A rod of length 40 cm is cut at a point chosen at random along its length…

Step 1. Both at least $8\ \text{cm}$. The cut must be between $8\ \text{cm}$ and $32\ \text{cm}$ from the left end, a length of $24\ \text{cm}$. So $\dfrac{24}{40} = 0.6$.

Step 2. Longer at least three times the shorter. If the shorter piece is $s$, the longer is $40\ \text{cm} - s$. We need $40\ \text{cm} - s \ge 3s$, so $s \le 10\ \text{cm}$. The cut must lie within $10\ \text{cm}$ of either end: $10\ \text{cm} + 10\ \text{cm} = 20\ \text{cm}$ out of $40\ \text{cm}$, so $0.5$.

Step 3. Shorter at least $15\ \text{cm}$. The cut must lie between $15\ \text{cm}$ and $25\ \text{cm}$, a length of $10\ \text{cm}$. So $\dfrac{10}{40} = 0.25$.