A train travelling at 48 kmph completely crosses another train having half its length and travelling in opposite direction at 42 kmph, in 12 seconds. It also passes a railway platform in 45 seconds. The length of the platform is

### Concept & Logic This problem requires applying the relative speed concept twice. First, to find the lengths of the trains by setting up their relative crossing in opposite directions. Second, using the found length of the first train to determine the length of the platform it crosses. $$ \text{Distance} = \text{Speed} \times \text{Time} $$ ### Step-by-Step Solution 1. **Convert Speeds to m/s:** - Speed of first train = $48$ kmph = $48 \times \frac{5}{18} = \frac{40}{3}$ m/s. - Speed of second train = $42$ kmph = $42 \times \frac{5}{18} = \frac{35}{3}$ m/s. - Relative speed (opposite directions) = $48 + 42 = 90$ kmph = $90 \times \frac{5}{18} = 25$ m/s. 2. **Find Lengths of Trains:** - Let the length of the first train be $2x$. The length of the second train is $x$. - Total distance = $2x + x = 3x$. - $3x = \text{Relative Speed} \times \text{Time} = 25 \times 12 = 300$ m. - $x = 100$ m. So, the first train's length is $200$ m. 3. **Find Length of the Platform:** - The first train passes a platform of length $P$ in 45 seconds. - Total distance = $200 + P$. - Speed of first train = $\frac{40}{3}$ m/s. - $200 + P = \frac{40}{3} \times 45$. - $200 + P = 40 \times 15 = 600$. - $P = 600 - 200 = 400$ m. ### Exam Strategy & Shortcut Perform quick mental conversions for kmph to m/s ($90$ kmph is trivially $25$ m/s). Establish the distance $3x = 300$, so $2x = 200$. For the platform, $48$ kmph over $45$ seconds is $48 \times \frac{5}{18} \times 45 = 600$ m total distance. Subtract the train's length ($600 - 200 = 400$). ### Common Pitfall A frequent error is assigning the first train length as $x$ and the second as $x/2$, which introduces fractions. Using $2x$ and $x$ keeps calculations clean. Additionally, remember to subtract the train's length from the total distance to find the platform length. ### Final Answer Therefore, the correct answer is **400 m**.