Question
A group of 5 people can do a piece of work in 6 hours.calculate the time a group of 8 people working at half rate of first group would take to complete the same work
Answers
GPT-4o mini
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:
1. **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:
\[
\text{Total Work} = \text{Number of People} \times \text{Time} = 5 \text{ people} \times 6 \text{ hours} = 30 \text{ person-hours}.
\]
2. **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:
\[
\text{Work rate of 1 person} = \frac{5 \text{ person-hours/hour}}{5} = 1 \text{ person-hour/hour}.
\]
3. **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:
\[
\text{Work rate of 8 people at half the rate} = \frac{5 \text{ person-hours/hour}}{2} = 2.5 \text{ person-hours/hour}.
\]
4. **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.
1. **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:
\[
\text{Total Work} = \text{Number of People} \times \text{Time} = 5 \text{ people} \times 6 \text{ hours} = 30 \text{ person-hours}.
\]
2. **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:
\[
\text{Work rate of 1 person} = \frac{5 \text{ person-hours/hour}}{5} = 1 \text{ person-hour/hour}.
\]
3. **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:
\[
\text{Work rate of 8 people at half the rate} = \frac{5 \text{ person-hours/hour}}{2} = 2.5 \text{ person-hours/hour}.
\]
4. **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.