To find the time it would take for a group of 8 people working at half the rate of the first group to complete the same work, we can follow these steps:
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Determine the work rate of the first group.
- A group of 5 people can finish the work in 6 hours.
- Thus, the total work done in person-hours by this group is: \[ \text{Total Work} = \text{Number of People} \times \text{Time} = 5 \text{ people} \times 6 \text{ hours} = 30 \text{ person-hours}. \]
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Calculate the work rate per person for the first group.
- The work rate of the first group (5 people) is: \[ \text{Work rate of 5 people} = \frac{30 \text{ person-hours}}{6 \text{ hours}} = 5 \text{ person-hours/hour}. \]
- So, the work rate of one person is: \[ \text{Work rate of 1 person} = \frac{5 \text{ person-hours/hour}}{5} = 1 \text{ person-hour/hour}. \]
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Calculate the work rate of the second group.
- The second group consists of 8 people, but they are working at half the rate of the first group's total work rate.
- The work rate of the first group is 5 person-hours/hour. Therefore, the work rate of the second group is: \[ \text{Work rate of 8 people at half the rate} = \frac{5 \text{ person-hours/hour}}{2} = 2.5 \text{ person-hours/hour}. \]
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Calculate the time required for the second group to complete the work.
- We have already established that the total work is 30 person-hours. Now, we can use the work rate of the second group to find the time: \[ \text{Time} = \frac{\text{Total Work}}{\text{Work Rate of the Second Group}} = \frac{30 \text{ person-hours}}{2.5 \text{ person-hours/hour}} = 12 \text{ hours}. \]
Thus, it would take the group of 8 people working at half the rate of the first group 12 hours to complete the same work.
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