Number of steps in escalator : Time, Speed and Distance Problem 
An escalator is moving downwards with a speed of $4$ steps per minute.
A person takes $6$ minutes less to get down if he is coming down on
the moving escalator as compared to when he comes down via the
stationary escalator. Another person takes $6$ minutes more to get up
if he is going up on the moving escalator as compared to when he goes
up via the stationary escalator. If they start together from the top
and the bottom respectively, they meet after $4$ minutes on the moving
escalator. How many steps are there in the escalator?

My solution approach :- 
Let the total steps in the escalator be $x$.
Let the speed of the persone be $p$ steps/minute.
Escalator speed = $4$ steps/minute.
In the case of coming down and as per the question :-
$\frac{x}{p}-\frac{x}{p+4}=6$
In the case of going up and as per the question :-
$\frac{x}{p-4}-\frac{x}{p}=6$
Equating both the above equations, we get ;
$\frac{x}{p}-\frac{x}{p+4}=\frac{x}{p-4}-\frac{x}{p}$
$\Rightarrow\frac{1}{p}-\frac{1}{p+4}=\frac{1}{p-4}-\frac{1}{p} \text{ (Cancelling out $x$ from both sides.)}$
$\Rightarrow \frac{2}{p}=\frac{1}{p-4}+\frac{1}{p+4} $
$\Rightarrow p^2=p^2-16$
which doesn't lead to any solution. What am I doing wrong? Is there any different concept that is used in escalator's problem. I was trying to use the concepts that I learnt in boat and stream problems. 
Also the last bit of information that has been given in the problem i.e. " If they start together from the top and the bottom respectively, they meet after $4$ minutes on the moving escalator." doesn't come to use in my solution approach. 
Please help me on this !!!
Thanks in advance !!!

You need to solve,
$\frac{x}{p}-\frac{x}{p+4} = \frac{x}{q-4}-\frac{x}{q} = 6$
$\implies \frac{2x}{p(p+4)} = \frac{2x}{q(q-4)} = 3 \tag1$
$\frac{y}{p+4} = \frac{x-y}{q-4}= 4 \tag2$
where $x$ is total number of steps in the escalator, $y$ is number of steps from top where both persons meet, $p$ is the speed of the person coming down the escalator in steps per minute and $q$ is the speed of the person going up the escalator.
Solving both, we get $x = 48$.
