Understanding Work Rates
First, determine the individual work rates of the father and the son.
- Father's work rate: If the father completes the task in 8 days, his rate is $\frac{1}{8}$ of the task per day.
- Son's work rate: If the son completes the task in 7 days, his rate is $\frac{1}{7}$ of the task per day.
Calculating Work Over Alternate Days
They work on alternate days, with the father starting.
- Day 1 (Father): Completes $\frac{1}{8}$ of the task.
- Day 2 (Son): Completes $\frac{1}{7}$ of the task.
Consider a 2-day cycle (Father + Son):
Work done in 2 days = Father's work + Son's work
Work = $\frac{1}{8} + \frac{1}{7} = \frac{7 + 8}{56} = \frac{15}{56}$ of the task.
Step-by-Step Task Completion
Calculate the cumulative work done day by day:
- End of Day 1 (Father): $\frac{1}{8}$
- End of Day 2 (Son): $\frac{1}{8} + \frac{1}{7} = \frac{15}{56}$
- End of Day 3 (Father): $\frac{15}{56} + \frac{1}{8} = \frac{15 + 7}{56} = \frac{22}{56}$
- End of Day 4 (Son): $\frac{22}{56} + \frac{1}{7} = \frac{22 + 8}{56} = \frac{30}{56}$
- End of Day 5 (Father): $\frac{30}{56} + \frac{1}{8} = \frac{30 + 7}{56} = \frac{37}{56}$
- End of Day 6 (Son): $\frac{37}{56} + \frac{1}{7} = \frac{37 + 8}{56} = \frac{45}{56}$
- End of Day 7 (Father): $\frac{45}{56} + \frac{1}{8} = \frac{45 + 7}{56} = \frac{52}{56}$
Calculating Final Time
At the end of Day 7, $\frac{52}{56}$ of the task is completed.
Remaining work = $1 - \frac{52}{56} = \frac{4}{56} = \frac{1}{14}$ of the task.
Day 8 is the son's turn to work. The son's rate is $\frac{1}{7}$ task per day.
Time needed for the son to complete the remaining work:
Time = $\frac{\text{Remaining Work}}{\text{Son's Rate}} = \frac{1/14}{1/7} = \frac{1}{14} \times 7 = \frac{7}{14} = \frac{1}{2}$ day.
Total time to complete the task = Days completed by Father + Days completed by Son
Total time = 7 days + $\frac{1}{2}$ day = $7 \frac{1}{2}$ days.
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Important Questions from Time and Work
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A can do a piece of work in 16 hours, B and C can do it in 8 hours while A and C can do it 12 hours. How long will B alone take to do it?
- lf 12 men can do a work in 20 days, in how many days will the work be done by 15 men-
