1. **State the problem:** Find the value of $k$ in the equation $$\frac{7}{18} \times \frac{k}{5} \times \frac{1}{3} = \frac{14}{45}.$$\n\n2. **Write the multiplication of fractions:** When multiplying fractions, multiply the numerators together and the denominators together. So, $$\frac{7}{18} \times \frac{k}{5} \times \frac{1}{3} = \frac{7 \times k \times 1}{18 \times 5 \times 3} = \frac{7k}{270}.$$\n\n3. **Set the equation:** We have $$\frac{7k}{270} = \frac{14}{45}.$$\n\n4. **Solve for $k$ by cross-multiplying:** $$7k \times 45 = 14 \times 270.$$\n\n5. **Calculate each side:** $$315k = 3780.$$\n\n6. **Isolate $k$ by dividing both sides by 315:** $$k = \frac{3780}{315}.$$\n\n7. **Simplify the fraction:** $$k = \frac{\cancel{3780}^{12} \times 315}{\cancel{315}^{12} \times 26.25} = 12.$$\n\n**Final answer:** $$k = 12.$$