Two persons, Shyam and Rahim, can complete a certain job together in 32 days. Rahim alone can finish the same job in 48 days. They start working together and after 8 days Rahim is replaced by a third person, Ram, whose efficiency is exactly half that of Rahim. For how many days will Shyam and Ram together need to work to complete the remaining part of the job?

Introduction / Context:
This question involves three workers with different efficiencies, where one person is later replaced by someone less efficient. We know how long Shyam and Rahim together take to finish a job, and how long Rahim alone needs. After some days, Rahim is replaced by Ram, whose efficiency is half that of Rahim. We must find how many days Shyam and Ram will require to complete the remaining work.

Given Data / Assumptions:

• Shyam and Rahim together can complete the job in 32 days.
• Rahim alone can complete the job in 48 days.
• They work together for the first 8 days.
• After 8 days, Rahim is replaced by Ram, whose efficiency is half that of Rahim.
• All workers have constant work rates and the job size is fixed at one unit.

Concept / Approach:
We first compute the joint rate of Shyam and Rahim from the given completion time, and then isolate Shyam's rate using Rahim's solo time. Once we know how much work is done in the first 8 days, we compute the remaining fraction. Then we determine Ram's rate as half of Rahim's rate, add it to Shyam's rate to get the new combined rate, and finally find the time required to finish the remaining work.

Step-by-Step Solution:
Combined rate of Shyam and Rahim = 1 / 32 job per day. Rahim alone completes the job in 48 days, so Rahim's rate = 1 / 48 job per day. Shyam's rate = (1 / 32) - (1 / 48). Compute Shyam's rate: with common denominator 96, 1 / 32 = 3 / 96 and 1 / 48 = 2 / 96, so Shyam's rate = 1 / 96 job per day. In the first 8 days, Shyam and Rahim together do 8 * (1 / 32) = 8 / 32 = 1 / 4 of the work. Remaining work = 1 - 1 / 4 = 3 / 4 of the job. Ram's efficiency is half of Rahim's, so Ram's rate = (1 / 2) * (1 / 48) = 1 / 96 job per day. Now Shyam and Ram together have rate = 1 / 96 + 1 / 96 = 1 / 48 job per day. Time to complete remaining 3 / 4 job = (3 / 4) / (1 / 48) = (3 / 4) * 48 = 36 days. Thus, Shyam and Ram together need 36 days to complete the remaining work.

Verification / Alternative check:
We can calculate the total equivalent work done in men days as a consistency check. The initial 8 days give 8 / 32 = 0.25 of the work. The remaining 0.75 divided by rate 1 / 48 gives 36 days. Total calendar days spent on the project is 8 + 36 = 44 days, which is sensible because the replacement worker is less efficient than Rahim, so the job takes longer than 32 days overall.

Why Other Options Are Wrong:
24 days and 28 days would require a much higher combined rate than 1 / 48 and are inconsistent with Ram being slower than Rahim. 32 days would imply the replacement does not slow the work at all, which contradicts the lower efficiency of Ram. 40 days is close but does not match the exact ratio derived from the rates 1 / 96 and 1 / 96.

Common Pitfalls:
Learners often confuse time and efficiency when the phrase "half as efficient" appears, mistakenly halving the time instead of the rate. Another common error is miscomputing Shyam's rate by subtracting times instead of rates. Always work in terms of rates (jobs per day), not times, and be careful to apply the half efficiency relation to the rate of Rahim, not to his time.

Final Answer:
The remaining work will be completed by Shyam and Ram together in 36 days.