If A and B can finish a work in 10 days, B and C can finish the same work in 12 days, C and A can finish the same work in 15 days; then in how many days can A, B and C together finish half of the work?
4 days
Solving Work and Time Problems with Multiple Workers
This problem involves calculating the time taken by A, B, and C to complete a certain amount of work, given the time it takes for them to complete the same work in pairs. These types of problems can be solved by determining the work rate of each individual or group per day.
Understanding Work Rate
Work rate is the amount of work done per unit of time. If someone can finish a whole work in 'd' days, their work rate per day is \(\frac{1}{d}\) of the work.
Step-by-Step Solution
1. Calculate the Work Rate of Each Pair
We are given the time taken by pairs to finish the full work:
- A and B together finish the work in 10 days.
- B and C together finish the work in 12 days.
- C and A together finish the work in 15 days.
From this, we can find their combined work rate per day:
- Work done by (A + B) in 1 day = \(\frac{1}{10}\) of the work.
- Work done by (B + C) in 1 day = \(\frac{1}{12}\) of the work.
- Work done by (C + A) in 1 day = \(\frac{1}{15}\) of the work.
2. Calculate the Combined Work Rate of A, B, and C Together
If we add the daily work rates of the three pairs, we get the combined rate of (A+B) + (B+C) + (C+A). Notice that each person's work rate (A, B, and C) is included twice in this sum:
Combined rate of (A+B) + (B+C) + (C+A) per day = \(\frac{1}{10} + \frac{1}{12} + \frac{1}{15}\)
To add these fractions, we find a common denominator. The least common multiple (LCM) of 10, 12, and 15 is 60.
Sum of daily rates = \(\frac{6}{60} + \frac{5}{60} + \frac{4}{60} = \frac{6+5+4}{60} = \frac{15}{60}\)
Simplifying the fraction: \(\frac{15}{60} = \frac{1}{4}\)
So, the combined work rate of (A + B + B + C + C + A), which is 2 times the work rate of (A + B + C), is \(\frac{1}{4}\) of the work per day.
This means 2 \(\times\) (Combined work rate of A, B, C) = \(\frac{1}{4}\) of the work per day.
To find the combined work rate of A, B, and C together, we divide this sum by 2:
Combined work rate of (A + B + C) per day = \(\frac{1}{4} \div 2 = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}\) of the work per day.
3. Calculate the Time Taken for A, B, and C to Finish the Full Work
If A, B, and C together can finish \(\frac{1}{8}\) of the work in 1 day, they can finish the full work (which is 1) in the reciprocal of this rate.
Time taken by (A + B + C) to finish the full work = \(\frac{1}{\text{Combined daily rate}}\)
Time taken = \(\frac{1}{1/8} = 1 \times 8 = 8\) days.
So, A, B, and C working together can finish the entire work in 8 days.
4. Calculate the Time Taken for A, B, and C to Finish Half of the Work
The question asks for the time taken to finish half of the work. Since they can finish the full work in 8 days, they will take half the time to finish half the work.
Time taken to finish half work = \(\frac{1}{2} \times\) (Time taken for full work)
Time taken to finish half work = \(\frac{1}{2} \times 8\) days = 4 days.
Thus, A, B, and C together can finish half of the work in 4 days.
Summary of Calculations
| Group | Time for Full Work | Daily Work Rate (per day) |
|---|---|---|
| A + B | 10 days | \(\frac{1}{10}\) |
| B + C | 12 days | \(\frac{1}{12}\) |
| C + A | 15 days | \(\frac{1}{15}\) |
| (A+B) + (B+C) + (C+A) = 2(A+B+C) | Not Applicable | \(\frac{1}{10} + \frac{1}{12} + \frac{1}{15} = \frac{15}{60} = \frac{1}{4}\) |
| A + B + C | 8 days | \(\frac{1}{4} \div 2 = \frac{1}{8}\) |
Time for A, B, C together to finish half work = Time for full work \(\times \frac{1}{2} = 8 \times \frac{1}{2} = 4\) days.
The final answer is 4 days.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula/Relation |
|---|---|---|
| Work Rate | Amount of work done per unit of time (e.g., per day). | Work Rate = \(\frac{1}{\text{Time taken}}\) |
| Time Taken | Total time required to complete the work. | Time Taken = \(\frac{1}{\text{Work Rate}}\) |
| Combined Work Rate | Sum of individual work rates when people work together. | Rate\(_{\text{total}}\) = Rate\(_1\) + Rate\(_2\) + ... |
| Work Done | Total amount of work completed. Can be fraction or a whole unit (1 for full work). | Work Done = Work Rate \(\times\) Time |
Additional Information: Solving Paired Work Problems
Problems where people work in pairs (like A+B, B+C, C+A) are common in work and time topics. The key to solving these is to:
- Calculate the daily work rate for each given pair.
- Summing these rates gives you twice the combined rate of all individuals working together.
- Divide the sum by two to get the combined daily rate of everyone together.
- From the combined rate, you can find the time taken for everyone to finish the full work.
- Adjust the time based on the fraction of work requested (e.g., half work, double work).
- The LCM method for adding fractions representing work rates simplifies calculations significantly.
This approach helps break down complex problems into simpler, manageable steps based on daily work rates.
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