A, B and C can complete a piece of work alone in 72 days, 24 days and 36 days respectively. If all three start the job together and continue working at their constant rates, in how many days will they finish the entire work?

Introduction / Context:
This question uses the standard time and work concept where total work is considered as one complete job and individual rates are added to get a combined rate. It checks the ability to convert days per job into jobs per day and manipulate fractions correctly.

Given Data / Assumptions:

• A alone completes the work in 72 days
• B alone completes the work in 24 days
• C alone completes the work in 36 days
• All three work together from the beginning
• Their work rates are constant and additive

Concept / Approach:
If a person finishes one job in T days, the work rate is 1/T jobs per day. When multiple people work together, their rates add. We first find the combined rate of A, B and C, then take the reciprocal to get the total time taken to finish one full job.

Step-by-Step Solution:
Rate of A = 1/72 job per day Rate of B = 1/24 job per day Rate of C = 1/36 job per day Combined rate R = 1/72 + 1/24 + 1/36 Take LCM of 72, 24 and 36 which is 72 1/72 + 1/24 + 1/36 = 1/72 + 3/72 + 2/72 = 6/72 So R = 6/72 = 1/12 jobs per day Time taken T = 1 / R = 1 / (1/12) = 12 days

Verification / Alternative check:
We can check by imagining the amount of work done in 12 days. At 1/12 job per day, in 12 days the team completes (1/12)*12 = 1 full job. This confirms that 12 days is consistent with the combined rate and completes exactly one job with no extra or missing work.

Why Other Options Are Wrong:
9 days would imply a rate of 1/9 which is greater than 1/12 and inconsistent with the calculated sum. 15 and 18 days correspond to combined rates that are too slow. 10 days also produces too large a rate. Only 12 days matches the exact sum of individual rates.

Common Pitfalls:
Typical mistakes include adding the times directly instead of adding rates, or computing the LCM incorrectly. Some students may invert the final rate incorrectly or stop at a fractional answer without simplifying. It is important to remember that time and rate are reciprocals of each other in these problems.

Final Answer:
Therefore, all three working together will finish the work in 12 days.