A, B, and C can complete a job working alone in 12 days, 8 days, and 24 days respectively. If all three work together from start to finish, in how many days will they complete the entire job?

Introduction / Context:
This is a standard time and work aptitude question where three people A, B, and C can each complete the same job working alone in different numbers of days. The task is to find how long they will take to complete the job if they work together from start to finish. Such problems are very common in competitive exams and interview tests because they check the understanding of work rate addition.

Given Data / Assumptions:

• A alone finishes the job in 12 days.
• B alone finishes the job in 8 days.
• C alone finishes the job in 24 days.
• All three work together for the entire duration.
• The total work is treated as one complete job.

Concept / Approach:
The key concept is that if a person can complete a job in N days, then the work rate of that person is 1/N of the job per day. When multiple people work together, their individual work rates are added to get the combined rate. Finally, time is calculated using the relation time = total work / combined rate. Here, total work is taken as 1 job.

Step-by-Step Solution:
Step 1: Let the total work be 1 job. Step 2: Work rate of A = 1/12 job per day. Step 3: Work rate of B = 1/8 job per day. Step 4: Work rate of C = 1/24 job per day. Step 5: Combined work rate = 1/12 + 1/8 + 1/24. Step 6: Take LCM of 12, 8, and 24 which is 24. Step 7: 1/12 = 2/24, 1/8 = 3/24, and 1/24 = 1/24. Step 8: Combined rate = (2 + 3 + 1) / 24 = 6/24 = 1/4 job per day. Step 9: Time taken = total work / combined rate = 1 / (1/4) = 4 days.

Verification / Alternative check:
If the team completes 1/4 of the job per day, then in 4 days they will complete 4 * (1/4) = 1 full job. This confirms the calculation. Another way to check is to compute the fraction of work done by each person in 4 days. A does 4 * (1/12) = 1/3, B does 4 * (1/8) = 1/2, and C does 4 * (1/24) = 1/6. Adding these gives 1/3 + 1/2 + 1/6 = 1 job, which fully verifies the answer.

Why Other Options Are Wrong:
3 days is too short because at a rate of 1/4 job per day, in 3 days they would complete only 3/4 of the job. 5 days and 6 days are too long because they would give total work greater than 1 job. For example, in 5 days they would complete 5 * (1/4) = 5/4 jobs which exceeds the required work. Similarly, 8 days would be even more unrealistic for such high combined efficiency.

Common Pitfalls:
A common mistake is to average the days directly instead of working with rates. Another error is to forget to use a proper common denominator when adding fractions. Some learners also confuse time and rate, attempting to add days instead of adding fractions of work per day. Always convert days into rates first and then perform addition.

Final Answer:
The three workers A, B, and C together will complete the job in 4 days.