In an 800 meter race, Kaushik gives Piyush a head start of 200 meters. Even after this head start, Kaushik loses the race by 20 seconds. If the ratio of the speeds of Piyush and Kaushik is 3 : 2, what are the speeds of Piyush and Kaushik respectively in m/s?

Introduction / Context:
This is a more challenging race problem combining head start, time difference at the finish, and a given ratio of speeds. It tests whether students can model the situation with algebraic equations, convert word statements into mathematical relationships, and handle ratios and units correctly. These types of problems are common in higher level aptitude and quantitative exams.

Given Data / Assumptions:

• Total race distance from the starting line = 800 meters for Kaushik.
• Piyush is given a head start of 200 meters, so Piyush runs only 600 meters.
• Kaushik loses the race by 20 seconds, meaning Kaushik finishes 20 seconds after Piyush.
• The ratio of speeds of Piyush to Kaushik is 3 : 2.
• We assume constant speeds for both runners throughout the race.
• We need the speeds of Piyush and Kaushik in meters per second.

Concept / Approach:
Let the speed of Piyush be 3k and the speed of Kaushik be 2k, in m/s, according to the given ratio. The time taken by each runner is distance divided by speed. Since Kaushik loses by 20 seconds, his time is 20 seconds more than that of Piyush. We set up an equation relating the two times and then solve for k. Once k is known, the speeds follow directly from the ratio.

Step-by-Step Solution:
Let speed of Piyush = 3k m/s. Let speed of Kaushik = 2k m/s. Distance run by Piyush = 600 m. Distance run by Kaushik = 800 m. Time taken by Piyush = 600 / (3k) seconds. Time taken by Kaushik = 800 / (2k) seconds. Kaushik loses by 20 seconds, so 800 / (2k) = 600 / (3k) + 20. Simplify: 800 / (2k) = 400 / k and 600 / (3k) = 200 / k. So 400 / k = 200 / k + 20. Subtract 200 / k from both sides: 200 / k = 20. Therefore k = 200 / 20 = 10. Speed of Piyush = 3k = 3 * 10 = 30 m/s. Speed of Kaushik = 2k = 2 * 10 = 20 m/s.

Verification / Alternative check:
Check the times with these speeds. Time of Piyush = 600 / 30 = 20 seconds. Time of Kaushik = 800 / 20 = 40 seconds. The difference is 40 - 20 = 20 seconds, which matches the condition that Kaushik loses by 20 seconds. The ratio of speeds is 30 : 20, which simplifies to 3 : 2, matching the given ratio. Therefore, the speeds are consistent with the entire problem statement.

Why Other Options Are Wrong:
The other speed pairs do not satisfy all conditions simultaneously. For example, 24 m/s and 16 m/s still have a ratio of 3 : 2 but do not produce the correct 20 second time difference when applied to the distances of 600 m and 800 m. Similarly, 27 m/s and 18 m/s, 36 m/s and 24 m/s, and 18 m/s and 12 m/s fail either the head start condition, the time difference, or both. Only 30 m/s and 20 m/s satisfy all given constraints.

Common Pitfalls:
Students may misinterpret who is faster or misread the ratio as Kaushik to Piyush instead of Piyush to Kaushik. Others forget that the race distances are different due to the head start and incorrectly use 800 meters for both runners. Careful attention to the wording of the problem and systematic use of the ratio in the equations prevents these mistakes.

Final Answer:
The speeds of Piyush and Kaushik are 30 m/s and 20 m/s respectively.