To solve this expression, we follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

First, let's simplify the expression within the innermost parentheses:
5/8 + 3/8 = 8/8 = 1

Then, we can rewrite the expression with the simplified innermost parentheses:
0.9 * (1 5/9 - 4/9: 1)

Next, let's simplify the expression within the outer parentheses:
1 5/9 = 14/9
4/9 ÷ 1 = 4/9

Now we rewrite the expression with the simplified components:
0.9 * (14/9 - 4/9)

Subtracting the fractions inside the parentheses:
14/9 - 4/9 = 10/9

Now we rewrite the expression with the simplified components:
0.9 * 10/9

Multiplying 0.9 by 10/9:
0.9 10/9 = 9/10 10/9 = 1

Therefore, the answer is 1.