X and Y can complete a work in 9 days and 36 days, respectively. X begins to do the work and they work alternately one at a time for one day each. The whole work will be complete in:
Solving Work and Time Problems with Alternating Days
This problem involves two individuals, X and Y, working on a task alternately. To solve this type of work and time problem, we first need to determine their individual work rates and then calculate the amount of work done in one complete cycle of alternation.
Understanding Individual Work Rates
The work rate of a person is the amount of work they can complete in one day. It is the reciprocal of the number of days they take to complete the whole work.
- X can complete the work in 9 days. So, X's work rate is $\frac{1}{9}$ of the work per day.
- Y can complete the work in 36 days. So, Y's work rate is $\frac{1}{36}$ of the work per day.
Calculating Work Done in One Alternating Cycle
They work alternately, with X starting. A full cycle consists of X working for one day and then Y working for one day. This cycle spans two days.
- On Day 1, X works and completes $\frac{1}{9}$ of the work.
- On Day 2, Y works and completes $\frac{1}{36}$ of the work.
Work done in one cycle (2 days) = Work done by X on Day 1 + Work done by Y on Day 2
Work done in one cycle = $\frac{1}{9} + \frac{1}{36}$
To add these fractions, we find a common denominator, which is 36.
$\frac{1}{9} = \frac{1 \times 4}{9 \times 4} = \frac{4}{36}$
Work done in one cycle = $\frac{4}{36} + \frac{1}{36} = \frac{4+1}{36} = \frac{5}{36}$ of the work.
So, $\frac{5}{36}$ of the work is completed in every 2-day cycle.
Determining the Number of Full Cycles
We need to find out how many full cycles are required to complete as much work as possible without exceeding the total work (which is 1). The work done per cycle is $\frac{5}{36}$.
Let $n$ be the number of cycles. The work done after $n$ cycles is $n \times \frac{5}{36}$. We want to find the largest integer $n$ such that $n \times \frac{5}{36} < 1$.
Multiply both sides by 36:
$5n < 36$
$n < \frac{36}{5} = 7.2$
The largest integer $n$ less than 7.2 is 7. So, 7 full cycles will be completed.
Time taken for 7 cycles = 7 cycles $\times$ 2 days/cycle = 14 days.
Work completed after 7 cycles = 7 $\times \frac{5}{36} = \frac{35}{36}$ of the work.
Calculating Remaining Work
After 7 cycles, the remaining work is:
Remaining work = Total work - Work done after 7 cycles
Remaining work = $1 - \frac{35}{36} = \frac{36}{36} - \frac{35}{36} = \frac{1}{36}$ of the work.
Completing the Remaining Work
After 7 full cycles (14 days), X just finished their turn as the last person in the 7th cycle (since X starts the first cycle). Thus, it is X's turn to work on the 15th day.
X's work rate is $\frac{1}{9}$ of the work per day.
The remaining work is $\frac{1}{36}$.
Time taken by X to complete the remaining work = $\frac{\text{Remaining Work}}{\text{X's Work Rate}}$
Time taken by X = $\frac{1/36}{1/9} = \frac{1}{36} \times \frac{9}{1} = \frac{9}{36} = \frac{1}{4}$ day.
Total Time Taken
The total time taken to complete the work is the time for 7 cycles plus the time taken by X to complete the remaining work.
Total time = Time for 7 cycles + Time for remaining work
Total time = 14 days + $\frac{1}{4}$ day = $14 \frac{1}{4}$ days.
Thus, the whole work will be complete in $14 \frac{1}{4}$ days.
| Day | Worker | Work Done on Day | Total Work Done |
|---|---|---|---|
| 1 | X | $\frac{1}{9}$ | $\frac{1}{9}$ |
| 2 | Y | $\frac{1}{36}$ | $\frac{1}{9} + \frac{1}{36} = \frac{5}{36}$ (End of Cycle 1) |
| 3 | X | $\frac{1}{9}$ | $\frac{5}{36} + \frac{1}{9} = \frac{5}{36} + \frac{4}{36} = \frac{9}{36}$ |
| 4 | Y | $\frac{1}{36}$ | $\frac{9}{36} + \frac{1}{36} = \frac{10}{36}$ (End of Cycle 2) |
| ... | ... | ... | ... |
| 13 | X | $\frac{1}{9}$ | (Work after 6 cycles) + $\frac{1}{9} = \frac{6 \times 5}{36} + \frac{1}{9} = \frac{30}{36} + \frac{4}{36} = \frac{34}{36}$ |
| 14 | Y | $\frac{1}{36}$ | $\frac{34}{36} + \frac{1}{36} = \frac{35}{36}$ (End of Cycle 7) |
| 15 | X | Work needed: $\frac{1}{36}$ | $\frac{35}{36} + \frac{1}{36} = \frac{36}{36} = 1$ (Work complete) |
| Time taken by X on Day 15 | $\frac{\text{Remaining Work}}{\text{X's Rate}} = \frac{1/36}{1/9} = \frac{1}{4}$ day | ||
| Total Time | 14 days (for 7 cycles) + $\frac{1}{4}$ day = $14 \frac{1}{4}$ days |
Revision Table: Key Concepts in Work and Time
| Concept | Explanation | Formula |
|---|---|---|
| Work Rate | Amount of work done by a person in one unit of time (e.g., per day). | Work Rate = $\frac{1}{\text{Time taken to complete work}}$ |
| Total Work | Usually considered as 1 unit or a common multiple of individual times. | |
| Work Done | Work Done = Work Rate $\times$ Time | |
| Alternating Work | People work on consecutive days or time units. Calculate work done in one cycle of turns. | Work per cycle = Sum of work rates for one turn of each person in the cycle |
Additional Information: Alternate Work Problems
When solving alternating work problems, it is crucial to identify:
- The individual work rates.
- The composition of one full work cycle (who works in what order and for how long in one turn).
- The total work done in one cycle.
- The number of full cycles needed to complete most of the work without exceeding the total.
- The remaining work after the full cycles.
- Who is scheduled to work after the full cycles are complete.
- The time taken by that person to finish the remaining work.
- The total time by adding the time for full cycles and the time for remaining work.
Always pay attention to who starts the work, as this determines whose turn it is after a certain number of days or cycles.
Similar Questions
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Important Questions from Work Efficiency
A and B working together can complete a job in 30 days. The ratio of their efficiencies is 3 : 2. In how many days can the faster person complete the job?
A takes 15 days to complete \(\frac{5}{7} \) of a work. With the help of B, they finish the whole work in 12 days. In how many days, B alone will complete the same work
A alone can complete a work in 14 days and B alone can complete the same work in 21 days. A and B start the work together but A leaves the work after 4 days of the starting of work. In how many days B will complete the remaining work?
For completing a certain work, A is 50% less efficient than B and B is 50% more efficient than C. Working together A, B and C can complete the work in 48 days. A alone can complete the same work in:
30 persons can do a piece of work in 24 days. How many more persons are required to complete the work in 20 days?
