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 $$ = 84 \text{ Km/hour} $$
Speed of train Y $$ = 84 \text{ Km/hour} $$

Both trains are crossing each other in the time $$t = 12 \text{ Sec}$$
Both the trains are running in the opposite direction, so the net relative speed $$= 84 + 52 = 136 \text{ Km/hour} = \dfrac{136 \times 1000}{}$$

$$\dfrac{340}{9} \text{ m/s}$$ 

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 $$=l+\dfrac{2l}{3} =\dfrac{5l}{3}$$

$$\Rightarrow \text{time } (t) = \dfrac{\text{distance}}{\text{speed}}$$

$$\Rightarrow 12=\dfrac{5l}{3\times(\dfrac{340}{9})} $$

$$\Rightarrow 12=\dfrac{5l\times3}{340} $$

$$\Rightarrow 4080=15l$$

Hence, $$l=272m$$