Given Information:
- The number consists of two digits.
- The number is four times the sum of its digits.
- When 27 is added to the number, a new number is formed.

Let's break down the problem:

Let the two-digit number be represented as AB:
- A is the digit in the tens place.
- B is the digit in the ones place.

Formulas:
- The number can be represented as 10A + B.
- The sum of the digits is A + B.

Given Conditions:
- The number is four times the sum of its digits: 10A + B = 4(A + B).

Solving for A and B:
10A + B = 4A + 4B
6A = 3B
2A = B

Possible Values:
Since A and B are digits, the possible values for A and B are limited:
- A can be 1, 2, or 3 (as it is the tens place digit).
- B can be 2, 4, or 6 (based on the relationship 2A = B).

Checking the Conditions:
Let's try the possible values:
- If A = 1, B = 2 (12). But 12 is not four times the sum of its digits.
- If A = 2, B = 4 (24). 24 is four times the sum of its digits (2 + 4 = 6, and 24 = 4 * 6).

Adding 27 to the Number:
- Adding 27 to the number 24 results in 51.
- 51 is not four times the sum of its digits (5 + 1 = 6, but 51 ≠ 4 * 6).

Conclusion:
The two-digit number that satisfies the given conditions is 24, but adding 27 to it does not result in a number that is four times the sum of its digits.