Asif is twice as efficient a workman as Bashir. Together they finish a piece of work in 30 days. If they both work at their constant efficiencies, in how many days will Asif alone complete the entire work?

Introduction / Context:
This is a standard work and efficiency problem, where we compare two workers, Asif and Bashir. We are told that one is twice as efficient as the other and given the time they take together to complete the job. The question asks how long the more efficient worker, Asif, would take alone. Such problems test your understanding of efficiency ratios and their relation to time taken.

Given Data / Assumptions:

• Asif is twice as efficient as Bashir.
• Asif and Bashir together complete the work in 30 days.
• Both work at constant efficiencies.
• We need the time Asif alone would require to finish the job.

Concept / Approach:
Efficiency is directly proportional to work rate and inversely proportional to time for the same work. If Asif is twice as efficient as Bashir, then his rate is twice Bashir's rate. We set Bashir's rate as a base variable, express Asif's rate, add them to get the combined rate, and then solve for Bashir's rate using the given combined time. From this we compute Asif's rate and finally his time to complete the work alone.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Let Bashir's daily rate = b units per day. Step 3: Asif is twice as efficient, so Asif's daily rate = 2b units per day. Step 4: Together, Asif and Bashir have a combined rate = b + 2b = 3b. Step 5: They complete the whole work in 30 days, so combined rate = 1/30 units per day. Step 6: Hence, 3b = 1/30, so b = 1/90. Step 7: Asif's rate = 2b = 2 × 1/90 = 1/45 of the work per day. Step 8: Time for Asif alone = 1 / (1/45) = 45 days.

Verification / Alternative check:
If Asif alone takes 45 days, then Bashir, being half as efficient, would take 90 days. The combined rate is then 1/45 + 1/90 = 2/90 + 1/90 = 3/90 = 1/30 of the work per day, which means 30 days to complete the job together. This matches the given condition perfectly, confirming that the answer is consistent.

Why Other Options Are Wrong:
90 days would be the time for Bashir, not Asif, since he is less efficient. 60 and 75 days are larger than 45 and would imply Asif is slower than calculated from the efficiency ratio. 30 days is the time they take together, not Asif alone. Therefore, only 45 days correctly reflects the doubled efficiency relationship and the joint completion time.

Common Pitfalls:
Many students confuse the ratio of efficiencies with the ratio of times. If Asif is twice as efficient as Bashir, Asif's time should be half of Bashir's time, not double. Another mistake is trying to split the 30 days directly in the ratio 2:1 without first expressing rates. Always use variables for rates, set up equations correctly, and then derive the required time from the rate.

Final Answer:
Asif alone will complete the work in 45 days.