Escalator puzzle equation I'm trying to understand the escalator puzzle.

A man visits a shopping mall almost every day and he walks up an
  up-going escalator that connects the ground and the first floor. If he
  walks up the escalator step by step it takes him 16 steps to reach the
  first floor. One day he doubles his stride length (walks up climbing
  two steps at a time) and it takes him 12 steps to reach the first
  floor. If the escalator stood still, how many steps would there be on
  sight?

The solution, apparently, is as follows:
$16x = 12(x+1)$, so $x=3$, so the answer is 48.
But why can we say $12(x+1)$?
First, he covers 16 steps and the motion of the escalator gives him a multiplier of $x$ to cover a total of $16x$ steps. That makes sense.
But why is this the same as 12 steps with a multiplier of $(x+1)$?

Let $d$ be the distance traveled, which remains same in both the cases. if $v$ is the speed of the man and $x$ is the speed of elevator, in case 1 the number of steps taken is $$\frac d{v+x}=16$$ In case 2 it is $$\frac d{2v+x}=12$$ because now he is traveling at double the speed; eliminating $d$, we get $x=2v$; therefore $d=48v$; when stationery $x=0$, we get no. of steps as $48v/v=48$.
