Simplify : d. `(1)/(6)` of `(1 (1)/(9) div 7 (1)/(2) )`

Answer

To simplify the expression \( \frac{1}{6} \) of \( \left( 1 \frac{1}{9} \div 7 \frac{1}{2} \right) \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers \( 1 \frac{1}{9} \) and \( 7 \frac{1}{2} \) into improper fractions.

- For \( 1 \frac{1}{9} \):
\[ 1 \frac{1}{9} = \frac{1 \times 9 + 1}{9} = \frac{10}{9} \]

- For \( 7 \frac{1}{2} \):
\[ 7 \frac{1}{2} = \frac{7 \times 2 + 1}{2} = \frac{15}{2} \]

### Step 2: Rewrite the Expression
Now, we can rewrite the expression:
\[ \frac{1}{6} \text{ of } \left( \frac{10}{9} \div \frac{15}{2} \right) \]

### Step 3: Change Division to Multiplication
Next, we change the division to multiplication by taking the reciprocal of the second fraction:
\[ \frac{10}{9} \div \frac{15}{2} = \frac{10}{9} \times \frac{2}{15} \]

### Step 4: Multiply the Fractions
Now, we multiply the fractions:
\[ \frac{10 \times 2}{9 \times 15} = \frac{20}{135} \]

### Step 5: Simplify the Fraction
Now we need to simplify \( \frac{20}{135} \):
- The greatest common divisor (GCD) of 20 and 135 is 5.
- Dividing both the numerator and the denominator by 5:
\[ \frac{20 \div 5}{135 \div 5} = \frac{4}{27} \]

### Step 6: Multiply by \( \frac{1}{6} \)
Now we multiply \( \frac{1}{6} \) by \( \frac{4}{27} \):
\[ \frac{1}{6} \times \frac{4}{27} = \frac{1 \times 4}{6 \times 27} = \frac{4}{162} \]

### Step 7: Final Simplification
Now we simplify \( \frac{4}{162} \):
- The GCD of 4 and 162 is 2.
- Dividing both the numerator and the denominator by 2:
\[ \frac{4 \div 2}{162 \div 2} = \frac{2}{81} \]

### Final Answer
The simplified result is:
\[ \frac{2}{81} \]