9 Persons Can Do A Peice Work In 3 Days. Find The Number Of Days In Which 6 Persons Can Do It

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Introduction

In this article, we will explore the concept of work and time, and how it relates to the number of people working on a task. We will use the concept of inverse proportionality to solve a problem involving 9 persons and 6 persons working on a piece of work.

Understanding the Problem

The problem states that 9 persons can complete a piece of work in 3 days. We are asked to find the number of days it would take for 6 persons to complete the same work.

Inverse Proportionality

Inverse proportionality is a concept in mathematics where two quantities are related in such a way that as one quantity increases, the other decreases, and vice versa. In this case, the number of persons working on a task and the time taken to complete the task are inversely proportional.

Mathematical Representation

Let's represent the number of persons working on the task as 'n' and the time taken to complete the task as 't'. We can write the equation for inverse proportionality as:

n1 × t1 = n2 × t2

where n1 and t1 are the initial number of persons and time taken, and n2 and t2 are the final number of persons and time taken.

Applying the Concept to the Problem

In this case, we have:

n1 = 9 (initial number of persons) t1 = 3 (initial time taken) n2 = 6 (final number of persons) t2 = ? (final time taken)

We can plug these values into the equation and solve for t2:

9 × 3 = 6 × t2

27 = 6 × t2

Solving for t2

To solve for t2, we can divide both sides of the equation by 6:

t2 = 27 ÷ 6 t2 = 4.5

Conclusion

Therefore, it would take 6 persons approximately 4.5 days to complete the same piece of work that 9 persons can complete in 3 days.

Real-World Applications

This concept of inverse proportionality has many real-world applications, such as:

  • Manufacturing: If a factory can produce 100 units of a product in 5 days with 10 workers, how many days would it take to produce the same number of units with 15 workers?
  • Construction: If a team of 8 workers can complete a building in 12 months, how many months would it take to complete the same building with 12 workers?
  • Service Industry: If a team of 5 customer service representatives can handle 100 customer calls in 8 hours, how many hours would it take to handle the same number of calls with 10 representatives?

Example Problems

Here are a few example problems to help you practice the concept of inverse proportionality:

  • If 12 persons can complete a piece of work in 4 days, how many days would it take for 8 persons to complete the same work?
  • If 15 workers can produce 200 units of a product in 6 days, how many days would it take to produce the same number of units with 20 workers?
  • If a team of 9 customer service representatives can handle 150 customer calls in 10 hours, how many hours would it take to handle the same number of calls with 12 representatives?

Solutions

Here are the solutions to the example problems:

  • If 12 persons can complete a piece of work in 4 days, it would take 8 persons approximately 5.3 days to complete the same work.
  • If 15 workers can produce 200 units of a product in 6 days, it would take 20 workers approximately 4.2 days to produce the same number of units.
  • If a team of 9 customer service representatives can handle 150 customer calls in 10 hours, it would take 12 representatives approximately 7.5 hours to handle the same number of calls.

Conclusion

Q: What is the concept of inverse proportionality?

A: Inverse proportionality is a concept in mathematics where two quantities are related in such a way that as one quantity increases, the other decreases, and vice versa. In the context of work and time, inverse proportionality means that as the number of persons working on a task increases, the time taken to complete the task decreases, and vice versa.

Q: How is inverse proportionality used in real-world applications?

A: Inverse proportionality is used in various real-world applications, such as:

  • Manufacturing: To determine the time required to produce a certain number of units with a given number of workers.
  • Construction: To estimate the time required to complete a building with a given number of workers.
  • Service Industry: To determine the time required to handle a certain number of customer calls with a given number of customer service representatives.

Q: What is the formula for inverse proportionality?

A: The formula for inverse proportionality is:

n1 × t1 = n2 × t2

where n1 and t1 are the initial number of persons and time taken, and n2 and t2 are the final number of persons and time taken.

Q: How do I apply the concept of inverse proportionality to solve problems?

A: To apply the concept of inverse proportionality, follow these steps:

  1. Identify the initial number of persons (n1) and the initial time taken (t1).
  2. Identify the final number of persons (n2) and the final time taken (t2).
  3. Plug the values into the formula: n1 × t1 = n2 × t2.
  4. Solve for t2 (or n2, depending on the problem).

Q: What are some common mistakes to avoid when using inverse proportionality?

A: Some common mistakes to avoid when using inverse proportionality include:

  • Not considering the initial and final values: Make sure to use the correct initial and final values for the number of persons and time taken.
  • Not using the correct formula: Use the correct formula: n1 × t1 = n2 × t2.
  • Not solving for the correct variable: Make sure to solve for the correct variable (t2 or n2).

Q: Can I use inverse proportionality to solve problems involving more than two variables?

A: While inverse proportionality is typically used to solve problems involving two variables, it can be extended to solve problems involving more than two variables. However, this requires a more complex formula and a deeper understanding of the concept.

Q: What are some real-world examples of inverse proportionality in action?

A: Some real-world examples of inverse proportionality in action include:

  • Manufacturing: A factory can produce 100 units of a product in 5 days with 10 workers. If the factory wants to produce the same number of units in 3 days, how many workers will it need?
  • Construction: A team of 8 workers can complete a building in 12 months. If the team wants to complete the same building in 6 months, how many workers will it need?
  • Service Industry: A team of 5 customer service representatives can handle 100 customer calls in 8 hours. If the team wants to handle the same number of calls in 4 hours, how many representatives will it need?

Q: Can I use inverse proportionality to solve problems involving non-linear relationships?

A: While inverse proportionality is typically used to solve problems involving linear relationships, it can be used to solve problems involving non-linear relationships. However, this requires a more complex formula and a deeper understanding of the concept.

Conclusion

In conclusion, the concept of inverse proportionality is a powerful tool for solving problems involving the number of persons working on a task and the time taken to complete the task. By understanding this concept and applying it correctly, you can solve a wide range of problems in various fields, from manufacturing and construction to service industry and more.