A and B alone can complete a piece of work in 7 and 14 days, respectively. A and B work together for 2 days, after which C joins them. Now, the work is completed in one day. In how many days can C alone do that work?

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Keywords: Work, days, individual rates, combined work, simultaneous work.

Required Approach: Factual and Analytical. This question requires a mathematical approach to solve for an unknown variable (the number of days C takes to complete the work alone).

Points to Remember:

  • Individual work rates are inversely proportional to the number of days taken to complete the work.
  • The combined work rate of multiple individuals working together is the sum of their individual work rates.
  • The total work can be represented as a unit (1).

Introduction:

This problem involves calculating the individual work rate of person C, given the work rates of A and B and the time taken to complete the work when all three work together. We will use the concept of work rate, which is the fraction of work completed per day. A’s work rate is 1/7 (completing 1/7th of the work per day), and B’s work rate is 1/14 (completing 1/14th of the work per day).

Body:

1. Calculating the combined work of A and B:

A and B work together for 2 days. Their combined work rate is (1/7) + (1/14) = 3/14. In 2 days, they complete (3/14) * 2 = 3/7 of the work.

2. Remaining work after A and B’s contribution:

After 2 days, the remaining work is 1 – (3/7) = 4/7.

3. Combined work of A, B, and C:

A, B, and C complete the remaining 4/7 of the work in 1 day. Let C’s work rate be 1/x, where x is the number of days C takes to complete the work alone. Therefore, their combined work rate is (1/7) + (1/14) + (1/x) = 3/14 + 1/x.

4. Equation and Solution:

Since they complete 4/7 of the work in 1 day, we can set up the equation:

(3/14) + (1/x) = 4/7

Solving for x:

1/x = 4/7 – 3/14 = 5/14

x = 14/5 = 2.8 days

5. Verification:

In 2 days, A and B complete 3/7 of the work. In the next day, A, B, and C together complete 4/7 of the work. C’s contribution in that one day is (4/7) – (3/14) = 5/14. This aligns with the calculated work rate of C (1/2.8 = 5/14).

Conclusion:

C alone can complete the work in 2.8 days (or 14/5 days). This problem demonstrates the application of basic work-rate calculations to solve for an unknown variable. The solution highlights the importance of breaking down complex problems into smaller, manageable steps. Understanding individual work rates and their combined effect is crucial in efficiently managing projects and allocating resources. This approach can be applied to various scenarios involving teamwork and project management, promoting better resource allocation and timely project completion. A holistic approach to project management, considering individual strengths and weaknesses, is essential for optimal efficiency.

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