Amar can complete a certain piece of work in 30 days, while Raman can finish the same work in 15 days when working alone. If both Amar and Raman work together continuously for 4 days, what percentage of the total work will be completed by them in that time?

Introduction / Context:
This time and work problem checks your understanding of how to add individual work rates and then convert the completed fraction of work into a percentage. Two workers, Amar and Raman, have different capacities, and you are asked to determine how much of the job they complete together in a fixed period of time.

Given Data / Assumptions:

Amar alone completes the work in 30 days.
Raman alone completes the same work in 15 days.
They work together for exactly 4 days at constant rates.
The total work is considered as one complete unit of job.

Concept / Approach:
We use the basic relationship Work = Rate * Time. First, convert the given times into daily work rates for Amar and Raman. Since they are working together, their combined rate is the sum of their individual rates. Multiply that combined rate by the time they work together (4 days) to find the fraction of the work completed. Finally, convert that fraction into a percentage to match the requirement of the question.

Step-by-Step Solution:
Let the total work be 1 unit. Daily work rate of Amar = 1/30 of the work per day. Daily work rate of Raman = 1/15 of the work per day. Combined daily work rate when they work together = 1/30 + 1/15. Compute this: 1/30 + 1/15 = 1/30 + 2/30 = 3/30 = 1/10 of the work per day. In 4 days, the fraction of work completed = combined rate * time = (1/10) * 4 = 4/10. Simplify 4/10 to 2/5 of the work. Convert this fraction to a percentage: (2/5) * 100% = 40% of the total work.

Verification / Alternative check:
You can double check by imagining the total work as 30 units (the least common multiple of 30 and 15). Amar does 1 unit per day and Raman does 2 units per day, so together they do 3 units per day. In 4 days they will complete 3 * 4 = 12 units. Since the total is 30 units, the portion completed is 12/30 = 2/5, which again is 40%. This confirms that our earlier calculation is accurate.

Why Other Options Are Wrong:
Option 15% is too low because even Amar alone would complete more than that in 4 days.
Option 37% does not correspond to a simple fraction of the total work and does not match any correct calculation.
Option 45% is greater than the actual 40% and results from overestimating their combined rate.
Option 50% would mean half of the work is done, which would require 5 days at a rate of 1/10 per day, not 4 days.

Common Pitfalls:
A common error is to average the days (30 and 15) instead of adding their rates. Another frequent mistake is converting the fraction into a percentage incorrectly. Always remember to sum rates, not times, when people work together. Also be careful not to confuse 4/10 with 40/100; in both cases the simplified form is 2/5, which is exactly 40%.

Final Answer:
Thus, Amar and Raman together complete 40% of the total work in 4 days, so the correct option is 40%.