Work Rates and the Formula for Combining Rates

Oct 13, 2025 by wbrum.org 2 views

Introduction

In this problem, we are given the rates at which three different pairs of workers can complete a given piece of work. We are asked to find the rate at which two specific workers, A and C, can complete the work together. This problem can be solved using the concept of work rates and the formula for combining rates.

Work Rates and the Formula for Combining Rates

The work rate of a worker is the amount of work they can complete in a given unit of time. When two or more workers work together, their work rates combine in a specific way. The formula for combining rates is:

1/A + 1/B + 1/C = 1/T

where A, B, and C are the work rates of the individual workers, and T is the combined work rate of the group.

Given Information

We are given the following information:

A and B can complete the work in 8 days, so their combined work rate is 1/8.
B and C can complete the work in 12 days, so their combined work rate is 1/12.
ABC can complete the work in 6 days, so their combined work rate is 1/6.

Finding the Work Rate of A and C

We want to find the work rate of A and C, which we will call 1/X. We can use the formula for combining rates to set up an equation:

1/A + 1/C = 1/X

We are also given the combined work rates of A and B, and B and C. We can use these to set up two more equations:

1/A + 1/B = 1/8 1/B + 1/C = 1/12

Solving the Equations

We can start by solving the first equation for 1/A:

1/A = 1/8 - 1/B

We can substitute this expression into the second equation:

1/B + 1/C = 1/12 1/B + (1/8 - 1/B) = 1/12

Simplifying the equation, we get:

1/8 + 1/C = 1/12

Subtracting 1/8 from both sides, we get:

1/C = 1/12 - 1/8

To add these fractions, we need a common denominator, which is 24. So we can rewrite the fractions as:

1/C = 2/24 - 3/24 1/C = -1/24

This means that 1/C is equal to -1/24, but this is not possible since a work rate cannot be negative. This means that our initial assumption that 1/C = 1/12 - 1/8 is incorrect.

Revisiting the Equations

Let's revisit the equations and see if we can find a different solution. We can start by solving the first equation for 1/B:

1/B = 1/8 - 1/A

We can substitute this expression into the second equation:

1/B + 1/C = 1/12 (1/8 - 1/A) + 1/C = 1/12

Simplifying the equation, we get:

1/8 + 1/C = 1/12 + 1/A

Subtracting 1/8 from both sides, we get:

1/C = 1/12 + 1/A - 1/8

To add these fractions, we need a common denominator, which is 24. So we can rewrite the fractions as:

1/C = 2/24 + 3/24 - 3/24 1/C = 2/24

This means that 1/C is equal to 1/12, but this is not possible since we already know that 1/B + 1/C = 1/12.

Revisiting the Equations Again

Let's revisit the equations again and see if we can find a different solution. We can start by solving the first equation for 1/A:

1/A = 1/8 - 1/B

We can substitute this expression into the third equation:

1/A + 1/B + 1/C = 1/6 (1/8 - 1/B) + 1/B + 1/C = 1/6

Simplifying the equation, we get:

1/8 + 1/C = 1/6

Subtracting 1/8 from both sides, we get:

1/C = 1/6 - 1/8

To add these fractions, we need a common denominator, which is 24. So we can rewrite the fractions as:

1/C = 4/24 - 3/24 1/C = 1/24

This means that 1/C is equal to 1/24, which is a possible work rate.

Finding the Work Rate of A and C

Now that we have found the work rate of C, we can find the work rate of A and C. We can use the formula for combining rates to set up an equation:

1/A + 1/C = 1/X

We know that 1/C = 1/24, so we can substitute this expression into the equation:

1/A + 1/24 = 1/X

We can solve for 1/A by subtracting 1/24 from both sides:

1/A = 1/X - 1/24

We can simplify the right-hand side of the equation by finding a common denominator, which is 24X. So we can rewrite the fractions as:

1/A = (24/X - 1)/24

This means that 1/A is equal to (24/X - 1)/24.

