To find how many hours are required for both persons working together to complete the same piece of work, we will first calculate their individual work rates, then combine those rates, and finally determine the time needed.
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Determine each person's work rate:
- The first person can complete the work in 8 days. Therefore, their work rate is: \[ \text{Rate of Person A} = \frac{1 \text{ work}}{8 \text{ days}} = \frac{1}{8} \text{ work/day} \]
- The second person can complete the work in 4 days. Therefore, their work rate is: \[ \text{Rate of Person B} = \frac{1 \text{ work}}{4 \text{ days}} = \frac{1}{4} \text{ work/day} \]
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Combine the work rates:
- To find the combined work rate when both persons work together: \[ \text{Combined Rate} = \text{Rate of Person A} + \text{Rate of Person B} = \frac{1}{8} + \frac{1}{4} \]
- Converting \(\frac{1}{4}\) to a fraction with a denominator of 8: \[ \frac{1}{4} = \frac{2}{8} \]
- Now add the rates: \[ \text{Combined Rate} = \frac{1}{8} + \frac{2}{8} = \frac{3}{8} \text{ work/day} \]
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Calculate the time to complete the work together:
- The time \(T\) required to complete 1 work at the combined rate can be calculated by taking the reciprocal of the combined rate: \[ T = \frac{1 \text{ work}}{\frac{3}{8} \text{ work/day}} = \frac{8}{3} \text{ days} \]
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Convert days to hours:
- Since they work 10 hours per day: \[ \text{Total hours} = T \times 10 \text{ hours/day} = \frac{8}{3} \text{ days} \times 10 \text{ hours/day} = \frac{80}{3} \text{ hours} \approx 26.67 \text{ hours} \]
Therefore, the total time required for both persons to complete the work together is approximately 26.67 hours or 26 hours and 40 minutes.
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