Two circular paths P1 and P2 of radii 150 m and 50 m, respectively touch at a point B. Starting from B at the same time, Alex and Andy are walking on path P1 and path P2 at speeds 20 km/hr and 10 km/hr, respectively. The number of full rounds that Alex will make before he meets Andy again for the first time is:

Let's solve this problem step by step:

Step 1: Calculate the circumference of each circular path. The circumference of a circle is given by the formula C = 2πr, where r is the radius. For path P1 with a radius of 150 m, the circumference is C1 = 2π(150) = 300π m. For path P2 with a radius of 50 m, the circumference is C2 = 2π(50) = 100π m.

Step 2: Calculate the time it takes for Alex and Andy to meet again. Since they start at the same time and meet for the first time, the time taken by both of them will be the same. We can use the formula Time = Distance / Speed to calculate the time. For Alex on path P1, the distance covered will be equal to the circumference of path P1, which is 300π m. His speed is given as 20 km/hr, which we need to convert to m/s. 1 km = 1000 m and 1 hr = 3600 s, so 20 km/hr = (20 * 1000) / 3600 m/s = 50/9 m/s. Using the formula, Time1 = Distance1 / Speed1 = (300π) / (50/9) = (300π * 9) / 50 = 54π/5 s.

For Andy on path P2, the distance covered will be equal to the circumference of path P2, which is 100π m. His speed is given as 10 km/hr, which we need to convert to m/s. Using the same conversion, 10 km/hr = (10 * 1000) / 3600 m/s = 25/9 m/s. Using the formula, Time2 = Distance2 / Speed2 = (100π) / (25/9) = (100π * 9) / 25 = 36π/5 s.

Step 3: Calculate the number of full rounds Alex will make before meeting Andy again. Since the time taken by both of them is the same, we can equate Time1 and Time2. 54π/5 = 36π/5 Dividing both sides by π/5, we get: 54 = 36 This is not possible, so there is no solution.

Therefore, Alex and Andy will not meet again on their respective paths.