Consider the first number to be (10x+y) .
When u reverse the digits the new two digit number is (10y + x).
Now we know the two numbers differ by 27

(10y + x) - (10x + y) = 27

9y - 9x = 27
y - x = 3

I got the answer by plugging in the answers provided. Question states must be 2 positive Digits and when reversed the difference is 27. So I started with choice C.

So at random i picked the number 6 and 1 b/c they had a difference of 5. (could have used 7 and 2, 8 and 3, etc. as long as they were difference of 5)

So , 6 and 1 would be 61 or 16 reversed. 61-16= 45
Number is bigger than 27 so we know that it must be a number smaller than a difference of 5.

Next I tried the smallest number. Choice A which is 3.

Again picked two random numbers that had a difference of 3. 4 and 1 was my choice, could have used 5 and 2, 6 and 3 etc.

So, 4 and 1 would be 41 or 14 reversed. 41-14= 27 Correct Answer is A.
You could double check your answer using 2 other number different by 3 such as 5 and 2.

Using digits 5 and 2 you get 52 and 25 reversed= 27

Aside from picking numbers or using the algebra..there must be an easier way to do this question. It baffles me. ANy help on other ways to do this?

There is a pattern that develops here:

10 - 01 = 9

20 - 02 = 18

30 - 03 = 27

40 - 04 = 36

50 - 05 = 45

60 - 06 = 54

70 - 07 = 63

80 - 08 = 72

90 - 09 = 81

As you can see the difference between the 2 digits is 9*(difference of digits)

So if you ever get a question like this again, the quick way is to ask yourself if the difference of AB - BA is a multiple of 9, then that difference/9 = the difference between the 2 digits.

NOTE: this probably won't work in too many situations.

*Edit* To add: For all combinations of AB -BA, the difference will be a multiple of 9. And 9*(difference of digits) will be the difference of AB-BA.