1. **State the problem:** Solve for $k$ in the equation $$\frac{K+k}{k} = 6.$$ 2. **Rewrite the equation:** Multiply both sides by $k$ to eliminate the denominator: $$\cancel{k} \times \frac{K+k}{\cancel{k}} = 6k \implies K + k = 6k.$$ 3. **Isolate $k$:** Subtract $k$ from both sides: $$K + k - k = 6k - k \implies K = 5k.$$ 4. **Solve for $k$:** Divide both sides by 5: $$\frac{K}{\cancel{5}} = \frac{5k}{\cancel{5}} \implies k = \frac{K}{5}.$$ **Final answer:** $$k = \frac{K}{5}.$$