Prabodh completes one-half of a job in 30 days when working alone, and Sapan completes the remaining half of the same job in 45 days. If Prabodh and Sapan now work together from the start at their usual constant rates, in how many days will they complete the entire job?

Introduction / Context:
This question concerns two workers, Prabodh and Sapan, whose partial completion times for different halves of a job are given. The problem asks how long they will take to complete the entire job if they work together from the beginning. It tests your ability to convert information about partial work into full-work rates and then combine those rates correctly.

Given Data / Assumptions:

Prabodh completes 1/2 of the job in 30 days when working alone.
Sapan completes the remaining 1/2 of the job in 45 days when working alone.
Both are assumed to work at constant individual rates.
We want the time for Prabodh and Sapan working together from the start to finish the whole job.

Concept / Approach:
From the given partial times, we first compute the daily work rates of Prabodh and Sapan for the full job. Once we know their individual rates as fractions of the whole job per day, we add those rates to get the combined daily rate. Finally, we find the total time by dividing 1 (the whole job) by the combined rate.

Step-by-Step Solution:
Let the total work be 1 unit. Prabodh completes 1/2 of the job in 30 days, so his daily rate p satisfies 30p = 1/2. Therefore, p = (1/2) / 30 = 1/60 of the job per day. Sapan completes the other 1/2 of the job in 45 days, so his daily rate s satisfies 45s = 1/2. Therefore, s = (1/2) / 45 = 1/90 of the job per day. Now, if they work together from the start, combined rate = p + s = 1/60 + 1/90. Find LCM of 60 and 90, which is 180. 1/60 = 3/180 and 1/90 = 2/180. Combined rate = 3/180 + 2/180 = 5/180 = 1/36 of the job per day. Time taken together = total work / combined rate = 1 / (1/36) = 36 days.

Verification / Alternative check:
If together they do 1/36 of the work per day, in 36 days they will complete exactly 1 full job, which matches the requirement. Also, note that Prabodh alone would take 60 days to finish the full job (since his rate is 1/60), and Sapan alone would take 90 days. It is reasonable that together they require less than 60 days, and 36 days fits within this logical range.

Why Other Options Are Wrong:
Option 18 days is too small; it would require a combined rate of 1/18, which is higher than 1/36 and contradicts the individual rates.
Option 48 days is too large; it implies a very slow combined rate and conflicts with the fact that even Prabodh alone could do the full job in 60 days.
Option 27 days suggests a higher combined rate than the actual 1/36 and does not match the computed sum of 1/60 and 1/90.
Option 30 days is also inconsistent because in 30 days they would complete 30 * (1/36) = 5/6 of the job, not the full job.

Common Pitfalls:
Students often misinterpret “half of the job in 30 days” as “full job in 30 days” and forget to scale the rate. Another pitfall is adding the times directly or averaging them instead of adding work rates. Always convert partial work times into full-job daily rates and then combine those rates carefully using a common denominator.

Final Answer:
Thus, working together from the beginning, Prabodh and Sapan can complete the entire job in 36 days, so the correct option is 36 days.