To evaluate the expression \((\frac{3}{4} - \frac{1}{36}) \div \frac{1}{6}\), we'll follow these steps:
- Evaluate \(\frac{3}{4} - \frac{1}{36}\).
- Divide the result by \(\frac{1}{6}\).
Step 1: Find a common denominator for \(\frac{3}{4}\) and \(\frac{1}{36}\).
The denominators are 4 and 36. The least common multiple of 4 and 36 is 36.
Convert \(\frac{3}{4}\) to a fraction with a denominator of 36: \[ \frac{3}{4} = \frac{3 \times 9}{4 \times 9} = \frac{27}{36} \]
Now we can perform the subtraction: \[ \frac{27}{36} - \frac{1}{36} = \frac{27 - 1}{36} = \frac{26}{36} \]
Now simplify \(\frac{26}{36}\): \[ \frac{26}{36} = \frac{13}{18} \quad \text{(dividing the numerator and denominator by 2)} \]
Step 2: Now divide \(\frac{13}{18}\) by \(\frac{1}{6}\).
Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{13}{18} \div \frac{1}{6} = \frac{13}{18} \times 6 \]
Now perform the multiplication: \[ \frac{13 \times 6}{18} = \frac{78}{18} \]
Now simplify \(\frac{78}{18}\): \[ \frac{78}{18} = \frac{39}{9} = \frac{13}{3} \quad \text{(dividing the numerator and denominator by 3)} \]
Thus, the final answer is: \[ \boxed{\frac{13}{3}} \]