A, B and C can each complete a certain job working alone in 12 days, 16 days and 24 days respectively. If all three work together continuously at their usual constant rates, in how many days will they complete the entire job?

Introduction / Context:
This is a straightforward time and work question where three workers, A, B and C, have known individual times to complete a job. You must determine the time required when all three work together simultaneously. Such questions test your ability to convert individual times into work rates and then combine those rates correctly.

Given Data / Assumptions:

A alone can complete the job in 12 days.
B alone can complete the job in 16 days.
C alone can complete the job in 24 days.
All of them work at constant rates and there are no breaks or interruptions.
The total work is treated as 1 complete unit.

Concept / Approach:
The key idea is that if a person can complete a job in T days, the daily work rate of that person is 1/T of the job per day. When multiple people work together, their rates simply add. After finding the combined rate of A, B and C, we use the formula Time = Total Work / Combined Rate to find how many days they need to finish the job together.

Step-by-Step Solution:
Let total work = 1 unit. Daily rate of A = 1/12 of the job per day. Daily rate of B = 1/16 of the job per day. Daily rate of C = 1/24 of the job per day. Combined daily rate of A, B and C = 1/12 + 1/16 + 1/24. Take the LCM of 12, 16 and 24 which is 48. Convert each fraction: 1/12 = 4/48, 1/16 = 3/48, 1/24 = 2/48. Add them: 4/48 + 3/48 + 2/48 = 9/48. So combined rate = 9/48 = 3/16 of the job per day. Time taken together = total work / combined rate = 1 / (3/16) = 16/3 days.

Verification / Alternative check:
To verify, think of the job as 48 equal parts (the LCM). A completes 4 parts per day, B completes 3 parts per day and C completes 2 parts per day. Together they complete 4 + 3 + 2 = 9 parts per day. Total parts are 48, so days needed = 48 / 9 = 16/3 days, which matches our earlier calculation. This double check confirms the accuracy of the result.

Why Other Options Are Wrong:
Option 15/4 days corresponds to 3.75 days, which would require a combined rate greater than 3/16, not supported by the data.
Option 17/3 days is slightly more than the correct value and comes from incorrect fraction addition or mistaken LCM usage.
Option 19/4 days is 4.75 days, again incompatible with the calculated combined rate of 3/16 per day.
Option 4 days would require an extremely high combined rate of 1/4 per day, which exceeds the sum of the three given rates.

Common Pitfalls:
A common mistake is to average the days instead of adding the rates. Another error is to mishandle the fractions when converting them to a common denominator. Always remember that time values cannot be added directly; only work rates (reciprocals of time) combine linearly. Using the LCM of denominators makes addition of fractions easier and less error prone.

Final Answer:
Hence, A, B and C together can complete the work in 16/3 days, making 16/3 days the correct option.