Amar is twice as efficient a workman as Badal and therefore finishes a certain job in 15 days less time than Badal would take alone. In how many days will Amar and Badal together complete the job working at their constant individual rates?

Introduction / Context:
This problem combines relative efficiency and time difference. Amar is described as being twice as good a worker as Badal and, as a result, he finishes a job in 15 days less than Badal. We must use this information to find each worker's individual time and then compute the time if both work together on the same job.

Given Data / Assumptions:

• Amar is twice as efficient as Badal.
• Amar takes 15 days less than Badal to finish the job.
• Each worker's efficiency is constant over time.
• The entire job is of fixed size.
• We need the time when Amar and Badal work together.

Concept / Approach:
If Amar is twice as efficient as Badal, then Amar's daily work rate is twice Badal's rate. Since time is inversely proportional to rate, Amar's time is half Badal's time. We set up an equation relating their times using the given 15 day difference. Once their individual times are found, we add their rates to get the combined rate and then invert this rate to find the time taken together.

Step-by-Step Solution:
Let Badal's time be TB days and Amar's time be TA days.Amar is twice as efficient as Badal, so TA = TB / 2.It is given that Amar finishes 15 days earlier: TB - TA = 15.Substitute TA = TB / 2 into the difference equation: TB - TB / 2 = 15.This simplifies to TB / 2 = 15, so TB = 30 days.Then TA = TB / 2 = 30 / 2 = 15 days.Rates: Amar = 1 / 15 per day, Badal = 1 / 30 per day.Combined rate = 1 / 15 + 1 / 30 = 2 / 30 + 1 / 30 = 3 / 30 = 1 / 10 per day.Time taken together = 1 / (1 / 10) = 10 days.

Verification / Alternative check:
Check the time difference: Amar needs 15 days, Badal needs 30 days, so the difference is indeed 30 - 15 = 15 days as stated. Together, in 10 days, Amar completes 10 / 15 = 2 / 3 of the job and Badal completes 10 / 30 = 1 / 3 of the job. The sum is 2 / 3 + 1 / 3 = 1 complete job, confirming that 10 days is accurate.

Why Other Options Are Wrong:
Thirty days is just Badal's solo time, not their combined time. Forty days is even longer and clearly impossible when combining two positive work rates. Twenty days would mean that two workers working together are slower than Amar alone, which contradicts basic logic. Only 10 days matches the derived combined rate correctly.

Common Pitfalls:
Some learners interpret twice as good incorrectly and treat Amar's time as twice Badal's instead of half. Others overlook the 15 day difference condition or set up the equation with the wrong sign. Carefully translating verbal descriptions of relative efficiency into correct algebraic relationships is critical in such problems.

Final Answer:
Amar and Badal together will complete the job in 10 days.