Find distance between two stations given travel times of trains

Question

Problem

Two trains A and B start from two points P1 and P2 respectively at the same time and travel towards each other. The difference between their speed is 10 kmph and train A takes one hour more to cover the distance between P1 and P2 as compared to train B. Also by the time they meet, train B has covered 200/9 km more as compared to train A. What is the distance between P1 and P2?

Progress

In my attempt I have these. Consider u and v be the speed of train A and B respectively. and train A travels x distance and train B travels y distance before they meet. Also train A takes time t1 to reach point P2 and train B takes t2 to reach point P1. and d be the total distance between P1 and P2: y = x + 200/9, d = x + y, t2 = t1 - 1, v - u = 10.. with these information, all i can find is total time taken when they meet, which would be equal to 20/9 hrs.

Answer 1

You have three equations in three unknowns: $$\frac du = \frac dv +1\\ v-u=10\\ \frac {20}9(u+v)=d$$ The third seems to be the one you are missing. It comes from the fact that when they meet, the total of the distances traveled is the whole distance from $P1$ to $P2$

Added: $$v+u=\frac{9d}{20}\\v=\frac{9d}{40}+5\\u=\frac{9d}{40}-5\\d(v-u)=10d=uv\\10d=\frac {81d^2}{1600}-25\\d=200 (or -\frac{200}{81})$$

Answer 2

The tagged answer is not correct because the expression: $$\frac{20}{9}(u+v)=d$$

should actually be

$$\frac{d/2-200/9}{u}\cdot(u+v) = d$$

where the first fraction is just the time it takes for train A to meet B. I'm actually going to use the following equation instead:

$$\frac{d+400/9}{v} = \frac{d-400/9}{u}$$ where the left and right sides are the time until the first meet.

So the 3 equations: $$\frac{d}{u} = \frac{d}{v} + 1$$ $$v-u=10$$ $$\frac{d+400/9}{v} = \frac{d-400/9}{u}$$

Using the 3rd equation and the 2nd equation, I get $$\frac{d+400/9}{d-400/9} = \frac{v}{u} = \frac{u+10}{u} = 1 + \frac{10}{u}$$

which simplifies to $$u = \frac{d-400/9}{80/9} = \frac{9d-400}{80}$$

Using the 1st equation, I get $$\frac{d}{u} = \frac{d+v}{v}$$

Combining the two equations above, I get $$\frac{80d}{9d-400} = \frac{d+v}{v}$$

Using the fact that $v=u+10$ and $u=\frac{9d-400}{80}$ and combining with the above, I get

$$\frac{80d}{9d-400} = \frac{d+\frac{9d-400}{80}+10}{\frac{9d-400}{80}+10} = \frac{89d+400}{9d+400}$$

If we simplify, we get the quadratic equation $$81d^2-64000d-160000=0$$ whose root is $$d=\frac{32000}{81} + \frac{400\cdot \sqrt{6481}}{81} \approx 792.62$$