Determine the Time Required for the Third Son, C, to Complete a Piece of Work

When tackling complex work problems, understanding the work rate of individuals involved is crucial. In this scenario, we explore how to determine the time required for the third son, C, to complete a piece of work when the rates of his brothers and the father are provided.

Understanding the Problem

Let's break down the information given:

Son A can complete the work in 24 days. Son B can complete the work in 36 days. The father can complete the work in 4 days. The father can do double the work than the combined effort of his three sons.

Calculations and Reasoning

Let's denote:

W as the total piece of work. c as the time in days required for C to complete the work alone.

The equation based on the rates of work can be set up as:

[frac{W}{4} 2left(frac{W}{24} frac{W}{36} frac{W}{c}right)]

This simplifies to:

[frac{1}{4} 2left(frac{1}{24} frac{1}{36} frac{1}{c}right)]

Further simplification leads to:

[frac{1}{4} frac{1}{12} frac{2}{36} frac{2}{c}]

This simplifies to:

[frac{1}{4} frac{1}{12} frac{1}{18} frac{2}{c}]

Combining the terms:

[frac{1}{4} frac{1}{12} frac{1}{18} frac{2}{c}]

This can be further simplified to:

[frac{1}{4} - frac{1}{12} - frac{1}{18} frac{2}{c}]

Calculating the right-hand side:

[frac{1}{4} - frac{1}{12} - frac{1}{18} frac{9 - 3 - 2}{36} frac{4}{36} frac{1}{9}]

Thus:

[frac{2}{c} frac{1}{9}]

Therefore:

[c 18]

Conclusion

Based on the above calculations, the third son, C, will take 18 days to complete the piece of work alone under the given conditions.

Further Insights

This problem showcases the importance of understanding work rates and combining them to solve more complex scenarios. It is a common type of question in project management and time management, where understanding how different individuals or teams contribute to a project is crucial.

By applying the principles of work rate, we can efficiently solve such problems and plan for project timelines more accurately. Understanding these concepts can be particularly useful in various fields such as construction, manufacturing, and software development.