Solution:

The number is 45, simply because
45 = 5 (4 + 5).

How does one find this number?

Let t be the digit in the tens place and u be the digit in the units place. Then, the number is 10 t + u, and the sum of its digits is t + u. The following equation can be readily written:

10 t + u = 5 t + 5 u or
5 t = 4 u.

Thus, t / u = 4 / 5. Since t and u are digits, t must be 4 and u must be 5.

Food for thought:

Is the assumption of a 2-digit number a valid one?