Using the Formula for Combined Work Rate

Oct 13, 2025 by miodeya.com 98 views

The Power of Collaboration: A Math Problem Involving Victor and Dennis

In this article, we will explore a math problem involving Victor and Dennis, who are working together to weed a garden. The problem requires us to find the time it would take Dennis to complete the task alone, given that Victor can finish it in 30 minutes and they can complete it together in 12 minutes. We will use mathematical concepts and formulas to solve this problem and understand the power of collaboration.

Victor and Dennis are working together to weed a garden. If Victor worked alone, it would take him 30 minutes to complete the task. On the other hand, when they work together, they can finish the task in 12 minutes. We are asked to find the time it would take Dennis to complete the task alone, represented by the variable $x$ .

Let's denote the time it would take Victor to complete the task alone as $V$ , and the time it would take Dennis to complete the task alone as $D$ . We are given that $V = 30$ minutes and $V + D = 12$ minutes. We need to find the value of $D$ , which represents the time it would take Dennis to complete the task alone.

Using the Formula for Combined Work Rate

When two people work together, their combined work rate is the sum of their individual work rates. In this case, Victor's work rate is $\frac{1}{V}$ , and Dennis's work rate is $\frac{1}{D}$ . When they work together, their combined work rate is $\frac{1}{12}$ , since they can complete the task in 12 minutes.

We can set up the equation:

\frac{1}{V} + \frac{1}{D} = \frac{1}{12}

Substituting $V = 30$ , we get:

\frac{1}{30} + \frac{1}{D} = \frac{1}{12}

To solve for $D$ , we can start by multiplying both sides of the equation by $30D$ to eliminate the fractions:

D + 30 = \frac{30D}{12}

Simplifying the equation, we get:

D + 30 = \frac{5D}{2}

Multiplying both sides by 2 to eliminate the fraction, we get:

2D + 60 = 5D

Subtracting $2D$ from both sides, we get:

60 = 3D

Dividing both sides by 3, we get:

D = 20

In this article, we used mathematical concepts and formulas to solve a problem involving Victor and Dennis, who are working together to weed a garden. We found that if Victor worked alone, it would take him 30 minutes to complete the task, and if they worked together, they can finish the task in 12 minutes. We used the formula for combined work rate to set up an equation and solve for the time it would take Dennis to complete the task alone, represented by the variable $x$ . The solution to the problem is $D = 20$ minutes.

This problem has real-world applications in various fields, such as:

Project Management: Understanding the concept of combined work rate is crucial in project management, where teams work together to complete tasks.
Time Management: Knowing how to calculate the time it would take to complete a task alone or with others is essential in time management.
Collaboration: This problem highlights the importance of collaboration and teamwork in achieving common goals.

In conclusion, this problem requires us to think critically and use mathematical concepts to solve a real-world problem. By understanding the concept of combined work rate and using formulas to solve the problem, we can gain insights into the power of collaboration and time management.
Q&A: Understanding the Power of Collaboration and Time Management

In our previous article, we explored a math problem involving Victor and Dennis, who are working together to weed a garden. We used mathematical concepts and formulas to solve the problem and understand the power of collaboration. In this article, we will answer some frequently asked questions (FAQs) related to the problem and provide additional insights into the concept of combined work rate and time management.

A: The combined work rate is the sum of the individual work rates of two or more people working together. In this problem, Victor's work rate is $\frac{1}{V}$ , and Dennis's work rate is $\frac{1}{D}$ . When they work together, their combined work rate is $\frac{1}{12}$ , since they can complete the task in 12 minutes.

A: To calculate the time it would take for one person to complete a task alone, you need to know their individual work rate. In this problem, Victor's work rate is $\frac{1}{30}$ , which means he can complete the task in 30 minutes.

A: The formula for combined work rate is:

\frac{1}{V} + \frac{1}{D} = \frac{1}{12}

where $V$ is the time it would take Victor to complete the task alone, and $D$ is the time it would take Dennis to complete the task alone.

A: To solve for $D$ , we can start by multiplying both sides of the equation by $30D$ to eliminate the fractions:

D + 30 = \frac{30D}{12}

Simplifying the equation, we get:

D + 30 = \frac{5D}{2}

Multiplying both sides by 2 to eliminate the fraction, we get:

2D + 60 = 5D

Subtracting $2D$ from both sides, we get:

60 = 3D

Dividing both sides by 3, we get:

D = 20

A: The concept of combined work rate has real-world applications in various fields, such as:

Project Management: Understanding the concept of combined work rate is crucial in project management, where teams work together to complete tasks.
Time Management: Knowing how to calculate the time it would take to complete a task alone or with others is essential in time management.
Collaboration: This problem highlights the importance of collaboration and teamwork in achieving common goals.

A: You can apply the concept of combined work rate in your daily life by:

Estimating the time it would take to complete a task alone: Use the formula for combined work rate to estimate the time it would take to complete a task alone.
Calculating the time it would take to complete a task with others: Use the formula for combined work rate to calculate the time it would take to complete a task with others.
Improving time management: Use the concept of combined work rate to improve your time management skills and achieve your goals more efficiently.

In this article, we answered some frequently asked questions related to the problem and provided additional insights into the concept of combined work rate and time management. We hope that this article has helped you understand the power of collaboration and time management, and how to apply the concept of combined work rate in your daily life.