1. **Problem statement:** A train passes a 50 m long platform in 4 \frac{1}{2} seconds and a pole in 2 seconds. Find the length of the train and its speed. 2. **Formulas and rules:** - Speed = \frac{Distance}{Time} - When a train passes a pole, the distance covered is the length of the train. - When a train passes a platform, the distance covered is the length of the train plus the length of the platform. 3. **Step 1: Find the length of the train.** - Let the length of the train be $L$ meters. - Time to pass pole = 2 s - Speed of train = \frac{L}{2} 4. **Step 2: Use the time to pass the platform to find $L$.** - Length of platform = 50 m - Time to pass platform = 4 \frac{1}{2} = \frac{9}{2} s - Distance covered passing platform = $L + 50$ - Speed = \frac{L + 50}{\frac{9}{2}} = \frac{2(L + 50)}{9}$ 5. **Step 3: Equate speeds from steps 3 and 4:** $$\frac{L}{2} = \frac{2(L + 50)}{9}$$ 6. **Step 4: Solve for $L$: ** $$9L = 4(L + 50)$$ $$9L = 4L + 200$$ $$9L - 4L = 200$$ $$5L = 200$$ $$L = \frac{200}{5} = 40$$ 7. **Step 5: Find speed of the train:** - Speed = \frac{L}{2} = \frac{40}{2} = 20 \text{ m/s}$$ - Convert to km/h: $$20 \times \frac{18}{5} = 72 \text{ km/h}$$ **Final answers:** - Length of the train = 40 m - Speed of the train = 72 km/h