Aptitude - Problems on Trains - Discussion

That is nice. Thanks all for the explanation.

Well done, Thanks all for explaining the answer.

Total distance = x.
Time is taken to cross = t.
Speed of both trains = U and V.
Time taken after t seconds to reach Patna = t+9.
Time taken after t seconds to reach Howrah = t+16.

x=(U+V)t-----------(1)
after time t total distance covered by train A.

x=U(t+9)------------(2)
After time t total distance covered by train B.

x=V(t+16)-----------(3)
By solving eq(1) and eq(2).

we get Vt=9U-----------(4)
by solving eq(1) and eq(3).

we get Ut=16V----------(5)
by solving eq(4) and eq(5).

We get U/V=4/3
therefore the ratio is 4:3.

Divide distance into two parts based on the point where they meet.

Let first half of journey took x hrs which is same for both trains.

Total time for train 1 = 9+x.
Total time for train 2 = 16+ x.

Let the speed of the first train(took 9 hrs to complete the second half of its journey) be a

Speed of the second train( took 16 hrs to complete the second half of its journey) be b

The total distance between the cities is 9a+16b.

Now,

Speed = distance/ time.

a = 9a+16b/9+x.
b= 9a+16b/16+x.

eliminate x.
ax= 16 b,
bx = 9a.

16b/a =9a/b.

a/b = 4/3.

Let say the distance between them is d, and it takes time t when both the trains meet each other.

So relative speed = v1+v2.

So t = d/ (v1+v2) -----> (1).

Now for the train 1 distance remains = d - t*v1.
For the train 2 distance remains = d-t*v2.

According to the question, it takes 9 hrs for the first train and 16 hrs for the second train.
So, 9 = (d-t*v1) /v1 and 16 = (d-t*v2) /v2.

Now from the equation 1, d = t*v1+t*v2 substitue above.
So 9 = t*v2/v1 and 16 = t*v1/v2.

Equating t, 9v1/v2 = 16v2/v1 so v1/v2 = 4/3.

Here we can use the ratio method:
Train 1 ( Howrah to Patna) = A.
Train 2 ( Patna to Howrah) = B/

Ratio of speed = √speed of A/ √speed of B
= √16/√9.
= 4/3.

That is the answer is 4 : 3.

Nice, Thanks all.

Good. Thanks everyone for explaining the answer.

Here in this question, one train had reached in 9 hrs and another one in 16 hrs.The first train reached faster than the second one ,clearly it should have more speed than the second one, as we know that time and speed are inversely proportional and comparing all answers, 4:3 is most optimal.

Let's say.

Train A travels 'x' is the distance from Patna with speed 'V1' till the point of meet and
train B travels 'y' is the distance from Patna with speed 'V2' till the point of the meet.

the time taken for them to meet 't' = x/V1= y/V2 ---- (1)

Now Train A will travel 'y' distance that was previously travelled by train 'B' and vice-versa.

As per given info;
y/V1 = 9 and x/V2 = 16.

Taking ratio:
y/V1*V2/x = 9/16.

but from (1) -----> y/x = V1/V2,

=> ( V1/V2)^2 = 9/16.
=> V1:V2 = 3:4.