Solution:

The number is 81, simply because
81 = 9 (8 + 1).

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 = 9 t + 9 u or
1 t = 8 u.

Thus, t / u = 8 / 1. Since t and u are digits, t must be 8 and u must be 1.

Food for thought:

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