Forty-five men or sixty boys can complete a piece of work in 20 days when working separately. If a group consists of 15 men and 20 boys working together on the same job, in how many days will they finish the entire work?

Introduction / Context:
This is a mixed time and work problem that involves two different types of workers, men and boys, with different individual efficiencies. You are given how long a group of only men or only boys would take to complete a piece of work. Using this information, you must determine how long a mixture of both men and boys takes to finish the same work.

Given Data / Assumptions:

Forty-five men alone can complete the work in 20 days.
Sixty boys alone can complete the same work in 20 days.
We assume all men have identical efficiency and all boys have identical efficiency among themselves.
We have a new group of 15 men and 20 boys working together.\nAll workers maintain constant rates, and no one joins or leaves during the job.

Concept / Approach:
We treat the total work as 1 unit and compute the work done per day by one man and one boy separately using the given information. Since 45 men finish in 20 days, we get the per-man rate. Similarly, we obtain the per-boy rate. Then we combine these rates for a group of 15 men and 20 boys and finally compute how many days this combined group needs to finish the entire job.

Step-by-Step Solution:
Let total work = 1 unit. Since 45 men finish in 20 days, total man-days = 45 * 20 = 900 man-days. So, work rate of one man per day = 1/900. Similarly, 60 boys finish in 20 days, so boy-days = 60 * 20 = 1200 boy-days. Work rate of one boy per day = 1/1200. Now for 15 men and 20 boys together: daily work rate = 15*(1/900) + 20*(1/1200). Compute 15/900 = 1/60 and 20/1200 = 1/60. So combined rate = 1/60 + 1/60 = 2/60 = 1/30 of the work per day. Time taken to complete the whole work = 1 / (1/30) = 30 days.

Verification / Alternative check:
To verify, in 30 days at a rate of 1/30 per day, the group completes exactly 1 unit of work, which matches the definition of total work. Also, notice that 15 men and 20 boys are less than half of the original 45 men and 60 boys combined, so it is reasonable that they take more than 20 days. The derived time of 30 days is therefore logically consistent with the relative group sizes and efficiencies.

Why Other Options Are Wrong:
Option 23 days is too small because the new group is significantly smaller than either original full group, so it cannot complete the work faster than 20 days.
Option 45 days overestimates the time; at the computed rate of 1/30 per day, 45 days would correspond to 1.5 units of work, which is impossible for a single job.
Option 25 days is also too low and does not match the calculated combined rate of 1/30 per day.
Option 20 days assumes that the efficiency of the new mixed group is equal to one of the original larger groups, which is not true because the group is much smaller.

Common Pitfalls:
Students sometimes treat 45 men and 60 boys as if they had equal efficiency, which leads to incorrect averaging. Another error is to average the times instead of computing individual rates. Always convert to per-person work rates first, then sum up the contributions of all members of the new group. Keeping track of units like man-days and boy-days helps to avoid confusion.

Final Answer:
Thus, 15 men and 20 boys working together will complete the work in 30 days, so the correct option is 30 days.