- Determine individual work rates: Shawn's rate is \(R_S = \frac{1}{9}\) and Hector's rate is \(r\).
- Set up the combined rate equation: \(6 \times (\frac{1}{9} + r) = 1\).
- Isolate \(r\) by solving the algebraic equation: \(\frac{1}{9} + r = \frac{1}{6}\), which leads to \(r = \frac{1}{6} - \frac{1}{9}\).
- Calculate the final rate: \(r = \frac{3}{18} - \frac{2}{18} = \frac{1}{18}\), matching the first option.
- Final Answer: \(\boxed{\frac{1}{18}}\).
Explanation
- Analyzing the Rates of Work
To determine Hector's rate of work, we start by analyzing the information provided. The work done is measured in "parts" of the job completed. If Shawn can install the tile floor in 9 hours, his rate of work is the reciprocal of his time:
\[R_S = \frac{1}{9} \text{ part per hour}.\]
Let Hector's rate of work be \(r\) parts per hour. When Shawn and Hector work together, their rates are additive. Thus, their combined rate of work is:
\[R_{combined} = R_S + r = \frac{1}{9} + r.\]
Given that they complete the job in 6 hours, we can relate their combined rate to the total work (1 job) using the formula: \(\text{Rate} \times \text{Time} = \text{Work}\). Substituting the known values, we have:
\[6 \times \left(\frac{1}{9} + r\right) = 1.\]
- Solving for Hector's Rate of Work
To find Hector's rate \(r\), we solve the equation:
\[6 \times \left(\frac{1}{9} + r\right) = 1.\]
First, divide both sides of the equation by 6 to isolate the combined rate:
\[\frac{1}{9} + r = \frac{1}{6}.\]
Next, subtract \(\frac{1}{9}\) from both sides:
\[r = \frac{1}{6} - \frac{1}{9}.\]
To subtract these fractions, we find a common denominator. The least common multiple of 6 and 9 is 18. Rewriting the fractions:
\[r = \frac{3}{18} - \frac{2}{18} = \frac{1}{18}.\]
Thus, Hector's rate of work is \(\frac{1}{18}\) part per hour.
- Conclusion and Final Result
Comparing our result \(r = \frac{1}{18}\) to the given options:
- \(\frac{1}{18}\)
- \(\frac{1}{9}\)
- \(\frac{1}{3}\)
- \(\frac{2}{3}\)
Our calculated rate of \(\frac{1}{18}\) matches the first option. The final answer is \(\boxed{\frac{1}{18}}\).
Examples
In many real-life scenarios, such as project management or home improvement, it is common to calculate how long tasks take when multiple people collaborate. Just like in this problem where Shawn and Hector combined their efforts to finish a flooring installation faster, project managers calculate combined work rates to estimate completion times for complex projects. Understanding how to add individual productivity rates to find a collective efficiency is a fundamental skill in ensuring projects are finished on schedule.