Question 893067
<pre>
Let the number be XY.  Then when you add 9 you have this:

                     XY
                    + 9
                   ----
                     YX

9 is so large that when you add 9 to Y you are bound to have to carry 1.  
So let's put a 1 to carry above the X in the left column:

                     1
                     XY
                    + 9
                   ----
                     YX

 The 1 to carry shows that Y is 1 more than X, so do a little trial and error:

Is X 1 and Y 2?  No because 1+2 isn't 9.

                     1
                     12
                    + 9
                   ----
                     21

But the digits are reversed.

Is X 2 and Y 3?  No because 2+3 isn't 9.

                     1
                     23
                    + 9
                   ----
                     32

But the digits are reversed.

Is X 3 and Y 4?  No because 3+4 isn't 9.

                     1
                     34
                    + 9
                   ----
                     43

But the digits are reversed.

Is X 4 and Y 5?  4+5=9.  HEY! RIGHT

                     1
                     45
                    + 9
                   ----
                     54

And the digits are reversed. HEY!  THAT'S IT!!!!

Phil</pre>