Worker A can make a cupboard in 10 days and worker B can make it in 50 days. Together with a third worker C, they finish making one cupboard in 6.25 days. In how many days can C alone make the cupboard?

Introduction / Context:
This question is a typical work and time problem involving three workers with different efficiencies. Two workers have known individual times to complete the job, and the third works with them so that together they finish in a given number of days. We need to determine the time taken by the third worker when working alone.

Given Data / Assumptions:

• Worker A alone can make the cupboard in 10 days.
• Worker B alone can make the cupboard in 50 days.
• Workers A, B and C working together complete the cupboard in 6.25 days.
• The work rate of each worker remains constant over time and there are no interruptions.

Concept / Approach:
The key approach is to convert the given times to rates of work, since rate is equal to 1 divided by the time taken for the whole job. The combined rate of A, B and C is known from the joint time. We subtract the known rates of A and B from this combined rate to obtain the rate of C. Finally, we take the reciprocal of C's rate to find the number of days C would need alone.

Step-by-Step Solution:
Step 1: Let the total work be 1 cupboard.Step 2: Rate of A = 1 / 10 cupboard per day.Step 3: Rate of B = 1 / 50 cupboard per day.Step 4: Time taken by A, B and C together = 6.25 days = 25 / 4 days.Step 5: Therefore combined rate of A, B and C = 1 / (25 / 4) = 4 / 25 cupboard per day.Step 6: Combined rate of A and B = 1 / 10 + 1 / 50 = (5 + 1) / 50 = 6 / 50 = 3 / 25 cupboard per day.Step 7: Rate of C = combined rate of A, B and C minus rate of A and B = 4 / 25 - 3 / 25 = 1 / 25 cupboard per day.Step 8: Time taken by C alone = 1 divided by C's rate = 1 / (1 / 25) = 25 days.

Verification / Alternative check:
We can verify by calculating the total work done in 6.25 days. In one day, A, B and C together do 4 / 25 of the work. In 25 / 4 days, work done = (4 / 25) * (25 / 4) = 1 cupboard. This matches exactly with the full work, confirming that the rates are consistent, and that C needing 25 days alone is correct.

Why Other Options Are Wrong:

• 20 days: This would correspond to a rate of 1 / 20, which is faster than the rate we obtained and would make the combined work finish earlier than 6.25 days.
• 16 days: This implies an even faster rate, giving a total combined rate larger than 4 / 25 and so reducing the total time below 6.25 days.
• 15 days: This is also too fast and would not match the given joint completion time.

Common Pitfalls:
A frequent mistake is to try to take a simple average of the given days, which is incorrect because rates, not times, add for simultaneous work. Another error is mishandling the fractional time 6.25 days. Converting 6.25 to the fraction 25 / 4 makes algebra clearer and reduces calculation errors.

Final Answer:
C alone can make the cupboard in 25 days.