To do a certain work, Ajay and Bharat work on alternate days, with Bharat starting the work on the first day. Ajay can finish the work alone in 32 days. If the work gets completed in exactly 8 days, then Bharat alone can finish 7 times the same work in ____________ days.
32
Let's break down this work and time problem involving Ajay and Bharat working on alternate days.
Understanding the Alternate Day Work Problem
We are told that Ajay and Bharat work on a certain job on alternate days, with Bharat starting the work on the first day. The total time taken to complete the work is exactly 8 days.
The sequence of work days looks like this:
- Day 1: Bharat works
- Day 2: Ajay works
- Day 3: Bharat works
- Day 4: Ajay works
- Day 5: Bharat works
- Day 6: Ajay works
- Day 7: Bharat works
- Day 8: Ajay works
In a total duration of 8 days, each person works for 4 days.
- Number of days Bharat works = 4 days
- Number of days Ajay works = 4 days
Calculating Ajay's Contribution to the Work
We are given that Ajay can finish the entire work alone in 32 days. This means Ajay's per-day work rate is:
\( \text{Ajay's work rate} = \frac{1}{\text{Time Ajay takes alone}} = \frac{1}{32} \) of the work per day.
Since Ajay works for 4 days in the 8-day period, the amount of work done by Ajay is:
\( \text{Work done by Ajay in 4 days} = \text{Ajay's work rate} \times \text{Number of days Ajay worked} \)
\( \text{Work done by Ajay} = \frac{1}{32} \times 4 = \frac{4}{32} = \frac{1}{8} \) of the total work.
Finding Bharat's Contribution
The total work completed in 8 days is the sum of the work done by Bharat and Ajay. Since the work is completed, the total work done is considered as 1 unit.
\( \text{Total Work} = \text{Work done by Bharat} + \text{Work done by Ajay} \)
\( 1 = \text{Work done by Bharat} + \frac{1}{8} \)
So, the work done by Bharat in his 4 days of work is:
\( \text{Work done by Bharat} = 1 - \frac{1}{8} = \frac{8}{8} - \frac{1}{8} = \frac{7}{8} \) of the total work.
Determining Bharat's Individual Work Rate
Bharat completed \( \frac{7}{8} \) of the work by working for 4 days. We can now find Bharat's per-day work rate:
\( \text{Bharat's work rate} = \frac{\text{Work done by Bharat}}{\text{Number of days Bharat worked}} \)
\( \text{Bharat's work rate} = \frac{7/8}{4} = \frac{7}{8} \times \frac{1}{4} = \frac{7}{32} \) of the work per day.
Calculating Time for Bharat to Complete the Original Work Alone
If Bharat completes \( \frac{7}{32} \) of the work in 1 day, the time taken by Bharat to complete the entire work (1 unit) alone is the reciprocal of his work rate:
\( \text{Time Bharat takes alone} = \frac{1}{\text{Bharat's work rate}} = \frac{1}{7/32} = \frac{32}{7} \) days.
Calculating Time for Bharat to Complete 7 Times the Work
The question asks for the time Bharat alone can finish 7 times the same work. If Bharat takes \( \frac{32}{7} \) days to complete 1 unit of work, the time taken to complete 7 units of work will be:
\( \text{Time for 7 times work} = \text{Time Bharat takes for 1 unit} \times 7 \)
\( \text{Time for 7 times work} = \frac{32}{7} \times 7 = 32 \) days.
Therefore, Bharat alone can finish 7 times the same work in 32 days.
| Person | Time to complete 1 unit of work (alone) | Days worked in 8-day cycle | Work done in 8-day cycle | Per day work rate |
|---|---|---|---|---|
| Ajay | 32 days | 4 days | \( 4 \times \frac{1}{32} = \frac{1}{8} \) | \( \frac{1}{32} \) |
| Bharat | \( \frac{32}{7} \) days (Calculated) | 4 days | \( 1 - \frac{1}{8} = \frac{7}{8} \) | \( \frac{7/8}{4} = \frac{7}{32} \) |
| Ajay & Bharat (Alternate) | 8 days | - | 1 (Complete Work) | - |
The final answer is 32 days.
Revision Table: Work and Time Concepts
| Concept | Description | Formula |
|---|---|---|
| Work Rate | Amount of work done by a person in one unit of time (e.g., per day, per hour). | \( \text{Work Rate} = \frac{1}{\text{Total Time to complete work alone}} \) |
| Work Done | Work completed by a person in a given time. | \( \text{Work Done} = \text{Work Rate} \times \text{Time Worked} \) |
| Total Time (Individual) | Time taken by a person to complete the entire work alone. | \( \text{Total Time} = \frac{1}{\text{Work Rate}} \) |
| Alternate Days Work | Individuals work on consecutive days one after another. The work done in one cycle (e.g., 2 days for two people) is the sum of their individual work rates for that cycle. | Work done in a cycle = Sum of work done by each person in their turn during that cycle. |
Additional Information: Solving Work and Time Problems
Work and time problems often involve calculating the efficiency or work rate of individuals or groups and then determining the time required to complete a certain amount of work. Key steps often include:
- Identify the total amount of work (usually considered as 1 unit for the original task).
- Determine the work rate of each individual or entity involved. Work rate is typically expressed as "fraction of work per unit time".
- If people work together, add their work rates to find the combined work rate (assuming they work simultaneously).
- If people work on alternate days or in shifts, calculate the work done in one cycle of their turns.
- Use the work rate and total work to find the time taken: \( \text{Time} = \frac{\text{Total Work}}{\text{Work Rate}} \).
- If the amount of work changes (e.g., 7 times the work), adjust the "Total Work" value accordingly in the calculation.
Always pay attention to who starts the work and the total duration when dealing with alternate day problems, as this determines how many days each person contributes.
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