1. **State the problem:** We have a circle with radius $7$ cm divided into two sectors in the ratio $3:7$. We need to find the length of the minor arc. 2. **Understand the ratio:** The ratio $3:7$ means the circle is divided into parts where the minor sector corresponds to $3$ parts and the major sector to $7$ parts, total parts = $3 + 7 = 10$. 3. **Calculate the central angle of the minor sector:** The full circle has $360^\circ$. The minor sector's angle is $$\frac{3}{10} \times 360^\circ = 108^\circ.$$ 4. **Formula for arc length:** The length $L$ of an arc with radius $r$ and central angle $\theta$ (in degrees) is $$L = 2 \pi r \times \frac{\theta}{360}.$$ 5. **Substitute values:** $$L = 2 \pi \times 7 \times \frac{108}{360}.$$ 6. **Simplify:** $$L = 14 \pi \times \frac{108}{360} = 14 \pi \times \frac{3}{10}.$$ 7. **Calculate:** $$L = \cancel{14} \pi \times \cancel{\frac{3}{10}} = \frac{42}{10} \pi = 4.2 \pi.$$ 8. **Final answer:** Using $\pi \approx 3.1416$, $$L \approx 4.2 \times 3.1416 = 13.19 \text{ cm}.$$ So, the length of the minor arc is approximately $13.19$ cm.