To evaluate the expression \((\frac{9}{10} - \frac{1}{6}) \div \frac{2}{9}\), we will follow these steps:
Step 1: Calculate \(\frac{9}{10} - \frac{1}{6}\)
First, we need a common denominator for the fractions \(\frac{9}{10}\) and \(\frac{1}{6}\). The least common multiple (LCM) of 10 and 6 is 30.
Convert each fraction:
\[ \frac{9}{10} = \frac{9 \times 3}{10 \times 3} = \frac{27}{30} \]
\[ \frac{1}{6} = \frac{1 \times 5}{6 \times 5} = \frac{5}{30} \]
Now we can perform the subtraction:
\[ \frac{27}{30} - \frac{5}{30} = \frac{27 - 5}{30} = \frac{22}{30} \]
Next, simplify \(\frac{22}{30}\):
The greatest common divisor (GCD) of 22 and 30 is 2.
\[ \frac{22 \div 2}{30 \div 2} = \frac{11}{15} \]
Step 2: Divide by \(\frac{2}{9}\)
Now we need to calculate \(\frac{11}{15} \div \frac{2}{9}\). Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{11}{15} \div \frac{2}{9} = \frac{11}{15} \times \frac{9}{2} \]
Now, multiply the fractions:
\[ \frac{11 \times 9}{15 \times 2} = \frac{99}{30} \]
Step 3: Simplify \(\frac{99}{30}\)
The GCD of 99 and 30 is 3.
\[ \frac{99 \div 3}{30 \div 3} = \frac{33}{10} \]
Conclusion
The final answer as a fraction in simplest form is:
\[ \boxed{\frac{33}{10}} \]