To do a certain work, A and B work on alternate days, with B beginning the work on the first day. A can finish the work alone in 48 days. If the work gets completed in \(11\frac{1}{3}\) days, then B alone can finish 4 times the same work in:
27 days
Understanding the Work and Time Problem
This problem involves two individuals, A and B, working on alternate days to complete a task. We are given the time A takes to finish the work alone and the total time taken when A and B work on alternate days, with B starting the work. We need to find the time B takes to complete four times the same work alone.
Calculating Individual Work Rates
The work rate of a person is the amount of work they can do in one day. If a person can complete a work in \(d\) days, their daily work rate is \(1/d\) of the total work.
- A can finish the work alone in 48 days.
- Therefore, A's daily work rate = \(\frac{1}{48}\) of the work per day.
Let B's daily work rate be \(r_B\). This means B can complete \(r_B\) fraction of the work per day.
Analyzing Work Done on Alternate Days
A and B work on alternate days, and B starts the work. The total time taken to complete the work is \(11\frac{1}{3}\) days. Let's analyze which person works on which day:
- Day 1: B works
- Day 2: A works
- Day 3: B works
- Day 4: A works
- ... and so on.
The total time is \(11\frac{1}{3}\) days, which means the work is completed during the 12th day. Since B starts, the odd-numbered days (1, 3, 5, ..., 11) are B's working days, and the even-numbered days (2, 4, 6, ..., 10) are A's working days. On day 12, it would be A's turn to work.
Let's count the number of full days worked by A and B within the 11 full days:
- B works on Day 1, 3, 5, 7, 9, 11. Total full days B worked = 6 days.
- A works on Day 2, 4, 6, 8, 10. Total full days A worked = 5 days.
The work is completed in \(11\frac{1}{3}\) days. This means after 11 full days, \(\frac{1}{3}\) of a day's work is needed on the 12th day. Day 12 is A's turn. So, A works for \(\frac{1}{3}\) of the 12th day.
Total time A worked = 5 full days + \(\frac{1}{3}\) day = \(5\frac{1}{3}\) days = \(\frac{16}{3}\) days.
Total time B worked = 6 full days = 6 days.
Setting up the Work Equation
The total work done is the sum of the work done by A and the work done by B. The total work is 1 (representing one complete work).
- Work done by A = A's daily rate \(\times\) Number of days A worked = \(\frac{1}{48} \times \frac{16}{3}\).
- Work done by B = B's daily rate \(\times\) Number of days B worked = \(r_B \times 6\).
The equation for the total work done is:
Work done by A + Work done by B = 1
\(\left(\frac{1}{48} \times \frac{16}{3}\right) + \left(r_B \times 6\right) = 1\)
Solving for B's Work Rate
Let's simplify the equation and solve for \(r_B\):
\(\frac{16}{48 \times 3} + 6 r_B = 1\)
\(\frac{16}{144} + 6 r_B = 1\)
Simplify the fraction:
\(\frac{1}{9} + 6 r_B = 1\)
Subtract \(\frac{1}{9}\) from both sides:
\(6 r_B = 1 - \frac{1}{9}\)
\(6 r_B = \frac{9 - 1}{9}\)
\(6 r_B = \frac{8}{9}\)
Divide by 6 to find \(r_B\):
\(r_B = \frac{8}{9 \times 6}\)
\(r_B = \frac{8}{54}\)
Simplify the fraction for \(r_B\):
\(r_B = \frac{4}{27}\)
So, B's daily work rate is \(\frac{4}{27}\) of the work per day.
Calculating Time for B to Finish the Work Alone
If B's daily work rate is \(\frac{4}{27}\) of the work, the time B takes to finish the entire work (1 whole work) alone is the reciprocal of the daily rate:
Time for B alone = \(\frac{1}{r_B} = \frac{1}{\frac{4}{27}} = \frac{27}{4}\) days.
Calculating Time for B to Finish 4 Times the Work
The question asks for the time B alone can finish 4 times the same work. If B takes \(\frac{27}{4}\) days for 1 work, for 4 times the work, it will take:
Time for 4 times the work = \(4 \times \text{Time for 1 work}\)
Time for 4 times the work = \(4 \times \frac{27}{4}\)
Time for 4 times the work = \(27\) days.
Conclusion
B alone can finish 4 times the same work in 27 days.
| Item | Value |
|---|---|
| A's time for 1 work | 48 days |
| A's daily work rate | \(\frac{1}{48}\) |
| Total time taken (alternate days) | \(11\frac{1}{3}\) days |
| Days A worked | \(5\frac{1}{3}\) or \(\frac{16}{3}\) days |
| Days B worked | 6 days |
| B's daily work rate (\(r_B\)) | \(\frac{4}{27}\) |
| B's time for 1 work | \(\frac{27}{4}\) days |
| B's time for 4 times the work | 27 days |
Revision Table: Key Concepts for Work and Time Problems
| Concept | Explanation | Formula/Idea |
|---|---|---|
| Work Rate | The amount of work done per unit of time (usually a day). | If time taken is T days, rate = \(\frac{1}{T}\) work/day |
| Total Work | Typically represented as '1' for one complete task. | Work = Rate \(\times\) Time |
| Combined Work Rate | Sum of individual work rates when working together. | \(R_{A+B} = R_A + R_B\) |
| Alternate Days Work | People work one after another, not simultaneously. Calculate work done in cycles or count individual days worked. | Total Work = Sum of (Individual Rate \(\times\) Days Worked by Individual) |
Additional Information: Solving Alternate Day Work Problems
Alternate day work problems require careful counting of the days each person works. When the total time is not a whole number, pay close attention to whose turn it is on the fractional part of the day.
Steps for solving alternate day problems:
- Calculate the individual daily work rates.
- Determine the sequence of work (who starts?).
- Count the exact number of days (or fractions of days) each person worked based on the total time given.
- Set up an equation: (Work done by Person 1) + (Work done by Person 2) + ... = 1 (or the specified amount of work).
- Solve the equation for the unknown variable (usually a work rate or time).
- Use the calculated work rate/time to answer the specific question posed.
In this problem, carefully identifying that A worked for \(5\frac{1}{3}\) days and B worked for 6 days was crucial because B started and the work finished during A's turn on the 12th day.
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