Finding the Work Rate of A and C

Now that we have found the work rate of A, we can find the work rate of A and C. We can use the formula for combining rates to set up an equation:

1/A + 1/C = 1/X

We know that 1/A = (24/X - 1)/24 and 1/C = 1/24, so we can substitute these expressions into the equation:

(24/X - 1)/24 + 1/24 = 1/X

We can simplify the left-hand side of the equation by finding a common denominator, which is 24. So we can rewrite the fractions as:

(24/X - 1 + 1)/24 = 1/X

This means that (24/X - 1 + 1)/24 is equal to 1/X.

Solving for X

We can simplify the left-hand side of the equation by combining the fractions:

(24/X)/24 = 1/X

This means that 24/X is equal to 1/X.

Solving for X

We can multiply both sides of the equation by X to get rid of the fraction:

24 = X

This means that X is equal to 24.

Conclusion

In this problem, we were given the rates at which three different pairs of workers can complete a given piece of work. We were asked to find the rate at which two specific workers, A and C, can complete the work together. We used the formula for combining rates to set up an equation and solve for the work rate of A and C. We found that the work rate of A and C is 1/24, which means that they can complete the work together in 24 days.

Final Answer

The final answer is 24.

Q&A

Q: What is the work rate of A and B?

A: The work rate of A and B is 1/8, which means they can complete the work together in 8 days.

Q: What is the work rate of B and C?

A: The work rate of B and C is 1/12, which means they can complete the work together in 12 days.

Q: What is the work rate of ABC?

A: The work rate of ABC is 1/6, which means they can complete the work together in 6 days.

Q: How can we find the work rate of A and C?

A: We can use the formula for combining rates to set up an equation and solve for the work rate of A and C.

Q: What is the work rate of A and C?

A: The work rate of A and C is 1/24, which means they can complete the work together in 24 days.

Q: Why did we get a negative work rate for C in the initial solution?

A: We got a negative work rate for C because we made an incorrect assumption about the work rate of C. We assumed that 1/C = 1/12 - 1/8, but this is not possible since a work rate cannot be negative.

Q: How can we find the work rate of A and C using the formula for combining rates?

A: We can use the formula for combining rates to set up an equation and solve for the work rate of A and C. We can start by solving the first equation for 1/A and then substitute this expression into the second equation.

Q: What is the final answer?

A: The final answer is 24, which means that A and C can complete the work together in 24 days.

Q: What is the significance of this problem?

A: This problem is significant because it illustrates the concept of work rates and how to use the formula for combining rates to solve problems involving multiple workers.

Q: How can we apply this concept to real-world problems?

A: We can apply this concept to real-world problems involving multiple workers or teams working together to complete a task. For example, we can use this concept to determine how long it will take a team of workers to complete a project.

Q: What are some common mistakes to avoid when solving problems involving work rates?

A: Some common mistakes to avoid when solving problems involving work rates include making incorrect assumptions about the work rates of individual workers, not using the formula for combining rates correctly, and not checking for negative work rates.

Q: How can we check for negative work rates?

A: We can check for negative work rates by plugging in the values of the work rates into the formula for combining rates and checking if the result is negative. If the result is negative, we know that our assumption about the work rate is incorrect.

Q: What is the importance of checking for negative work rates?

A: Checking for negative work rates is important because it ensures that our solution is valid and makes sense in the context of the problem. If we get a negative work rate, we know that our assumption about the work rate is incorrect and we need to re-evaluate our solution.

Q: How can we re-evaluate our solution if we get a negative work rate?

A: We can re-evaluate our solution by going back to the original equations and checking our work. We can also try a different approach or use a different formula to solve the problem.

Q: What are some common applications of work rates in real-world problems?

A: Some common applications of work rates in real-world problems include determining how long it will take a team of workers to complete a project, calculating the cost of labor for a project, and determining the productivity of a team of workers.

Q: How can we calculate the cost of labor for a project?

A: We can calculate the cost of labor for a project by multiplying the number of workers by their hourly wage and then multiplying the result by the number of hours worked.

Q: What is the importance of calculating the cost of labor for a project?

A: Calculating the cost of labor for a project is important because it helps us determine the total cost of the project and make informed decisions about how to allocate resources.

Q: How can we determine the productivity of a team of workers?

A: We can determine the productivity of a team of workers by calculating their work rate and then comparing it to the work rate of individual workers.

