A man standing on a platform finds that a train takes 3 seconds to pass him and another train of the same length moving in the opposite direction takes 4 seconds. The time taken by the trains to pass each other will be

### Concept & Logic When two trains of equal length cross a stationary point (a man), their speeds are inversely proportional to the time taken. When crossing each other in opposite directions, the total distance is the sum of their lengths, and the relative speed is the sum of their individual speeds. $$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$ ### Step-by-Step Solution 1. **Define Length and Speeds:** - Let the length of each train be $L$. - Speed of the first train ($v_1$) = $L / 3$. - Speed of the second train ($v_2$) = $L / 4$. 2. **Calculate Relative Speed:** - Since they move in opposite directions, relative speed = $v_1 + v_2 = \frac{L}{3} + \frac{L}{4}$. - Relative speed = $\frac{4L + 3L}{12} = \frac{7L}{12}$. 3. **Calculate Time to Cross Each Other:** - Total distance = $L + L = 2L$. - Time = $\frac{\text{Total Distance}}{\text{Relative Speed}} = \frac{2L}{\frac{7L}{12}}$. - Time = $2L \times \frac{12}{7L} = \frac{24}{7}$ seconds. - Converting to a mixed fraction: $\frac{24}{7} = 3 \frac{3}{7}$ seconds. ### Exam Strategy & Shortcut If two trains of equal length take $t_1$ and $t_2$ seconds to cross a stationary pole/man, the time taken to cross each other in opposite directions is given by the formula: $$ \text{Time} = \frac{2 \times t_1 \times t_2}{t_1 + t_2} $$ Substitute the values: $\frac{2 \times 3 \times 4}{3 + 4} = \frac{24}{7} = 3 \frac{3}{7}$ seconds. This skips all algebraic setup. ### Common Pitfall Students often average the speeds or the times, incorrectly answering $3.5$ seconds. Time to cross each other is a harmonic mean-related function, not an arithmetic mean. ### Final Answer Therefore, the correct answer is **$3 \frac{3}{7}$ seconds**.