To determine how many minutes of the 9-hour workday are devoted to routine work, we first need to break down the segments of the workday based on the given percentages for lunch and workshops.
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Calculate the total number of minutes in a 9-hour workday: \[ 9 \text{ hours} = 9 \times 60 \text{ minutes} = 540 \text{ minutes} \]
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Calculate the time taken for lunch: \[ \text{Lunch time} = 15% \text{ of } 540 \text{ minutes} = 0.15 \times 540 = 81 \text{ minutes} \]
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Calculate the time spent in off-site workshops: \[ \text{Workshop time} = \frac{3}{10} \text{ of } 540 \text{ minutes} = 0.3 \times 540 = 162 \text{ minutes} \]
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Now, add the time spent on lunch and workshops to find the total time accounted for: \[ \text{Total time for lunch and workshops} = 81 \text{ minutes} + 162 \text{ minutes} = 243 \text{ minutes} \]
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Finally, calculate the time devoted to routine work: \[ \text{Routine work time} = \text{Total workday} - \text{Total time for lunch and workshops} \] \[ \text{Routine work time} = 540 \text{ minutes} - 243 \text{ minutes} = 297 \text{ minutes} \]
Therefore, the number of minutes of the day that are devoted to routine work is \(\boxed{297}\).