To simplify \(1 \frac{2}{36}\), we first convert the mixed number into an improper fraction:
\[ 1 \frac{2}{36} = \frac{36}{36} + \frac{2}{36} = \frac{36 + 2}{36} = \frac{38}{36} \]
Now, we can simplify \(\frac{38}{36}\) by finding the greatest common divisor (GCD) of 38 and 36. Both numbers can be divided by 2:
\[ \frac{38 \div 2}{36 \div 2} = \frac{19}{18} \]
Thus, the simplified form of \(1 \frac{2}{36}\) is \(\frac{19}{18}\).