Ravi can do a piece of work in 40 days and Sudha can do the same piece of work in 60 days. If they work on alternative days starting with Sudha on the first day, then in how many days will the work be completed?
48
Solving Work and Time Problems with Alternative Days
This problem involves two individuals, Ravi and Sudha, working on the same task on alternate days. We need to determine the total time taken to complete the work when they follow this alternative day pattern, starting with Sudha.
Understanding Individual Work Rates
First, let's figure out how much work each person can do in a single day.
- Ravi can complete the work in 40 days. This means Ravi does \(\frac{1}{40}\) of the total work per day.
- Sudha can complete the same work in 60 days. This means Sudha does \(\frac{1}{60}\) of the total work per day.
Calculating Work Done in One Cycle
They work on alternative days, starting with Sudha. A cycle consists of two days:
- Day 1: Sudha works.
- Day 2: Ravi works.
Let's calculate the total work done in one complete cycle (2 days):
Work done in 1 cycle = (Work done by Sudha on Day 1) + (Work done by Ravi on Day 2)
Work done in 1 cycle = \(\frac{1}{60} + \frac{1}{40}\)
To add these fractions, we find a common denominator, which is the LCM of 60 and 40. LCM(60, 40) = 120.
Work done in 1 cycle = \(\frac{2}{120} + \frac{3}{120} = \frac{2+3}{120} = \frac{5}{120} = \frac{1}{24}\)
So, in every 2-day cycle, they complete \(\frac{1}{24}\) of the total work.
Finding the Number of Cycles
The entire work is represented as 1. Since they complete \(\frac{1}{24}\) of the work in one cycle, to complete the full work (1), they would ideally need 24 cycles.
Total cycles required = \(\frac{1}{\text{Work done in 1 cycle}} = \frac{1}{\frac{1}{24}} = 24\) cycles.
Calculating Total Days
Each cycle is 2 days long. So, if they complete the work in exactly 24 cycles, the total number of days would be:
Total days = Number of cycles \(\times\) Days per cycle
Total days = \(24 \times 2 = 48\) days.
Verifying the Work Completion
Let's check how the work is completed over these 48 days:
- There are 24 cycles in 48 days.
- Sudha works on the 1st, 3rd, ..., 47th day (the odd days). Total days Sudha works = 24 days.
- Ravi works on the 2nd, 4th, ..., 48th day (the even days). Total days Ravi works = 24 days.
Total work done by Sudha = 24 days \(\times\) \(\frac{1}{60}\) work/day = \(\frac{24}{60} = \frac{2}{5}\) of the work.
Total work done by Ravi = 24 days \(\times\) \(\frac{1}{40}\) work/day = \(\frac{24}{40} = \frac{3}{5}\) of the work.
Total work done together = \(\frac{2}{5} + \frac{3}{5} = \frac{5}{5} = 1\) (Complete work).
Since the total work is exactly completed at the end of an even number of days (which is Ravi's turn), the total time taken is indeed 48 days.
Alternative Method: Using Units of Work (LCM Method)
Let the total amount of work be the LCM of 40 and 60, which is 120 units.
- Ravi's daily work rate = \(\frac{120 \text{ units}}{40 \text{ days}} = 3\) units/day.
- Sudha's daily work rate = \(\frac{120 \text{ units}}{60 \text{ days}} = 2\) units/day.
They work on alternate days starting with Sudha:
- Day 1 (Sudha): 2 units worked.
- Day 2 (Ravi): 3 units worked.
Work done in a 2-day cycle = 2 units + 3 units = 5 units.
We need to complete 120 units of work. Let's find how many full 2-day cycles are needed.
Number of full cycles = \(\frac{\text{Total work}}{\text{Work per cycle}} = \frac{120 \text{ units}}{5 \text{ units/cycle}} = 24\) cycles.
Total days for 24 cycles = 24 cycles \(\times\) 2 days/cycle = 48 days.
At the end of 48 days (24 cycles), the total work done is \(24 \times 5 = 120\) units, which is the complete work.
Summary of Steps
- Calculate individual daily work rates for Ravi and Sudha.
- Calculate the total work done in one 2-day cycle (Sudha on day 1, Ravi on day 2).
- Determine how many such cycles are needed to complete the work.
- Multiply the number of cycles by 2 to find the total number of days.
- Alternatively, use the LCM of days as total work units and calculate daily efficiencies and cycle work.
| Worker | Time to Complete Work | Daily Work Rate |
|---|---|---|
| Ravi | 40 days | \(\frac{1}{40}\) |
| Sudha | 60 days | \(\frac{1}{60}\) |
| Day | Worker | Work Done on Day | Cumulative Work |
|---|---|---|---|
| 1 | Sudha | \(\frac{1}{60}\) | \(\frac{1}{60}\) |
| 2 | Ravi | \(\frac{1}{40}\) | \(\frac{1}{60} + \frac{1}{40} = \frac{5}{120} = \frac{1}{24}\) |
| ... (This 2-day pattern repeats) ... | |||
| 47 | Sudha | \(\frac{1}{60}\) | \(\frac{23}{24} + \frac{1}{60}\) (after 23 cycles) |
| 48 | Ravi | \(\frac{1}{40}\) | \(\frac{23}{24} + \frac{1}{60} + \frac{1}{40} = 1\) (after 24 cycles) |
The work is completed precisely at the end of 48 days.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula/Relation |
|---|---|---|
| Work Rate | Amount of work done per unit of time. | Work Rate = \(\frac{\text{Total Work}}{\text{Total Time}}\) |
| Total Work | The entire task to be completed, often considered as 1 unit or the LCM of individual times. | Total Work = Work Rate \(\times\) Total Time |
| Time Taken | The duration required to complete the work. | Time Taken = \(\frac{\text{Total Work}}{\text{Work Rate}}\) |
| Combined Work Rate | Sum of individual work rates when people work together. | Rate\(_{A+B}\) = Rate\(_A\) + Rate\(_B\) (when working together) |
| Alternative Days | Workers perform the task one after another on consecutive days. | Calculate work done in one full cycle (e.g., 2 days or more) |
Additional Information: Alternate Day Problems
Problems involving work done on alternate days require careful tracking of who works on which day and how much work is completed in each cycle. A cycle is typically formed by one turn of each worker involved. If there are two workers A and B, a cycle is A works, then B works (or vice versa). If there are three workers A, B, C, a cycle might be A, B, C in order.
Key steps often include:
- Calculating individual daily work rates.
- Calculating the total work done in one complete cycle.
- Determining how many full cycles can be completed before the work is nearly finished.
- Calculating the remaining work after the full cycles.
- Calculating the time taken by the worker whose turn it is to complete the remaining work.
- Adding the time for full cycles and the time for the remaining work to get the total time.
In this specific problem, the work finished exactly at the end of a cycle, simplifying the last steps.
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