1. **Problem:** Given the operation $m * n = \frac{m - n}{n}$, and that $3 * 9 = \frac{3}{2}$, find the value of $9$. 2. **Step 1: Write down the given operation and substitution.** $$3 * 9 = \frac{3 - 9}{9} = \frac{3}{2}$$ 3. **Step 2: Simplify the left side.** $$\frac{3 - 9}{9} = \frac{-6}{9} = -\frac{2}{3}$$ 4. **Step 3: Compare with the given value.** We have $-\frac{2}{3} \neq \frac{3}{2}$, so the given equality $3 * 9 = \frac{3}{2}$ is not true for $n=9$. 5. **Step 4: Interpret the problem carefully.** The problem states "if $3 * 9 = \frac{3}{2}$, find the value of 9." This suggests that the symbol "9" in the expression $3 * 9$ is not the number 9 but an unknown value to be found. 6. **Step 5: Let $n$ be the unknown value. Then:** $$3 * n = \frac{3 - n}{n} = \frac{3}{2}$$ 7. **Step 6: Solve for $n$.** Multiply both sides by $n$: $$3 - n = \frac{3}{2} n$$ Rewrite: $$3 = \frac{3}{2} n + n = \frac{3}{2} n + \frac{2}{2} n = \frac{5}{2} n$$ 8. **Step 7: Isolate $n$.** $$n = \frac{3}{\frac{5}{2}} = 3 \times \frac{2}{5} = \frac{6}{5}$$ 9. **Answer:** The value of $n$ (the unknown "9" in the problem) is $$\boxed{\frac{6}{5}}$$