Atraveller found himself in a strange country where the temperature rose sharply during the day and then fell again at night and made his watch act erratically. he noticed that his watch was 30 seconds fast at nightfall but at dawn it had lost 20 seconds. on the morning of may 1st the watch showed the right time.
by which date was it five minutes fast?

Wow !  This could be a real limbic tickler.
Let's try to find an easy way to do it . . .

If the watch is 30 seconds ahead at bedtime, and then it loses 20 of them
overnight, then from one morning to the next morning, it pulls ahead by
10 seconds.

So every morning, the watch is 10 seconds farther ahead than it was on
the previous morning.

That's the last time you have to worry about what it reads at night, or
what it loses overnight !  Just remember . . . 10 seconds farther ahead
every morning.


Now, how many times does it have to gain 10 seconds in order to add up
to 5 minutes ?

-- 10 seconds is (10/60) = 1/6 of a minute.  So from one morning until
6 mornings  later, the watch gains (6 x 1/6) = 1 whole minute.

-- And from one morning until (5 x 6) = 30 mornings later, it gains
(5 x 1) = 5 minutes.

The watch was correct on the morning of May 1.  It gained 10 more seconds
every morning after that. 

-- It was 1minute ahead on the morning of May (1 + 6) = 7.
-- It was 2 minutes ahead on the morning of May (7 + 6) = 13.
-- It was 3 minutes ahead on the morning of May (13 + 6) = 19.
-- It was 4 minutes ahead on the morning of May (19 + 6) = 25.

And it was 5 minutes ahead on the morning of May 31.