To do a certain work, A and B work on alternate days with B beginning the work on the first day. A alone can complete the same work in 24 days. If the work gets completed in \(11 \frac{1}{3}\) days, then B alone can complete \(\rm \frac{7}{9}^{th}\) part of the original work in:
6 days
Understanding the Work and Time Problem
This problem involves two individuals, A and B, working on a task on alternate days. We are given the time A takes to complete the entire work alone, the total time taken for the work when they work alternately, and we need to find the time B takes to complete a specific fraction of the work alone.
Let's break down the information given:
- A and B work on alternate days.
- B starts the work on the first day.
- A alone can complete the work in 24 days.
- The work is completed in \(11 \frac{1}{3}\) days.
- We need to find the time B alone takes to complete \(\rm \frac{7}{9}^{th}\) part of the original work.
Calculating Individual Work Rates
If A can complete the entire work in 24 days, A's work rate per day is the reciprocal of the total time taken.
- A's 1 day work = \(\frac{1}{\text{Time taken by A alone}}\) = \(\frac{1}{24}\).
Let B's 1 day work be \(\frac{1}{b}\), where \(b\) is the number of days B takes to complete the work alone.
Determining the Number of Days A and B Worked
The work is completed in \(11 \frac{1}{3}\) days. B starts the work.
In 11 full days, since B starts, the sequence of work is B, A, B, A, ...
Days 1, 3, 5, 7, 9, 11 are working days for B.
Days 2, 4, 6, 8, 10 are working days for A.
- Number of days B worked in the first 11 days = 6 days.
- Number of days A worked in the first 11 days = 5 days.
The work continues for an additional \(\frac{1}{3}\) day. On the 12th day, it would be A's turn (since B worked on the 11th day). So, A works for the remaining \(\frac{1}{3}\) day.
- Total number of days B worked = 6 days.
- Total number of days A worked = 5 days + \(\frac{1}{3}\) day = \(5 \frac{1}{3}\) days = \(\frac{16}{3}\) days.
Calculating the Work Done by A
A's total work done is the product of A's daily work rate and the total number of days A worked.
A's total work = (A's 1 day work) \(\times\) (Total days A worked)
A's total work = \(\frac{1}{24} \times \frac{16}{3}\)
A's total work = \(\frac{16}{24 \times 3}\) = \(\frac{16}{72}\)
Simplifying the fraction:
A's total work = \(\frac{16 \div 8}{72 \div 8}\) = \(\frac{2}{9}\).
Calculating the Work Done by B
The total work done is 1 (representing the complete work). The total work is the sum of the work done by A and the work done by B.
Total work = Work done by A + Work done by B
\(1 = \frac{2}{9} + \text{Work done by B}\)
Work done by B = \(1 - \frac{2}{9}\)
Work done by B = \(\frac{9}{9} - \frac{2}{9} = \frac{7}{9}\).
So, B completed \(\frac{7}{9}\) of the total work in the 6 days B worked.
Finding B's Work Rate and Time for Full Work
We know B did \(\frac{7}{9}\) of the work in 6 days.
B's 1 day work = \(\frac{\text{Work done by B}}{\text{Days B worked}}\) = \(\frac{7/9}{6}\)
B's 1 day work = \(\frac{7}{9 \times 6} = \frac{7}{54}\).
If B does \(\frac{7}{54}\) of the work in 1 day, B can complete the full work (1) in \(\frac{1}{7/54}\) days.
Time taken by B alone for full work = \(\frac{54}{7}\) days.
Calculating Time Taken by B for \(\frac{7}{9}\) Part of Work
We need to find the time B alone takes to complete \(\frac{7}{9}\) part of the original work. We already found that B alone takes \(\frac{54}{7}\) days to complete the full work.
Time taken by B for \(\frac{7}{9}\) work = (Time taken by B for full work) \(\times\) \(\frac{7}{9}\)
Time taken by B for \(\frac{7}{9}\) work = \(\frac{54}{7} \times \frac{7}{9}\)
Time taken by B for \(\frac{7}{9}\) work = \(\frac{54}{9}\)
Time taken by B for \(\frac{7}{9}\) work = 6 days.
Summary of Calculations
| Parameter | Value |
|---|---|
| A's time for full work | 24 days |
| A's 1 day work | \(\frac{1}{24}\) |
| Total time for alternate work | \(11 \frac{1}{3}\) days |
| Days B worked | 6 days |
| Days A worked | \(5 \frac{1}{3}\) days (\(\frac{16}{3}\) days) |
| Work done by A | \(\frac{16}{3} \times \frac{1}{24} = \frac{2}{9}\) |
| Work done by B | \(1 - \frac{2}{9} = \frac{7}{9}\) |
| B's 1 day work | \(\frac{7/9}{6} = \frac{7}{54}\) |
| B's time for full work | \(\frac{1}{7/54} = \frac{54}{7}\) days |
| Time for B to complete \(\frac{7}{9}\) work | \(\frac{54}{7} \times \frac{7}{9} = 6\) days |
Conclusion
Based on the calculations, B alone can complete \(\frac{7}{9}^{th}\) part of the original work in 6 days.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula/Relation |
|---|---|---|
| Work Rate | The amount of work done by a person in one unit of time (e.g., per day). | Work Rate = \(\frac{\text{Total Work}}{\text{Time Taken}}\) |
| Time Taken | The total time required to complete the work. | Time Taken = \(\frac{\text{Total Work}}{\text{Work Rate}}\) |
| Total Work | Often considered as 1 unit for calculation purposes. If work rate is per day, Total Work = Work Rate \(\times\) Total Days. | Total Work = Sum of work done by individuals |
| Work Done in 'x' days | Work Rate \(\times\) Number of days worked. | Part of Work = Work Rate \(\times\) Days |
| Alternate Days Work | Individuals work on consecutive days, alternating turns. The total work done in a cycle (usually 2 days) is the sum of their individual work rates for one day each. | Cycle Work = Work Rate A \(\times\) 1 day + Work Rate B \(\times\) 1 day (if A and B work in a cycle) |
Additional Information: Alternate Day Problems
Problems involving work done on alternate days require careful accounting of who works on which day and for how long. Key steps usually include:
- Determine the work done in one complete cycle (e.g., Day 1 by A, Day 2 by B).
- Calculate how many full cycles are completed within the total time, considering who starts the work.
- Calculate the work done in the full cycles.
- Determine the remaining work.
- Calculate the time taken to complete the remaining work by the person whose turn it is.
- Add the time for full cycles and the time for the remaining work to get the total time, or use the total time to deduce individual contributions as done in this problem.
- Always remember that if the total time is not an integer, the last fraction of a day is worked by only one person.
- The person who starts the work will work on days 1, 3, 5, ... and the other person on days 2, 4, 6, ...
Understanding the pattern of who works on which day is crucial for solving these types of problems accurately.
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