A, B, and C can complete a piece of work in 24 days, 6 days, and 12 days respectively. Working all together at their usual rates, in how many days will they complete the same work?

Introduction / Context:
This is a straightforward time and work problem involving three workers A, B, and C. Each worker has a known individual time to complete the whole job alone. The problem asks how long all three will take to complete the same job when working together at their usual constant rates. This is a classic example of adding work rates to find a combined time.

Given Data / Assumptions:
A alone can complete the work in 24 days. B alone can complete the work in 6 days. C alone can complete the work in 12 days. All three are assumed to work at constant rates, and the total work is considered as one complete unit. They work together for the entire duration until the job is done.

Concept / Approach:
The concept is that each person’s efficiency can be expressed as a rate equal to 1 divided by the number of days they take to complete the job. The combined rate of multiple workers is the sum of their individual rates. Once you know the combined rate, you can find the total time required by taking the reciprocal of that combined rate. Using fractional arithmetic allows an exact answer in terms of a simplified fraction of a day.

Step-by-Step Solution:
Step 1: Let the total work be 1 unit. Step 2: The rate of A is 1 / 24 work per day. Step 3: The rate of B is 1 / 6 work per day. Step 4: The rate of C is 1 / 12 work per day. Step 5: The combined rate when A, B, and C work together is the sum of these rates: 1 / 24 + 1 / 6 + 1 / 12. Step 6: Convert to a common denominator of 24: 1 / 24 remains 1 / 24; 1 / 6 = 4 / 24; 1 / 12 = 2 / 24. Step 7: Add the fractions: 1 / 24 + 4 / 24 + 2 / 24 = 7 / 24 work per day. Step 8: Time taken together is the reciprocal of the combined rate: time = 1 / (7 / 24) = 24 / 7 days.

Verification / Alternative check:
We can check by estimating decimals. 24 / 7 is approximately 3.4286 days. In one day, A does about 1 / 24 ≈ 0.0417 of the work, B does 1 / 6 ≈ 0.1667, and C does 1 / 12 ≈ 0.0833. Together they do about 0.2917 of the work per day, which is 7 / 24 ≈ 0.2917. Over 3.4286 days, the total work done is 0.2917 * 3.4286 ≈ 1 job, confirming the calculation is consistent.

Why Other Options Are Wrong:
1 / 24 and 7 / 24 days are not realistic total times because they are much less than even B’s individual time, meaning the job would be finished faster than the fastest worker alone in an implausible way. 4 days is longer than the exact time 24 / 7 days and corresponds to a combined rate too small compared to the given rates. 3 days implies a higher combined rate than 7 / 24, which is not supported by the individual efficiencies. The only correct value is 24 / 7 days.

Common Pitfalls:
Learners sometimes mistakenly average the times 24, 6, and 12 directly instead of working with rates. Others may forget to take the reciprocal at the end and might report the combined rate instead of the combined time. Always remember that time calculations in such problems rely on adding rates, not adding times.

Final Answer:
Working together, A, B, and C will complete the work in 24/7 days.