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.
Similar Questions
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Important Questions from Time and Work
A and B can do a piece of work in 18 days. B and C together can do it in 30 days. If A is twice as good a workman as C, find in how many days B alone can do the work?
Two men and 7 women can complete a work in 28 days, whereas 6 men and 16 women can do the same work in 11 days. In how many days will 5 men and 4 women, working together, complete the same work?
A man and a woman, working together can do a work in 66 days. The ratio of their working efficiencies is 3 ∶ 2. In how many days 6 men and 2 women together can do the same work?
Each one of five men independently can complete a work in 20 days. The work is started by one person. Next day one more person joins and every next day one more person joins. From the fifth day, five persons continued working as a team. In how many days, will the work be completed?
Numan does half the work as Gagan in 4/5 of the time. If together they take 16 days to complete a piece of work, then how long will it take Gagan to complete the work?
