Working together, Shawn and Hector can install a tile floor in 6 hours. It would take Shawn 9 hours to do the job alone.
| | Rate (part/hour) | Time (hours) | Part of Tile Installation |
| :----- | :--------------- | :----------- | :------------------------ |
| Shawn | [tex]$\frac{1}{9}$[/tex] | 6 | [tex]$\frac{6}{9}$[/tex] |
| Hector | r | 6 | 6r |
What is the value of [tex]$r$[/tex], Hector's rate of work in part per hour?
Answer (1)
Set up the equation: 9 6 + 6 r = 1 , where r is Hector's rate of work.
Simplify the equation: 6 r = 1 − 9 6 = 9 3 = 3 1 .
Solve for r : r = 3 1 ÷ 6 = 18 1 .
Hector's rate of work is 18 1 .
Explanation
Understanding the Problem We are given that Shawn and Hector, working together, can install a tile floor in 6 hours. Shawn can do the job alone in 9 hours. We need to find Hector's rate of work per hour.
Setting up the Equation Let's denote Hector's rate of work as r (part/hour). Since Shawn's rate is 9 1 (part/hour), and they work together for 6 hours, we can set up an equation representing the fraction of work done by each of them. The sum of the fractions of work done by Shawn and Hector must equal 1 (the whole job).
The Equation The equation is: 9 6 + 6 r = 1
Solving for r Now, we solve for r . First, subtract 9 6 from both sides of the equation: 6 r = 1 − 9 6 6 r = 9 9 − 9 6 6 r = 9 3 6 r = 3 1
Finding the Value of r Next, divide both sides by 6: r = 3 1 ÷ 6 r = 3 1 × 6 1 r = 18 1
Final Answer Therefore, Hector's rate of work is 18 1 part per hour.
Examples
This type of problem is useful in real-life scenarios such as project management. For example, if you know how fast two team members work together and how fast one works alone, you can determine how fast the other team member works alone. This can help in planning and allocating tasks efficiently. Understanding combined work rates is essential for optimizing team productivity and meeting deadlines.
