1. **State the problem:** Simplify the expression $6 - \frac{1}{7} \div \frac{3}{14}$. 2. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $\frac{1}{7} \div \frac{3}{14} = \frac{1}{7} \times \frac{14}{3}$. 3. **Perform the multiplication:** $$\frac{1}{7} \times \frac{14}{3} = \frac{1 \times 14}{7 \times 3} = \frac{14}{21}$$ 4. **Simplify the fraction:** $$\frac{14}{21} = \frac{\cancel{14}^{2 \times 7}}{\cancel{21}^{3 \times 7}} = \frac{2}{3}$$ 5. **Substitute back into the original expression:** $$6 - \frac{2}{3}$$ 6. **Convert 6 to a fraction with denominator 3:** $$6 = \frac{18}{3}$$ 7. **Subtract the fractions:** $$\frac{18}{3} - \frac{2}{3} = \frac{18 - 2}{3} = \frac{16}{3}$$ **Final answer:** $$\boxed{\frac{16}{3}}$$