What is the word or phrase?

Find a 9-digit number, which you will gradually round off starting with units, then tenth, hundred etc., until you get to the last numeral, which you do not round off. The rounding alternates (up, down, up ...). After rounding off 8 times, the final number is 500000000. The original number is commensurable by 6 and 7, all the numbers from 1 to 9 are used, and after rounding four times the sum of the not-rounded numerals equals 24.

Queen Elizabeth organised a royal party.

To Avoid uninvited guest royal family set a password.
Jack (an uninvited person) plan to enter the party. He stand nearby the door.

First guest comes, the security person said 'twelve' and guest replied with six.
Second guest comes , the security person said 'six' and guest replied with 'three'.

Jack thought is enough and he walked to the entry point. The security person said 'eight' , Jack replied smilingly 'four'.
He was immediately thrown out of the party. why ?

I start and end with 500 and carry a 5 in my heart. But I require the first letter and the first number to be complete. I am the name of a King.

Who am I ?

A woman lives in a skyscraper thirty-six floors high and is served by several elevators which stop at each floor going up and down. Each morning she leaves her apartment and goes to one of the elevators. Whichever one she takes is three times more likely to be going up than down. Why?

Can you name three consecutive days without using the words Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, or Sunday?

There was a strange boy who was born before his father. How can this be possible?

What's the next in the series?

H, Be, F, S, Mn, Kr, In, Gd, Tl, ?

You need to remove two matchsticks to make the below equation true. Can you do it ?

What word is pronounced the same if you take away four of its five letters?