A group of carpenters promised to finish a job in 15 days. However, 3 of the carpenters were absent from the beginning, and the remaining carpenters actually took 20 days to complete the job. Based on this information, what was the original number of carpenters in the group?

Introduction / Context:

This time and work problem addresses how changes in team size affect completion time. The carpenters initially plan to finish a job in a certain period with the full team, but due to absences, the effective workforce decreases and the completion time increases. The question asks us to back calculate the original team size, which is a typical application of the concept of man days and proportional reasoning.

Given Data / Assumptions:

• The original promise was to finish the job in 15 days with all carpenters.
• 3 carpenters are absent, so only (original number minus 3) carpenters actually work.
• With the reduced team, the job is finished in 20 days.
• We assume all carpenters work at the same constant rate and total work is one unit.

Concept / Approach:

We use the idea that total work equals number of workers multiplied by number of days multiplied by individual daily work. Since the total work is the same in both scenarios, we can equate the total man days in each case, taking into account that in the second scenario there are fewer workers and more days. This gives a simple equation in the unknown number of carpenters, which we can solve algebraically.

Step-by-Step Solution:

Verification / Alternative Check:

With 12 carpenters working for 15 days, the total man days are 12 * 15 = 180. This is the total workload in carpenter days. With 3 carpenters absent, there are 9 carpenters, and they take 20 days, giving 9 * 20 = 180 carpenter days, which matches the same total workload. Since the total man days are equal in both the planned and actual scenarios, the answer is consistent.

Why Other Options Are Wrong:

If the original number were 10, then with only 7 carpenters working, 7 * 20 would equal 140 man days, which does not match 10 * 15 = 150 man days. Similarly, 15 original carpenters would imply 15 * 15 = 225 man days, and with 12 carpenters the 20 days would produce 240 man days, again inconsistent. A value of 14 would give 14 * 15 = 210 man days versus 11 * 20 = 220 man days. Only 12 ensures that both planned and actual scenarios match in total work.

Common Pitfalls:

Some candidates incorrectly assume that days are inversely proportional to workers without setting up the full equation, leading to rounding errors. Others confuse which number to subtract the absent carpenters from or do not model the total work properly. Using clear expressions for total work and equating them is the safest route to avoid mistakes.

Final Answer:

The original number of carpenters in the group was 12.