See the questions here. Please do not read the solutions unless you have given a solid 100% attempt at all the questions. Reading the solutions without trying the questions means a lost opportunity to learn. You will not crack CAT by just reading solutions, you will crack CAT only when you attempt, succeed or fail. Learning then will be at a totally different level.

1. It is directly given that, in SAME time, (P – Escalator) takes 40 steps, and (Q + Escalator) takes 80 steps. Thus ratio of speeds of (P – Escalator) and of (Q + Escalator) will be 40 : 80 i.e. 1 : 2. Thus, to cover equal distances (P – escalator) & (Q + escalator) will take time in ratio 2 : 1.

2. While walking up, steps covered by escalator will be added to my 20 steps and will be the total number of steps.
While coming down, I would be walking more than the total number of steps, but the steps covered by escalator will be subtracted to result in the same total number of steps. See pic below:

Thus, part of the extra 40 steps needs to be added to 20 and remaining part of it needs to be subtracted from 60 such that (20 + part of 40) and (60 – remaining part of 40) end up being the same.
The key point is what part of 40?
Time taken for Me to take 20 steps and that to take 60 steps will be in ratio 1 : 3. In these time intervals, the Escalator will cover steps in the ratio 1 : 3. Thus, 40 needs to be divided in ratio 1 : 3 i.e. the two parts will be 10 & 30. And total number of steps will be 20 + 10 = 60 – 30 = 30.

3. This question has a lot of redundant data. We could only focus on any one of Amit or Bharat
Amit takes 60 secs to walk-up and 100 secs to walk down. Did you realise this question is same as Q2, except that here data is given in seconds and there it was in steps. So the logic is also same …
Amit walking up on his own would take more than 60 seconds. So he saved few seconds because of the escalator.
Amit walking down on his own would take less than 100 seconds. So he took few more seconds because escalator was against him.
Divide the extra 100 – 60 = 40 seconds in what ratio? Yes, in ratio of 60:100 i.e. 3 : 5. And the seconds saved is 15 and the extra seconds taken is 25. Thus Amit alone would take 60 + 15 = 100 – 25 = 75 seconds.

Amit alone takes 75 seconds, Amit + Escalator takes 60 seconds. Thus ratio of speed, Amit and (Amit + Escalator) is 4 : 5. Thus, ratio of Amit : Escalator is 4 : 1. At speed of 5, taken taken is 60 seconds, at speed of 1, tame taken is 300 seconds.

Now you try this approach by focussing on Bharat.

Alternate easier approach but not as enriching as above:
If the thinking is getting too much, you could also use the shorter approach of assuming total distance as LCM (all given timings).
Let Distance (escalator length) be 300 mts.
Thus, Amit walking up gives a + e = 300/60 = 5 m/s, and Amit walking down gives a – e = 300/100 = 3 m/s. From the two, e = 1 m/s. Thus only escalator will take 300/1 = 300 secs.

Yet another Alternate method, a ‘aisa bhi hota hai’ types …
@(a + e) time taken is 60 sec; @(a – e) time taken is 100 sec. Thus, @a alone i.e. Amit alone, time taken will be Harmonic Mean of 60 & 100 i.e. 2*60*100/160 = 75 secs.
Why? Speeds are (a + e), a, (a – e) i.e. in AP, so time taken will be in HP.
For SAME distance, Amit alone takes 75 secs; with the help of escalator, (a + e), takes 60 secs i.e. their speeds are in ratio 4 : 5 i.e. speed of Amit & Escalator is in ratio 4 : 1. Amit alone takes 75 seconds, so Escalator alone will take 300 seconds.

4. I am sure that you can solve the question using the equation 4d/x = d/(x – e) + d/(x + e) very comfortably (d is the distance of staircase/escalator, x is Amit’s & Bharat’s speed, e is Escalator’s speed)

Just as an mental exercise (not recommending it as a solution), try comprehending the following …
It will be far easier if you just consider Bharat (the one on escalator) is at first walking down the escalator and then next walking up the escalator. It would make not difference to the question if we change the order.
Let the ratio of speed of Amit (= Bharat) and that of escalator be k : 1.
When Bharat walks down the down-escalator and reaches the bottom, say he has taken k steps, the escalator would have taken 1 step.
And also Amit (on the staircase) would also have just taken k steps in the same time.
And total steps  of staircase or escalator would be (k + 1).
See left part of the following picture to understand.

From now onwards, Amit to complete 4 flights would need to take 4(k + 1) – k = 3k + 4 steps more. In the time Amit takes (3k + 4) steps, Bharat would also have taken (3k + 4) steps. But Bharat reaches a net distance of only (k + 1). Thus, escalator in this time would take (3k + 4) – (k + 1) = (2k + 3) steps.
See right part of the above picture to understand.
So what do we have?
In the time Bharat covers k steps, escalator covers 1 step.
In the time Bharat covers (3k + 4) steps, escalator covers (2k + 3) steps.
And the ratios has to be equal i.e. k/1 = (3k+4)/(2k+3) i.e. 2k^2 + 3k = 3k + 4  i.e.2k^2 = 4 i.e. k = root(2).

5. Time being same, distance covered by Me and distance covered by (Me – Escalator) is in the ratio 2 : 1. Thus, the respective speeds are also in ratio 2 : 1. Thus, speed of Me and speed of Escalator is in ratio 2 : (2 – 1) i.e. 2 : 1.
While walking down the down-escalator, ratio of speed of (Me + Escalator) and Me are in ratio 3 : 2. Thus, time being same, distance covered will also be in ratio 3 : 2. Thus, I will have covered 2/3rd the distance.