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Text Solution:

Let’s start by determining the rate of one machine.

First, we need to determine the rate of 6 machines:

$\Rightarrow {\text{Rate}}_{\text{6 Machines}}=\frac{Work}{Time}=\frac{\mathrm{1\; job}}{\text{3 hours}}=\frac{1}{3}\frac{job}{\text{hour}}$.

Next, we can determine the rate of 1 machine. To do so, we divide the rate of the 6 machines by 6:

$\Rightarrow {\text{Rate}}_{\text{1 Machine}}=\frac{{\text{Rate}}_{\text{6 Machines}}}{6}=\frac{\frac{1}{3}}{6}=\frac{1}{3}\times \frac{1}{6}=\frac{1}{18}\frac{job}{\text{hour}}$

Finally, we need to determine the rate of 9 machines, so we multiply the rate of one machine by 9:

$\Rightarrow {\text{Rate}}_{\text{9 Machines}}=\left({\text{Rate}}_{\text{1 Machine}}\right)\times 9=\frac{1}{18}\times 9=\frac{9}{18}=\frac{1}{2}\frac{job}{\text{hour}}$.

Keep in mind, we also could have used the proportion method:

$\begin{array}{l}\Rightarrow \frac{\text{6 machines}}{\text{combined rate of 6 machines}}=\frac{\text{9 machines}}{\text{combined rate of 9 machines}}\\ \Rightarrow \frac{6}{\frac{1}{3}}=\frac{9}{n}\\ \Rightarrow 6\times 3=\frac{9}{n}\to 18=\frac{9}{n}\\ \Rightarrow 18n=9\to n=\frac{9}{18}=\frac{1}{2}\frac{job}{hour}\end{array}$

Now that we have the rate of 9 machines, we see that we need to determine how long it will take the 9 machines to complete $\frac{1}{3}$ of the job, so we can put this into a matrix.

Since $time=\frac{work}{rate}$ we have the following:

$\Rightarrow Tim{\text{e}}_{\text{9 Machines}}=\frac{\frac{1}{3}}{\frac{1}{2}}=\frac{1}{3}\times 2=\frac{2}{3}hour=40\min utes$