(10 3/4 - 12 :1 1/5) *4/9 + 2 1/6

Let's break down the expression step by step.

Step 1: Convert mixed numbers to improper fractions

1. Convert \(10 \frac{3}{4}\) to an improper fraction: \[ 10 \frac{3}{4} = \frac{10 \times 4 + 3}{4} = \frac{40 + 3}{4} = \frac{43}{4} \]

2. Convert \(1 \frac{1}{5}\) to an improper fraction: \[ 1 \frac{1}{5} = \frac{1 \times 5 + 1}{5} = \frac{5 + 1}{5} = \frac{6}{5} \]

3. Convert \(2 \frac{1}{6}\) to an improper fraction: \[ 2 \frac{1}{6} = \frac{2 \times 6 + 1}{6} = \frac{12 + 1}{6} = \frac{13}{6} \]

Step 2: Rewrite the expression

The expression can now be rewritten as: \[ \left( \frac{43}{4} - 12 \div \frac{6}{5} \right) \times \frac{4}{9} + \frac{13}{6} \]

Step 3: Perform the division

Converting the division into multiplication by taking the reciprocal: \[ 12 \div \frac{6}{5} = 12 \times \frac{5}{6} = \frac{12 \times 5}{6} = \frac{60}{6} = 10 \]

Step 4: Perform the subtraction

Now we can compute the subtraction: \[ \frac{43}{4} - 10 = \frac{43}{4} - \frac{40}{4} = \frac{43 - 40}{4} = \frac{3}{4} \]

Step 5: Multiply by \(\frac{4}{9}\)

Now, multiply by \(\frac{4}{9}\): \[ \frac{3}{4} \times \frac{4}{9} = \frac{3 \times 4}{4 \times 9} = \frac{3}{9} = \frac{1}{3} \]

Step 6: Add \(\frac{13}{6}\)

Now add \(\frac{1}{3}\) and \(\frac{13}{6}\). We need a common denominator, which is 6: Convert \(\frac{1}{3}\): \[ \frac{1}{3} = \frac{2}{6} \] Now add: \[ \frac{2}{6} + \frac{13}{6} = \frac{2 + 13}{6} = \frac{15}{6} \]

Step 7: Simplifying the fraction

Simplify \(\frac{15}{6}\): \[ \frac{15}{6} = \frac{5}{2} = 2 \frac{1}{2} \]

Final Result

The final result is: \[ \boxed{2 \frac{1}{2}} \]