I believe the fastest approach here is to test the small handful of 2-digit numbers that satisfy the given information...

The ten’s digit of a two-digit number is twice the unit’s digit.
So the original number must be one of the four following numbers: 21, 42, 63 or 84

Reversing the digits yields a new number that is 27 less than the original number. Which one of the following is the original number?
Reverse 21 to get 12
So, the new number is 9 less than the original number.
No good. We need a difference of 27.

Reverse 42 to get 24
So, the new number is 18 less than the original number.
No good. We need a difference of 27.

Reverse 63 to get 36
So, the new number is 27 less than the original number.

Answer: D

ALTERNATE APPROACH: Algebra
Let x = the units digit in the original number
So 2x = the tens digit in the original number (since the tens digit is twice the units digit)
So, the VALUE of the original number = 10(2x) + x (in much the same way that (7)(10) + 3 is the VALUE of the number 73)
We can simplify 10(2x) + x to get 21x

When we REVERSE the digits, we have:
Let 2x = the units digit in the original number
So x = the tens digit in the original number
So, the VALUE of the new number = 10(x) + 2x = 10x + 2x = 12x

Since the new number is 27 less than the original number we can write: (VALUE of original number) - (VALUE of new number) = 27
Substitute to get: (21x) - (12x) = 27
Simplify: 9x = 27
Solve: x = 3

Since x = the units digit in the original number and 2x = the tens digit in the original number, the original number equals 63

Answer: D