A can complete a work in 60 days. B is 25% more efficient than A. They work together for 15 days. C alone completes the remaining work in 14 days. B and C together will complete \(\frac{5}{8}\)th part of the original work in:
12 days
Time and Work Problem Solution
This problem involves calculating the time taken to complete a certain amount of work based on individual efficiencies and combined work durations. We need to determine the work rates of individuals A, B, and C, the amount of work done by combinations, and finally the time required for B and C together to complete a specific fraction of the total work.
Calculating Individual Work Rates
First, let's find the daily work rate for A and B.
- A completes the work in 60 days.
A's work rate = \(\frac{1}{\text{Time taken by A}}\) = \(\frac{1}{60}\) work per day. - B is 25% more efficient than A. This means B can do 25% more work than A in the same amount of time.
B's efficiency = 100% + 25% = 125% of A's efficiency.
B's work rate = 125% of A's work rate = \(\frac{125}{100} \times \frac{1}{60} = \frac{5}{4} \times \frac{1}{60} = \frac{1}{48}\) work per day.
Work Done by A and B Together
A and B work together for 15 days. Let's find their combined work rate and the total work they complete in this period.
- Combined work rate of A and B = A's work rate + B's work rate = \(\frac{1}{60} + \frac{1}{48}\) work per day.
- To add these fractions, we find a common denominator. The least common multiple (LCM) of 60 and 48 is 240.
Combined work rate of A and B = \(\frac{4}{240} + \frac{5}{240} = \frac{9}{240} = \frac{3}{80}\) work per day. - Work done by A and B in 15 days = Combined rate \(\times\) Time = \(\frac{3}{80} \times 15 = \frac{45}{80} = \frac{9}{16}\) of the total work.
Calculating Remaining Work and C's Work Rate
After A and B work for 15 days, the remaining work is completed by C alone in 14 days.
- Remaining work = Total work - Work done by A and B = \(1 - \frac{9}{16} = \frac{16 - 9}{16} = \frac{7}{16}\) of the total work.
- C completes \(\frac{7}{16}\) of the work in 14 days.
C's work rate = \(\frac{\text{Remaining Work}}{\text{Time taken by C}}\) = \(\frac{7/16}{14} = \frac{7}{16 \times 14} = \frac{1}{16 \times 2} = \frac{1}{32}\) work per day.
Time for B and C to Complete a Specific Part of Work
We need to find the time taken by B and C together to complete \(\frac{5}{8}\)th part of the original work.
- First, calculate the combined work rate of B and C.
Combined work rate of B and C = B's work rate + C's work rate = \(\frac{1}{48} + \frac{1}{32}\) work per day. - To add these fractions, we find a common denominator. The LCM of 48 and 32 is 96.
Combined work rate of B and C = \(\frac{2}{96} + \frac{3}{96} = \frac{5}{96}\) work per day. - The amount of work B and C need to complete is \(\frac{5}{8}\)th of the original work.
- Time taken by B and C to complete \(\frac{5}{8}\)th work = \(\frac{\text{Amount of work to be done}}{\text{Combined work rate of B and C}}\) = \(\frac{5/8}{5/96}\) days.
- Time = \(\frac{5}{8} \times \frac{96}{5} = \frac{96}{8} = 12\) days.
Therefore, B and C together will complete \(\frac{5}{8}\)th part of the original work in 12 days.
| Person | Time to Complete Full Work (Days) | Work Rate (Work/Day) |
|---|---|---|
| A | 60 | \(\frac{1}{60}\) |
| B | 48 | \(\frac{1}{48}\) |
| C | \( \frac{7/16 \text{ work}}{1/32 \text{ work/day}} \) = 14 (for 7/16 work) or 32 (for full work) | \(\frac{1}{32}\) |
Revision Table: Key Calculations
| Step | Calculation | Result |
|---|---|---|
| A's Work Rate | \(1/60\) | \(\frac{1}{60}\) |
| B's Work Rate | \(1.25 \times 1/60\) | \(\frac{1}{48}\) |
| A & B Combined Rate | \(\frac{1}{60} + \frac{1}{48}\) | \(\frac{3}{80}\) |
| Work by A & B (15 days) | \(\frac{3}{80} \times 15\) | \(\frac{9}{16}\) |
| Remaining Work | \(1 - \frac{9}{16}\) | \(\frac{7}{16}\) |
| C's Work Rate | \(\frac{7/16}{14}\) | \(\frac{1}{32}\) |
| B & C Combined Rate | \(\frac{1}{48} + \frac{1}{32}\) | \(\frac{5}{96}\) |
| Time for B & C (5/8 work) | \(\frac{5/8}{5/96}\) | 12 days |
Additional Information on Time and Work Concepts
Time and work problems often rely on the concept of work rate, which is the amount of work done per unit of time. If a person can complete a work in \(D\) days, their work rate is \(\frac{1}{D}\) work per day.
- Inverse Relationship: Time and work rate are inversely proportional. More efficient workers have higher work rates and take less time to complete the same amount of work.
- Combined Work Rate: When multiple people work together, their individual work rates are added to find the combined work rate. If person 1 has rate \(R_1\) and person 2 has rate \(R_2\), their combined rate is \(R_1 + R_2\).
- Total Work: Often, the total work is considered as '1 unit' or '1 whole'.
- Work Done: Work done = Work Rate \(\times\) Time.
- Time Taken: Time taken = \(\frac{\text{Total Work}}{\text{Work Rate}}\).
- Efficiency: Efficiency is proportional to the work rate. If person B is X% more efficient than A, B's work rate is \((1 + X/100)\) times A's work rate.
These basic principles are applied to solve various time and work problems, including those involving multiple workers, varying efficiencies, and partial work completion.
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Important Questions from Work Efficiency
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