A train speeds past a pole in 20 seconds and speeds past a platform 100 metres in length in 30 seconds. What is the length of the train?

### Concept & Logic The speed of the train is constant in both scenarios. We can set up an equation by equating the speed formula for crossing a pole with the speed formula for crossing a platform. $$ \text{Speed} = \frac{\text{Length of Train}}{\text{Time to cross pole}} = \frac{\text{Length of Train} + \text{Length of Platform}}{\text{Time to cross platform}} $$ ### Step-by-Step Solution * **Given:** * Time to cross pole = $20$ seconds. * Time to cross platform = $30$ seconds. * Length of platform = $100$ m. * Let the length of the train be $L$. * **Calculation:** * Equating the speeds: $\frac{L}{20} = \frac{L + 100}{30}$. * Cross-multiply to solve for $L$: $30L = 20(L + 100)$. * $30L = 20L + 2000$. * $10L = 2000$. * $L = 200$ m. ### Exam Strategy & Shortcut The train takes an extra $10$ seconds ($30 - 20$) specifically to cover the extra $100$ m of the platform. This means the train's speed is $\frac{100}{10} = 10$ m/s. Since it takes $20$ seconds to cross its own length (the pole), its length is $20 \times 10 = 200$ m. ### Common Pitfall A standard trap is trying to find the speed first without fully realizing that setting up a direct ratio between distances and times provides an immediate path to the unknown length. ### Final Answer Therefore, the correct answer is **200 m**.