1. The first problem is to write $3 \frac{1}{6}$ as an improper fraction. 2. To convert a mixed number to an improper fraction, use the formula: $$\text{Improper fraction} = \frac{\text{whole number} \times \text{denominator} + \text{numerator}}{\text{denominator}}$$ 3. For $3 \frac{1}{6}$, the whole number is 3, the numerator is 1, and the denominator is 6. 4. Calculate the numerator of the improper fraction: $$3 \times 6 + 1 = 18 + 1 = 19$$ 5. So, the improper fraction is: $$\frac{19}{6}$$ Final answer: $\frac{19}{6}$