Two trains, one from Howrah to Patna and the other from Patna to Howrah, start simultaneously. After they meet, the trains reach their destinations after 9 hours and 16 hours respectively. The ratio of their speeds is :

### Concept & Post-Meeting Time Formula If two bodies start simultaneously from points A and B towards each other, and after crossing each other they take $a$ and $b$ hours respectively to reach their destinations, then the ratio of their speeds is inversely proportional to the square root of the times taken. $$ \frac{\text{Speed of 1st}}{\text{Speed of 2nd}} = \sqrt{\frac{b}{a}} $$ ### Step-by-Step Solution 1. **Identify the given variables:** - Time taken by first train after meeting ($t_1$) = $9$ hours. - Time taken by second train after meeting ($t_2$) = $16$ hours. 2. **Apply the standard formula:** - Ratio of speeds = $\sqrt{t_2} : \sqrt{t_1}$ 3. **Calculate the ratio:** - Ratio = $\sqrt{16} : \sqrt{9}$ - Ratio = $4 : 3$ ### Exam Strategy & Shortcut This is a standard pattern question. The moment you see "after they meet, reach in $t_1$ and $t_2$", instantly apply $\sqrt{t_2 / t_1}$. Just be careful about the order. The first train's speed corresponds to the square root of the second train's time. ### Common Pitfall The most common error is forgetting to invert the ratio, leading students to incorrectly choose $3 : 4$ or simply picking the direct ratio $9 : 16$. ### Final Answer Therefore, the correct answer is **4 : 3**.