UGC NET Questions (Paper – 1)

For a fixed distance d, time is given by t = d/speed, so it is inversely proportional to speed. The first train’s time is d/90 and the second’s time is d/60. Their time ratio is (d/90):(d/60) = 1/90:1/60 = 60:90, which simplifies to 2:3. Thus, the first train takes 2 parts of time for every 3 parts taken by the second.

Option A:
Option A, 3:2, reverses the correct ratio and would mean the faster train takes more time, which contradicts the fundamental relationship between speed and time for the same distance. A higher speed must correspond to a shorter time, not a longer one.

Option B:
Option B is correct because it arises from properly comparing the reciprocals of the speeds. When the distance is constant, the time ratio is the inverse of the speed ratio. Since the speed ratio is 90:60 = 3:2, the time ratio must be 2:3, matching the computed result.

Option C:
Option C, 1:2, would imply that the first train takes half the time of the second, meaning its speed would have to be double the second’s speed. However, 90 km/h is only 1.5 times 60 km/h, not twice, so 1:2 overstates the difference.

Option D:
Option D, 5:6, bears no direct relationship to 90 and 60 when we follow the inverse proportionality rule. It cannot be obtained by simplifying 60:90 or any equivalent fraction, so it does not represent the correct time ratio.

For the same distance, speed is inversely proportional to time. Let the common distance be d. The first cyclist’s speed is d/4 and the second’s speed is d/6. Their speed ratio is (d/4):(d/6) = 1/4:1/6 = 6:4, which simplifies to 3:2. Thus, the ratio of their speeds is 3:2.

Option A:
Option A, 2:3, corresponds to the ratio of their times, not their speeds. It would imply that the faster cyclist rides more slowly than the slower one, which contradicts the idea that less time for the same distance means higher speed.

Option B:
Option B is correct because it correctly applies the inverse proportionality between speed and time. The cyclist who takes 4 hours must be faster, and the computed ratio 3:2 quantifies exactly how much faster he is than the one who takes 6 hours.

Option C:
Option C, 4:3, might arise from directly comparing 4 and 6 but simplifying incorrectly. However, when we compute speeds rather than times, we get 3:2, not 4:3. Thus 4:3 is not compatible with the relationship between speed and time here.

Option D:
Option D, 5:4, has no clear basis in the given numbers and does not result from any correct manipulation of the times or speeds. It therefore does not represent the true speed ratio.