Paulo Hired Painters To Paint His House. 4 Painters Took 9 Hours To Paint A Wall. If Paulo Hires 2 More Painters, How Long Will It Take To Paint The Next Wall?

Jun 19, 2025 by ADMIN 160 views

Calculating Painting Time with Variable Painters

In this article, we'll explore a practical problem involving the time it takes to paint walls, considering the number of painters involved. This is a classic example of an inverse proportion problem, where increasing the number of workers decreases the time required to complete a task, assuming all workers contribute equally. We'll delve into the steps to solve this problem and provide a detailed explanation to help you understand the underlying concepts.

Problem Statement

Paulo hired a service to paint the walls of his house. One of the walls was painted by 4 painters, and it took them 9 hours to complete the job. Due to the time taken, Paulo decided to hire 2 more painters for the next wall. Assuming the walls are of the same size and the painters work at the same rate, how long will it take the 6 painters to paint the second wall?

Breaking Down the Problem

To solve this problem, we need to understand the relationship between the number of painters and the time taken to paint a wall. This is an inverse relationship; meaning, if you increase the number of painters, the time taken to paint the wall will decrease, and vice versa. This is because the workload is being distributed among more people.

Setting Up the Proportion

We can set up an inverse proportion to represent this relationship. Let's denote:

${ n_1 }$ = Number of painters for the first wall = 4
${ t_1 }$ = Time taken to paint the first wall = 9 hours
${ n_2 }$ = Number of painters for the second wall = 4 + 2 = 6
${ t_2 }$ = Time taken to paint the second wall (what we want to find)

The inverse proportion can be expressed as:

${ n_1 \times t_1 = n_2 \times t_2 }$

This equation states that the product of the number of painters and the time taken remains constant. Now, we can plug in the values we know:

${ 4 \times 9 = 6 \times t_2 }$

Solving for ${ t_2 }$

Now, we need to solve for ${ t_2 }$ . First, calculate the product on the left side:

${ 36 = 6 \times t_2 }$

Next, divide both sides by 6 to isolate ${ t_2 }$ :

${ t_2 = \frac{36}{6} }$

${ t_2 = 6 }$

So, it will take 6 painters 6 hours to paint the second wall.

Detailed Explanation and Concepts

Understanding Inverse Proportion

Inverse proportion is a fundamental concept in mathematics and real-world problem-solving. Two quantities are said to be inversely proportional if an increase in one quantity results in a decrease in the other quantity, and vice versa, in such a way that their product remains constant. In simpler terms, if one variable doubles, the other variable halves, and so on.

Real-World Examples

Speed and Time: If you increase the speed of a car, the time it takes to travel a certain distance decreases. This is a classic example of inverse proportion. For instance, if you double your speed, you'll halve the time it takes to reach your destination.
Workers and Time: As seen in our painting problem, the number of workers and the time taken to complete a task are inversely proportional. More workers mean less time, assuming each worker contributes equally.
Pressure and Volume (Boyle's Law): In physics, Boyle's Law states that the pressure and volume of a gas are inversely proportional when the temperature and amount of gas are kept constant. If you decrease the volume, the pressure increases, and vice versa.

Mathematical Representation

If two quantities, ${ x }$ and ${ y }$ , are inversely proportional, their relationship can be expressed as:

${ x \times y = k }$

where ${ k }$ is a constant. This constant represents the product of the two quantities and remains the same regardless of their individual values. In our painting problem, ${ x }$ represents the number of painters, ${ y }$ represents the time taken, and ${ k }$ represents the total work done (which remains constant).

Applying the Concept to Our Problem

In our scenario, the total work required to paint a wall remains constant. This work can be quantified as the product of the number of painters and the time they spend painting. For the first wall, this total work can be calculated as:

${ \text{Total Work} = 4 \text{ painters} \times 9 \text{ hours} = 36 \text{ painter-hours} }$

This means it takes 36 "painter-hours" to complete the task. For the second wall, the total work remains the same (36 painter-hours), but the number of painters has increased to 6. To find the time taken, we use the same principle:

${ 6 \text{ painters} \times t_2 \text{ hours} = 36 \text{ painter-hours} }$

Solving for ${ t_2 }$ gives us:

${ t_2 = \frac{36 \text{ painter-hours}}{6 \text{ painters}} = 6 \text{ hours} }$

This confirms our earlier calculation that it will take 6 hours for 6 painters to paint the second wall.

