Question

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
hector	\\(r\\)	6	\\(6r\\)

what is the value of \\(r\\), hector’s rate of work in part per hour?

options:
\\(\frac{1}{9}\\)
\\(\frac{1}{3}\\)
\\(\frac{1}{18}\\)
\\(\frac{2}{3}\\)

Explanation:

Step1: Recall work rate formula

The total work done together is the sum of the work done by each person. The work done is rate times time. Let the total work be 1 (installing the tile floor is 1 job). Shawn's rate is $\frac{1}{9}$ part per hour, and together their combined rate is $\frac{1}{6}$ part per hour (since they finish in 6 hours). So, Shawn's work + Hector's work = total work. Mathematically, $\frac{1}{9} \times 6 + r \times 6 = 1$, or alternatively, the sum of their rates equals the combined rate: $\frac{1}{9} + r = \frac{1}{6}$.

Step2: Solve for \( r \)

Subtract $\frac{1}{9}$ from both sides: \( r = \frac{1}{6} - \frac{1}{9} \). Find a common denominator, which is 18. So, $\frac{1}{6} = \frac{3}{18}$ and $\frac{1}{9} = \frac{2}{18}$. Then, \( r = \frac{3}{18} - \frac{2}{18} = \frac{1}{18} \).

Answer:

$\frac{1}{18}$