Train K crosses a pole in 30 seconds and train L crosses the same pole in 1 minute 20 seconds. The length of train K is three fourths of the length of train L. What is the ratio of the speed of train K to the speed of train L?

Introduction / Context:
This problem involves two trains crossing the same pole in different times and having a known relationship between their lengths. Such questions test understanding of the relation speed = distance / time and how proportional reasoning can be used when actual lengths are not given but ratios are. We are asked to find the ratio of speeds of the two trains based on their crossing times and relative lengths.

Given Data / Assumptions:

• Train K crosses a pole in 30 seconds.
• Train L crosses the same pole in 1 minute 20 seconds = 80 seconds.
• Length of train K = three fourths of the length of train L.
• Both trains move at constant speeds on straight tracks.
• When a train crosses a pole, the distance covered is equal to its own length.

Concept / Approach:
Speed is equal to distance divided by time. When a train crosses a pole, the distance is simply the length of the train. Therefore, the speed of each train is proportional to its length divided by its crossing time. With the ratio of lengths known and times given, we can find the ratio of speeds without needing the actual lengths in metres.

Step-by-Step Solution:
Step 1: Let the length of train L be L metres. Step 2: Then length of train K = (3/4) * L metres. Step 3: Speed of train K = distance / time = (3/4 * L) / 30. Step 4: Speed of train L = L / 80. Step 5: Ratio of speeds K : L = [(3/4 * L) / 30] : [L / 80]. Step 6: Cancel L and simplify: (3/4) / 30 divided by 1 / 80 = (3/4) * (80 / 30) = 3 * 80 / (4 * 30) = 240 / 120 = 2 / 1. Step 7: Therefore, the ratio of speeds is 2 : 1.

Verification / Alternative check:
We can assign a convenient value, for example L = 80 m. Then train L speed = 80 / 80 = 1 m/s. Train K length = 3/4 * 80 = 60 m. Its speed = 60 / 30 = 2 m/s. So the ratio of speeds is 2 : 1, which confirms the result. The absolute values were not necessary since ratios suffice.

Why Other Options Are Wrong:
Ratios like 1 : 3, 3 : 1, and 1 : 2 contradict the fact that train K is shorter but crosses the pole in much less time, so it must be faster, not slower. The distractor 3 : 2 does not match the simplified exact ratio obtained by algebraic cancellation.

Common Pitfalls:
A common mistake is to compare times directly without including the effect of different lengths. Another pitfall is to confuse three fourths with four thirds, which would reverse the ratio. Students may also forget to convert 1 minute 20 seconds to 80 seconds correctly, which can distort the final ratio.

Final Answer:
The ratio of the speed of train K to the speed of train L is 2 : 1.