In a two-digit number the tens digit and the units digit are in the ratio 2 : 3. Adding 27 to the number reverses its digits. The number is:

The digits in ratio 2 : 3 can only be (2, 3), (4, 6) or (6, 9). Reversal on adding 27 means $(10b+a)-(10a+b)=27$, i.e. $9(b-a)=27$, so the digits must differ by 3. Only 6 and 9 do, giving 69. Check: 69 + 27 = 96, the reverse. Answering 96 gives the reversed number rather than the original.