1. **Problem statement:** Alex has a 10-faced die with faces 1 to 10. The probability of each face is proportional to its number, so probabilities are in ratio $1:2:3:4:5:6:7:8:9:10$. We roll the die twice and want the probability that the sum of the two rolls is 9. 2. **Step 1: Find the probability of each face.** The sum of the ratios is $1+2+3+4+5+6+7+8+9+10 = 55$. So, the probability of face $i$ is $\frac{i}{55}$. 3. **Step 2: Identify pairs of rolls that sum to 9.** Possible pairs $(x,y)$ with $x,y \in \{1,...,10\}$ and $x+y=9$ are: $(1,8), (2,7), (3,6), (4,5), (5,4), (6,3), (7,2), (8,1)$. 4. **Step 3: Calculate the probability for each pair.** Since rolls are independent, $P(x,y) = P(x) \times P(y) = \frac{x}{55} \times \frac{y}{55} = \frac{xy}{3025}$. 5. **Step 4: Sum probabilities of all pairs:** $$ \sum P(x,y) = \frac{1\times8 + 2\times7 + 3\times6 + 4\times5 + 5\times4 + 6\times3 + 7\times2 + 8\times1}{3025} = \frac{8 + 14 + 18 + 20 + 20 + 18 + 14 + 8}{3025} = \frac{120}{3025}. $$ 6. **Step 5: Simplify the fraction:** $$ \frac{120}{3025} = \frac{\cancel{5} \times 24}{\cancel{5} \times 605} = \frac{24}{605}. $$ **Final answer:** The probability that the sum of the two rolls is 9 is $\boxed{\frac{24}{605}}$. This corresponds to option (E).