This was a tricky puzzle – I received two correct answers, one from Prakhar and another one from Suman Saraf. Thank you both!

Answer is that they will reach at the same time, provided certain conditions are met, see the detailed answer below.

Draw the escalators as semi-circles below, and indicate on each semicircle the direction of motion of the escalator.

Solution to Puzzle #190

When one of the girls steps off her escalator, she immediately steps on the other one. Thus we can assume that they are running on a moving circle. At the first, the girls are at one point of this circle and the hat is diametrically opposite. After that the girls start running towards the hat in opposite directions. Since their speeds relative to the escalators are equal, they will reach the hat simultaneously, unless the hat gets to the end of the escalator first. If this happens, the hat will leave the escalator altogether, and therefore Arushi, who started running down will get to it first.

Hope you all enjoyed the puzzle!