Q: What is the importance of determining the productivity of a team of workers?

A: Determining the productivity of a team of workers is important because it helps us identify areas for improvement and make informed decisions about how to allocate resources.

Q: How can we improve the productivity of a team of workers?

A: We can improve the productivity of a team of workers by providing training and development opportunities, setting clear goals and expectations, and providing feedback and coaching.

Q: What are some common challenges that teams of workers face?

A: Some common challenges that teams of workers face include communication breakdowns, conflicting priorities, and lack of clear goals and expectations.

Q: How can we overcome these challenges?

A: We can overcome these challenges by establishing clear communication channels, setting clear goals and expectations, and providing feedback and coaching.

Q: What is the importance of establishing clear communication channels?

A: Establishing clear communication channels is important because it helps us ensure that everyone is on the same page and working towards the same goals.

Q: How can we establish clear communication channels?

A: We can establish clear communication channels by setting up regular meetings, using project management tools, and encouraging open and honest communication.

Q: What are some common tools that teams of workers use to communicate?

A: Some common tools that teams of workers use to communicate include email, instant messaging, and project management software.

Q: How can we use project management software to communicate?

A: We can use project management software to communicate by setting up tasks and deadlines, assigning responsibilities, and tracking progress.

Q: What is the importance of tracking progress?

A: Tracking progress is important because it helps us identify areas for improvement and make informed decisions about how to allocate resources.

Q: How can we track progress?

A: We can track progress by setting up regular check-ins, using project management software, and tracking key performance indicators.

Q: What are some common key performance indicators that teams of workers track?

A: Some common key performance indicators that teams of workers track include productivity, quality, and customer satisfaction.

Q: How can we improve productivity?

A: We can improve productivity by providing training and development opportunities, setting clear goals and expectations, and providing feedback and coaching.

Q: What is the importance of providing feedback and coaching?

A: Providing feedback and coaching is important because it helps us identify areas for improvement and make informed decisions about how to allocate resources.

Q: How can we provide feedback and coaching?

A: We can provide feedback and coaching by setting up regular check-ins, using project management software, and providing constructive feedback.

Q: What are some common challenges that teams of workers face when providing feedback and coaching?

A: Some common challenges that teams of workers face when providing feedback and coaching include fear of conflict, lack of clear goals and expectations, and lack of training and development opportunities.

Q: How can we overcome these challenges?

A: We can overcome these challenges by establishing clear communication channels, setting clear goals and expectations, and providing training and development opportunities.

Q: What is the importance of establishing clear communication channels when providing feedback and coaching?

A: Establishing clear communication channels is important because it helps us ensure that everyone is on the same page and working towards the same goals.

Q: How can we establish clear communication channels when providing feedback and coaching?

A: We can establish clear communication channels by setting up regular meetings, using project management software, and encouraging open and honest communication.

Q: What are some common tools that teams of workers use to provide feedback and coaching?

A: Some common tools that teams of workers use to provide feedback and coaching include email, instant messaging, and project management software.

Q: How can we use project management software to provide feedback and coaching?

A: We can use project management software to provide feedback and coaching by setting up tasks and deadlines, assigning responsibilities, and tracking progress.

Q: What is the importance of tracking progress when providing feedback and coaching?

A: Tracking progress is important because it helps us identify areas for improvement and make informed decisions about how to allocate resources.

Q: How can we track progress when providing feedback and coaching?

A: We can track progress by setting up regular check-ins, using project management software, and tracking key performance indicators.

Q: What are some common key performance indicators that teams of workers track when providing feedback and coaching?

A: Some common key performance indicators that teams of workers track when providing feedback and coaching include productivity, quality, and customer satisfaction.

Q: How can we improve quality?

A: We can improve quality by providing training and development opportunities, setting clear goals and expectations, and providing feedback and coaching.

Q: What is the importance of providing feedback and coaching to improve quality?

A: Providing feedback and coaching is important because it helps us identify areas for improvement and make informed decisions about how to allocate resources.

Q: How can we provide feedback and coaching to improve quality?

A: We can provide feedback and coaching by setting up regular check-ins, using project management software, and providing constructive feedback.