1. **State the problem:** We are given the equation $\frac{3m+n}{n-m} = 9$ and need to find the ratio $m:n$. 2. **Write the equation:** $$\frac{3m+n}{n-m} = 9$$ 3. **Cross multiply to eliminate the fraction:** $$3m + n = 9(n - m)$$ 4. **Expand the right side:** $$3m + n = 9n - 9m$$ 5. **Bring all terms involving $m$ and $n$ to one side:** $$3m + n + 9m - 9n = 0$$ 6. **Combine like terms:** $$12m - 8n = 0$$ 7. **Rewrite the equation:** $$12m = 8n$$ 8. **Divide both sides by 4 to simplify:** $$\frac{\cancel{12}m}{\cancel{4}} = \frac{\cancel{8}n}{\cancel{4}} \Rightarrow 3m = 2n$$ 9. **Express the ratio $m:n$:** $$3m = 2n \Rightarrow \frac{m}{n} = \frac{2}{3}$$ 10. **Final answer:** The ratio $m:n$ is $2:3$.