If 10 people can complete a certain job in 20 days, then in how many days will 20 people, each working at twice the efficiency of one of the original people, complete the same job?

Introduction / Context:
This problem combines two important ideas in time and work questions: changing the number of workers and changing their efficiency. We know how long it takes an initial group to complete a job. Then we are asked how long a larger group, with each member being more efficient, will take to finish the same job. The question tests understanding of proportionality between work, rate and time.

Given Data / Assumptions:
- Ten people can complete the job in 20 days. - In the new scenario, there are 20 people. - Each of the 20 people works at twice the efficiency of one of the original people. - The nature of the job does not change; only manpower and efficiency change. - All workers work at constant rates.

Concept / Approach:
Total work can be expressed in person-days. Work = Number of workers * Efficiency * Time. First, we compute the total work in terms of the old people working at their original efficiency. Then, we calculate the combined rate of the new group, taking into account both more people and higher efficiency. Finally, we divide total work by this new combined rate to find the required time. Another way is to see how many times the total work rate increases in the second scenario and divide the original time by that factor.

Step-by-Step Solution:
Step 1: Let the work done by one original person in one day be 1 unit of efficiency. Step 2: Combined rate of 10 people at original efficiency = 10 * 1 = 10 units per day. Step 3: Time taken by them to complete the job = 20 days, so total work = 10 * 20 = 200 work units. Step 4: In the new scenario, there are 20 people, each with twice the original efficiency. Step 5: Combined rate of new group = 20 people * 2 units per person-day = 40 units per day. Step 6: Time required with the new group = total work / new rate = 200 / 40 = 5 days.

Verification / Alternative check:
We can compare rates directly. Original group rate is 10 units per day. New group rate is 40 units per day. The new rate is 4 times the original rate. Therefore, the time needed should be 1 / 4 of the original time: 20 days / 4 = 5 days. This quick reasoning confirms the detailed calculation and shows that 5 days is the correct answer.

Why Other Options Are Wrong:
10 days: This is only half of the original time and corresponds to a doubling of rate, not a fourfold increase.
20 days and 40 days: These completely ignore the increased manpower and improved efficiency, giving too large a duration.
2.5 days: This would require an eightfold increase in work rate, which is not supported by the data (we only have a fourfold increase).

Common Pitfalls:
Students may forget to incorporate both changes: the increase in number of people and the increase in efficiency per person. Another typical mistake is to simply double or halve the time without checking the combined factor of change in work rate. The safest approach is always to compute total work in units and then divide by the new combined rate.

Final Answer:
Twenty people with twice the efficiency will complete the job in 5 days.