Explanations:The problem involves two individuals, A and B, walking towards each other from opposite points, Jammu and Delhi. After they meet, A takes 361 hours to reach Delhi, and B takes 289 hours to reach Jammu. We need to find the ratio of their speeds.
The key concept here is that the ratio of their speeds is inversely proportional to the square root of the time taken after they meet. This is because the distance each person covers after meeting is the same, and the time taken is inversely proportional to their speed.
Let's denote the speed of A as \(v_A\) and the speed of B as \(v_B\). The time taken by A after meeting is 361 hours, and the time taken by B after meeting is 289 hours. The relationship between speed and time is given by the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
Since the distance covered by A and B after meeting is the same, we can write:
\[ v_A \times 361 = v_B \times 289 \]
However, to find the ratio of their speeds, we use the inverse of the square root of the times taken:
\[ \frac{v_A}{v_B} = \sqrt{\frac{289}{361}} \]
Calculating the square roots, we get:
\[ \sqrt{289} = 17 \]
\[ \sqrt{361} = 19 \]
Thus, the ratio of the speeds of A to B is:
\[ \frac{v_A}{v_B} = \frac{17}{19} \]
Therefore, the correct answer is 17:19.
It is important to note that the ratio of the speeds is not directly proportional to the times taken, but rather the inverse of the square root of the times taken. This is a common misconception that students often fall into, leading them to choose incorrect options such as 289:361 or 361:289.