The sum of the digits of a two-digit number is 6.  if the digits are reversed, the difference between the new number and the original number is 18.  find the original number

So:

There are only a few two numbers with a sum of 6. They are:

1 and 5
2 and 4
3 and 3

So basically, there are only three sets of possible two digit + reversed two digit numbers.

3 and 3, we can just eliminate that because it's the same thing both ways around.

1 and 5, the difference between 15 and 51 is way too much.

So therefore it would have to be the pairs of 2 and 4.

Since the difference between the new number and the original number is 18, the new number would have to be 42 and the old one 24.

So the original number would be 24.