A train travelling with a constant speed crosses a 96-metre long platform in 12 seconds and another 141-metre long platform in 15 seconds. The length of the train and its speed are

### Concept & Formula When a train crosses two platforms of different lengths, the difference in the time taken is entirely due to the difference in the lengths of the platforms. $$ \text{Speed} = \frac{\text{Difference in Platform Lengths}}{\text{Difference in Time}} $$ ### Step-by-Step Solution * **Given:** * Platform 1: $96$ m, Time 1: $12$ seconds. * Platform 2: $141$ m, Time 2: $15$ seconds. * **Calculation:** * Difference in lengths = $141 - 96 = 45$ m. * Difference in times = $15 - 12 = 3$ seconds. * Speed of train = $\frac{45}{3} = 15$ m/s. * Convert speed to km/hr = $15 \times \frac{18}{5} = 3 \times 18 = 54$ km/hr. * To find the length of the train ($L$): In the first scenario, total distance = $L + 96$. * $L + 96 = \text{Speed} \times \text{Time} = 15 \times 12 = 180$ m. * $L = 180 - 96 = 84$ m. ### Exam Strategy & Shortcut Find the speed immediately: $\frac{141 - 96}{15 - 12} = \frac{45}{3} = 15$ m/s. Eliminate options with incorrect speeds ($15$ m/s = $54$ km/hr, so options (b) and (d) are out). Then, simply calculate length: $(15 \times 12) - 96 = 84$ m. ### Common Pitfall Setting up two separate complex equations ($L + 96 = 12S$ and $L + 141 = 15S$) and trying to solve them simultaneously wastes precious time compared to using the difference method. ### Final Answer Therefore, the correct answer is **84 metres and 54 km/hr**.