Raghu can complete a certain piece of work in 12 days by working 9 hours per day. Arun can complete the same work in 8 days by working 11 hours per day. If Raghu and Arun now work together on this work, each working 12 hours per day, in how many days will they complete the work?

Introduction / Context:
This is a time and work problem where workers have different daily hours and different total days taken to finish the same job. To compare their productivity fairly, we must first convert their capabilities into a common unit, such as work done per hour. Once their hourly rates are known, we can find the combined daily output when both work longer hours and then compute the total time needed.

Given Data / Assumptions:
• Raghu can finish the work in 12 days, working 9 hours each day.
• Arun can finish the work in 8 days, working 11 hours each day.
• When they work together, each works 12 hours per day.
• Work rate of each person is constant and does not change with working hours.

Concept / Approach:
We first find how many hours each person would take individually to complete the work. The reciprocal of that gives an hourly work rate. Adding their hourly rates gives the combined hourly rate when both work together. Multiplying this combined hourly rate by the new daily working hours gives the combined daily work fraction. Finally, we divide the total work (1 job) by this combined daily fraction to find the number of days required.

Step-by-Step Solution:
Let the total work be 1 unit. Raghu works 12 days × 9 hours per day = 108 hours to finish the work. So Raghu hourly rate = 1 / 108 of the work per hour. Arun works 8 days × 11 hours per day = 88 hours to finish the same work. So Arun hourly rate = 1 / 88 of the work per hour. Combined hourly rate when both work together = 1 / 108 + 1 / 88. Compute this: 1 / 108 + 1 / 88 = (88 + 108) / (108 × 88) = 196 / 9504 = 49 / 2376. They now work 12 hours per day, so combined daily work = 12 × (49 / 2376) = 588 / 2376 = 49 / 198. Thus, in one day they finish 49 / 198 of the work. Total days required = 1 ÷ (49 / 198) = 198 / 49 days.

Verification / Alternative check:
Approximate 198 / 49 as a decimal: 198 ÷ 49 is a little more than 4 (49 × 4 = 196). So they need about 4.04 days, which is reasonable because working together for long hours should be faster than either working alone for many days. This confirms that the answer magnitude is logical.

Why Other Options Are Wrong:
4 days would correspond to exactly 1 quarter of work per day, which is slightly too high for their calculated combined daily output. Other fractional options like 49 / 12 days or 9 / 2 days are either too large or too small when compared with the true combined rate, and they do not match the arithmetic derived from hourly work rates.

Common Pitfalls:
A common error is to treat 12 days and 8 days as if both workers worked the same hours per day. This ignores different daily working hours and leads to wrong rates. Another frequent mistake is to compute daily rates first and then directly add them, forgetting to adjust for the change to 12 hours per day in the combined scenario.

Final Answer:
Raghu and Arun together will complete the work in 198/49 days.