Worker A can complete a piece of work alone in 10 days, and worker B can complete the same work alone in 12 days. With the additional help of worker C, all three together finish the work in 4 days. In how many days can C alone complete the same work?

Introduction / Context:
This question concerns three workers A, B, and C who together complete a single job faster than any of them could alone. We know the individual times for A and B but only the combined time when all three work together. Using this information, we are asked to determine how long C alone would take to complete the entire job, which requires subtracting known rates from the combined rate.

Given Data / Assumptions:

• A alone can complete the work in 10 days.
• B alone can complete the work in 12 days.
• A, B, and C together complete the work in 4 days.
• Total work is taken as 1 unit.
• Work rates of A, B, and C are constant.

Concept / Approach:
We convert A's and B's times into daily work rates. The combined time of 4 days for all three gives us the joint rate of A, B, and C together. Subtracting A's and B's individual rates from the total combined rate yields C's rate. Finally, we take the reciprocal of C's rate to find the number of days C would need to complete the work alone.

Step-by-Step Solution:
Step 1: Assume total work is 1 unit. Step 2: A's rate = 1/10 units per day. Step 3: B's rate = 1/12 units per day. Step 4: A, B, and C together complete the work in 4 days, so combined rate of A + B + C = 1/4 units per day. Step 5: Let C's rate be c units per day. Then 1/10 + 1/12 + c = 1/4. Step 6: Compute 1/10 + 1/12. The least common multiple of 10 and 12 is 60. So, 1/10 = 6/60 and 1/12 = 5/60. Step 7: Therefore, 1/10 + 1/12 = 6/60 + 5/60 = 11/60. Step 8: Substitute into the equation: 11/60 + c = 1/4. Step 9: Express 1/4 with denominator 60: 1/4 = 15/60. Step 10: So, 11/60 + c = 15/60, which gives c = (15/60) - (11/60) = 4/60 = 1/15 units per day. Step 11: Time taken by C alone to complete the work = 1 / (1/15) = 15 days.

Verification / Alternative check:
We can verify by checking the combined rate. A works at 1/10 units per day, B at 1/12 units per day, and C at 1/15 units per day. Their total rate is 1/10 + 1/12 + 1/15. Using a common denominator of 60, we get 6/60 + 5/60 + 4/60 = 15/60 = 1/4, meaning they finish the work in 4 days, which matches the given condition.

Why Other Options Are Wrong:

• 12 days: This would correspond to a faster rate of 1/12 and would not satisfy the combined rate equation.
• 18 days: This gives a slower rate of 1/18 and leads to a combined rate that is less than 1/4, so they would take more than 4 days.
• 24 days: Even slower, with a rate of 1/24, clearly contradicting the given 4-day completion time.

Common Pitfalls:
Some students incorrectly average the times of A and B instead of using work rates, or they forget to subtract both A and B's rates from the total combined rate, leading to an incorrect rate for C. Errors in fraction arithmetic, particularly when finding common denominators, can also lead to wrong answers. Systematic step-by-step calculation helps avoid these pitfalls.

Final Answer:
Worker C alone can complete the work in 15 days.