A, B, and C together can complete a job in 4 days. A and C together can complete the same job in 4.5 days, and B and C together can complete it in 12 days. In how many days can C alone complete the entire job?

Introduction / Context:
This problem is a classic application of time and work concepts involving three workers with different combinations of joint work durations. By using the given times for A + B + C, A + C, and B + C, we are asked to find how long C alone will take to complete the job. This kind of problem is very popular in quantitative aptitude tests for exams and job interviews.

Given Data / Assumptions:

• A, B, and C together complete the job in 4 days.
• A and C together complete the job in 4.5 days.
• B and C together complete the job in 12 days.
• We must find the time taken by C alone to complete the job.
• Total work is treated as 1 complete job.

Concept / Approach:
Let the daily work rates of A, B, and C be a, b, and c jobs per day respectively. Then, we translate the given information into equations using 1/time = rate. We set up three equations for the combined rates and then solve for c. Once we get the rate of C, the time taken by C alone is found by taking the reciprocal of that rate. This algebraic method is systematic and reliable.

Step-by-Step Solution:
Step 1: Let the total work be 1 job. Step 2: Let a, b, and c be the daily work rates of A, B, and C. Step 3: A + B + C together finish in 4 days, so a + b + c = 1/4. Step 4: A + C together finish in 4.5 days, so a + c = 1/4.5 = 2/9 job per day. Step 5: B + C together finish in 12 days, so b + c = 1/12 job per day. Step 6: From a + b + c = 1/4 and a + c = 2/9, subtract to get b = 1/4 - 2/9. Step 7: Compute b: 1/4 = 9/36 and 2/9 = 8/36 so b = (9/36 - 8/36) = 1/36 job per day. Step 8: Using b + c = 1/12 gives 1/36 + c = 1/12, so c = 1/12 - 1/36. Step 9: Compute c: 1/12 = 3/36, so c = (3/36 - 1/36) = 2/36 = 1/18 job per day. Step 10: Therefore, time taken by C alone = 1 / (1/18) = 18 days.

Verification / Alternative check:
We can verify by reconstructing all combined rates using the found value of c. If c = 1/18 and b = 1/36, then b + c = 1/36 + 1/18 = 3/36 = 1/12, which matches the given. Also, a + c = 2/9, and since a + b + c = 1/4, then a must satisfy that relation. The values are consistent and give back the same original conditions, confirming that C indeed completes the job in 18 days.

Why Other Options Are Wrong:
6 days and 12 days are much too small because they imply that C is faster than the combined groups, which contradicts the given data. 24 days and 36 days are too large and do not satisfy the algebraic equations derived from the joint work times. Only 18 days fits all three conditions simultaneously, making it the only correct choice.

Common Pitfalls:
Learners sometimes try to average the given times rather than forming equations, which leads to incorrect answers. Another common mistake is miscalculation with fractions, especially when converting mixed or decimal times. It is important to consistently use fractions and carefully simplify them step by step to avoid arithmetic errors.

Final Answer:
C alone will complete the entire job in 18 days.