1. The problem is to calculate the division of two fractions: $\frac{3}{14} \div \frac{6}{7}$.\n\n2. The formula for dividing fractions is:\n$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$\nThis means we multiply the first fraction by the reciprocal of the second fraction.\n\n3. Applying this to our problem:\n$$\frac{3}{14} \div \frac{6}{7} = \frac{3}{14} \times \frac{7}{6}$$\n\n4. Multiply the numerators and denominators:\n$$\frac{3 \times 7}{14 \times 6} = \frac{21}{84}$$\n\n5. Simplify the fraction $\frac{21}{84}$ by dividing numerator and denominator by their greatest common divisor, which is 21:\n$$\frac{21 \div 21}{84 \div 21} = \frac{1}{4}$$\n\n6. Therefore, the final answer is $\frac{1}{4}$.