of the train

Then, total length of the both the combine train $$=l+\dfrac{2l}{3} =\dfrac{5l}{3}$$
$$\Rightarrow time (t) =\dfrac{distance}{speed} $$
$$\Rightarrow 12=\dfrac{5l}{3\times(\dfrac{340}{9})} $$
$$\Rightarrow 12=\dfrac{5l\times3}{340} $$
$$\Rightarrow 4080=15l$$
Hence, $$l=272m$$