Why This Works: The Underlying Logic

The core idea behind inverse proportion is that the total amount of work to be done remains constant. In this case, the total work is the painting of the wall. When more painters are added, each painter contributes a smaller fraction of the total work, leading to a reduction in the time required to complete the task. Conversely, if fewer painters are available, each painter must contribute a larger fraction of the total work, increasing the time required.

Step-by-Step Solution

To recap, let's go through the steps to solve this type of problem:

Identify the Quantities: Determine the quantities that are inversely proportional. In this case, they are the number of painters and the time taken.
Set Up the Proportion: Use the formula ${ n_1 \times t_1 = n_2 \times t_2 }$ to represent the inverse proportion, where ${ n }$ is the number of painters and ${ t }$ is the time taken.
Plug in the Known Values: Substitute the given values into the formula. In our example, we had ${ n_1 = 4 }$ , ${ t_1 = 9 }$ , and ${ n_2 = 6 }$ .
Solve for the Unknown: Rearrange the equation to solve for the unknown quantity, ${ t_2 }$ in this case.
Check Your Answer: Ensure your answer makes sense in the context of the problem. If the number of painters increased, the time taken should decrease.

Practical Implications and Applications

Understanding inverse proportion is not just useful for solving mathematical problems; it also has practical applications in various real-world scenarios.

Project Management

In project management, resource allocation is a critical aspect. Managers often need to estimate the time required to complete a task based on the number of resources (e.g., workers, machines) available. Understanding inverse proportion helps in making accurate estimates and adjusting resources to meet deadlines.

For example, if a software development project is falling behind schedule, adding more developers can potentially reduce the time taken to complete the project. However, this needs to be balanced against factors like communication overhead and the complexity of the tasks. Similarly, in construction projects, the number of workers can be adjusted to speed up or slow down progress based on project requirements.

Resource Allocation in General

Inverse proportion principles apply to various resource allocation scenarios. In manufacturing, the number of machines and the time taken to produce a certain quantity of goods are inversely proportional. In agriculture, the number of workers and the time taken to harvest a field follow a similar relationship.

Everyday Life

Even in everyday life, we often encounter situations where inverse proportion applies. For example, if you are planning a road trip, the speed at which you drive and the time it takes to reach your destination are inversely proportional. If you increase your speed, you will reduce the travel time, assuming traffic and road conditions remain constant.

Potential Pitfalls and Considerations

While inverse proportion is a powerful concept, it's important to consider its limitations and potential pitfalls.

Assuming Equal Contribution

The inverse proportion model assumes that all workers (or resources) contribute equally to the task. In reality, this may not always be the case. Some workers may be more efficient or skilled than others, and this can affect the overall time taken to complete the task. Similarly, machines may have different capabilities or efficiencies.

Diminishing Returns

Adding more resources does not always linearly decrease the time taken. There can be a point of diminishing returns, where adding more workers or machines has a smaller and smaller impact on reducing the time. This is because of factors like coordination overhead, communication challenges, and the need for additional resources to manage the increased workforce.

Task Complexity

The complexity of the task can also affect the relationship between resources and time. For simple, repetitive tasks, the inverse proportion model may be a good approximation. However, for complex tasks that require coordination, communication, and problem-solving, adding more resources may not always lead to a proportional reduction in time.

Conclusion

In summary, understanding inverse proportion is crucial for solving problems involving the relationship between resources and time. In the case of Paulo's painting project, by doubling the number of painters from 4 to 6, the time taken to paint the second wall decreased from 9 hours to 6 hours. This illustrates the fundamental principle that when the number of workers increases, the time required to complete a task decreases, assuming the total work remains constant.

By applying the formula ${ n_1 \times t_1 = n_2 \times t_2 }$ , we can accurately calculate the time required for various scenarios. This concept is not only applicable to mathematical problems but also has significant practical implications in project management, resource allocation, and everyday life. However, it's important to consider the limitations of the model and factors like equal contribution, diminishing returns, and task complexity to make realistic estimations.

By mastering the concept of inverse proportion, you can solve a wide range of problems efficiently and effectively. Whether you are managing a project, allocating resources, or simply planning your day, understanding these principles will help you make informed decisions and achieve your goals more effectively.

Final Answer

It will take 6 painters 6 hours to paint the second wall.