A can complete a piece of work in 25 days while B can complete the same work in 30 days. They work on alternate basis, starting with A. Both A and B follow this pattern for 5 days and then A leaves the work. In how many days will B finish the remaining work?
Understanding the Work and Time Problem
This problem involves calculating the time taken to complete a piece of work when two individuals, A and B, work together under specific conditions. A and B have different efficiencies, work on alternate days initially, and then one leaves, leaving the other to finish the remaining task. We need to find how long B takes to finish the remaining work alone.
Calculating Individual Work Rates (Efficiency)
The efficiency of a person in completing work is usually measured as the fraction of work done per day.
- A can complete the work in 25 days.
- A's daily work rate = $\frac{1}{25}$ of the work per day.
- B can complete the same work in 30 days.
- B's daily work rate = $\frac{1}{30}$ of the work per day.
Work Done in the First 5 Days (Alternate Basis)
A and B work on alternate days, starting with A, for 5 days. Let's calculate the work done each day:
- Day 1: A works. Work done = $\frac{1}{25}$
- Day 2: B works. Work done = $\frac{1}{30}$
- Day 3: A works. Work done = $\frac{1}{25}$
- Day 4: B works. Work done = $\frac{1}{30}$
- Day 5: A works. Work done = $\frac{1}{25}$
Total work done in 5 days is the sum of the work done on each of these days.
Total work in 5 days = (Work by A on Day 1, 3, 5) + (Work by B on Day 2, 4)
Work done by A in 3 days = $3 \times \frac{1}{25} = \frac{3}{25}$
Work done by B in 2 days = $2 \times \frac{1}{30} = \frac{2}{30} = \frac{1}{15}$
Total work done in the first 5 days = $\frac{3}{25} + \frac{1}{15}$
To add these fractions, we find a common denominator, which is the Least Common Multiple (LCM) of 25 and 15. LCM(25, 15) = 75.
- $\frac{3}{25} = \frac{3 \times 3}{25 \times 3} = \frac{9}{75}$
- $\frac{1}{15} = \frac{1 \times 5}{15 \times 5} = \frac{5}{75}$
Total work done in 5 days = $\frac{9}{75} + \frac{5}{75} = \frac{14}{75}$
Calculating the Remaining Work
The total work is considered as 1 unit. The work remaining after the first 5 days is calculated by subtracting the work done from the total work.
Remaining work = Total work - Work done in 5 days
Remaining work = $1 - \frac{14}{75} = \frac{75}{75} - \frac{14}{75} = \frac{61}{75}$
Time Taken by B to Finish the Remaining Work
After 5 days, A leaves the work. B has to finish the remaining work alone. We know B's daily work rate is $\frac{1}{30}$ of the work.
Time taken by B to finish the remaining work = $\frac{\text{Remaining Work}}{\text{B's daily work rate}}$
Time taken by B = $\frac{61/75}{1/30}$
This can be calculated as:
Time taken by B = $\frac{61}{75} \times 30$
We can simplify the calculation by dividing 75 and 30 by their common factor, 15.
$\frac{61}{75} \times 30 = \frac{61}{5 \times 15} \times (2 \times 15) = \frac{61 \times 2}{5} = \frac{122}{5}$
Now, we convert the improper fraction $\frac{122}{5}$ into a mixed number. Divide 122 by 5:
$122 \div 5$
$122 = 5 \times 24 + 2$
So, $\frac{122}{5} = 24 \frac{2}{5}$
Therefore, B will take $24 \frac{2}{5}$ days to finish the remaining work.
Revision Table: Work and Time Calculations
| Worker | Time to complete work | Daily work rate |
|---|---|---|
| A | 25 days | $\frac{1}{25}$ |
| B | 30 days | $\frac{1}{30}$ |
| Activity | Duration | Work Done |
|---|---|---|
| A works (Days 1, 3, 5) | 3 days | $3 \times \frac{1}{25} = \frac{3}{25}$ |
| B works (Days 2, 4) | 2 days | $2 \times \frac{1}{30} = \frac{1}{15}$ |
| Total work in first 5 days | 5 days | $\frac{3}{25} + \frac{1}{15} = \frac{14}{75}$ |
| Remaining Work | - | $1 - \frac{14}{75} = \frac{61}{75}$ |
| Time for B to finish remaining work | ? | $\frac{61/75}{1/30} = 24 \frac{2}{5}$ days |
Additional Information on Work and Time Problems
Work and time problems are common in quantitative aptitude sections of various exams. Key concepts include:
- Work Rate: If a person can do a piece of work in 'n' days, their work rate is $\frac{1}{n}$ work per day.
- Total Work: Often considered as 1 unit or the LCM of the individual times to make calculations with whole numbers easier.
- Work Done: Work done = Work Rate $\times$ Time.
- Combined Work Rate: If two people A and B work together, their combined work rate is the sum of their individual work rates (A's rate + B's rate).
- Alternate Work: When individuals work on alternate days, calculate the work done over a cycle (usually 2 days if two people work) and see how many such cycles fit into the given time, or how many cycles are needed to complete the work.
- Remaining Work: Calculated as Total Work - Work Done. The person or persons finishing the remaining work will take time based on their work rate(s).
Understanding these basic principles helps in solving various types of work and time problems efficiently.
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