The answer to your question is 499.
First, we are told that we are dealing with a three digit number. At this point, this can be any number from 100 to 999. We are told that, if this number is doubled, the number will remain a three digit number. From this, we can deduce that the doubled number cannot be more than 999.
We are also told that, if we add 2 to the doubled number, the number immediately becomes a four digit number. To experiment, we can add 2 to 999 (999+2= 1001).
However, we soon realize that 999 does not divide by 2 without a remainder. So, now we know that we can't use 999 as the doubled number. At this point, the best thing to do is to try 998, which divides by 2 easily, giving us 499. Also, when we add 998 with 2, we get 1000 (a four digit number). So the original three digit number is 499.