A, B and C can separately complete a work in 12, 15 and 20 days, respectively. They worked together 4 days. What will be the remaining work?
Calculating Remaining Work: A Step-by-Step Approach
This problem involves calculating the portion of work remaining after multiple individuals have worked together for a certain period. We need to first determine the individual work rates, then their combined rate, the total work done, and finally, the remaining work.
Understanding Individual Work Rates
The work rate of a person is the amount of work they can complete in one day. If a person can complete a total work in 'n' days, their daily work rate is \(\frac{1}{n}\) of the total work.
- A can complete the work in 12 days. So, A's daily work rate is \(\frac{1}{12}\) of the work.
- B can complete the work in 15 days. So, B's daily work rate is \(\frac{1}{15}\) of the work.
- C can complete the work in 20 days. So, C's daily work rate is \(\frac{1}{20}\) of the work.
Calculating the Combined Daily Work Rate
When A, B, and C work together, their daily work rates add up to find their combined daily work rate.
Combined daily work rate = (A's daily rate) + (B's daily rate) + (C's daily rate)
Combined daily work rate = \(\frac{1}{12} + \frac{1}{15} + \frac{1}{20}\)
To add these fractions, we find a common denominator. The least common multiple (LCM) of 12, 15, and 20 is 60.
Convert each fraction to have a denominator of 60:
- \(\frac{1}{12} = \frac{1 \times 5}{12 \times 5} = \frac{5}{60}\)
- \(\frac{1}{15} = \frac{1 \times 4}{15 \times 4} = \frac{4}{60}\)
- \(\frac{1}{20} = \frac{1 \times 3}{20 \times 3} = \frac{3}{60}\)
Now, add the fractions:
Combined daily work rate = \(\frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{5 + 4 + 3}{60} = \frac{12}{60}\)
Simplify the combined daily work rate:
Combined daily work rate = \(\frac{12}{60} = \frac{1}{5}\) of the work per day.
Calculating Work Done in 4 Days
A, B, and C worked together for 4 days. To find the total work done in 4 days, we multiply their combined daily work rate by the number of days they worked.
Work done in 4 days = (Combined daily work rate) \(\times\) (Number of days worked)
Work done in 4 days = \(\frac{1}{5} \times 4 = \frac{4}{5}\) of the work.
Calculating Remaining Work
The total work is considered as 1 unit. The remaining work is the total work minus the work that has already been done.
Remaining work = Total work - Work done in 4 days
Remaining work = \(1 - \frac{4}{5}\)
To subtract the fraction from 1, write 1 as a fraction with the same denominator as the work done:
Remaining work = \(\frac{5}{5} - \frac{4}{5} = \frac{5 - 4}{5} = \frac{1}{5}\) of the work.
So, the remaining work is \(\frac{1}{5}\).
Summary of Calculations
| Entity | Time to Complete Work | Daily Work Rate |
|---|---|---|
| A | 12 days | \(\frac{1}{12}\) |
| B | 15 days | \(\frac{1}{15}\) |
| C | 20 days | \(\frac{1}{20}\) |
| A, B, C (Combined) | - | \(\frac{1}{12} + \frac{1}{15} + \frac{1}{20} = \frac{12}{60} = \frac{1}{5}\) |
Work done in 4 days = Combined rate \(\times\) Days worked = \(\frac{1}{5} \times 4 = \frac{4}{5}\)
Remaining work = Total work - Work done = \(1 - \frac{4}{5} = \frac{1}{5}\)
The remaining work is \(\frac{1}{5}\).
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula/Relation |
|---|---|---|
| Work Rate | The amount of work done per unit of time (e.g., per day, per hour). | Work Rate = \(\frac{1}{\text{Time taken to complete total work}}\) |
| Total Work | Usually considered as 1 unit when calculating fractions of work. Can also be the LCM of individual times. | Total Work = Work Rate \(\times\) Time |
| Combined Work Rate | The sum of individual work rates when multiple people work together. | Rate\(_{A+B}\) = Rate\(_A\) + Rate\(_B\) |
| Work Done | The portion of the total work completed in a given time. | Work Done = Combined Rate \(\times\) Time Worked |
| Remaining Work | The portion of work yet to be completed. | Remaining Work = Total Work - Work Done |
Additional Information: Work and Time Problems
Work and time problems are common in quantitative aptitude. They typically involve calculating how quickly individuals or groups can complete a task based on their rates of work. Key principles include:
- Assuming the total work is 1 unit or the LCM of the individual times.
- Calculating individual work rates (work done per day/hour).
- Calculating combined work rates when people work together.
- Understanding that if a person works for only a part of the time or leaves before completion, their contribution is calculated based on their work rate and the time they worked.
- Problems might also involve efficiency ratios or scenarios where people work on alternate days.
Practicing different variations of work and time problems helps in understanding the application of fractions, ratios, and LCM in real-world scenarios.
Similar Questions
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Important Questions from Work Efficiency
A and B working together can complete a job in 30 days. The ratio of their efficiencies is 3 : 2. In how many days can the faster person complete the job?
A takes 15 days to complete \(\frac{5}{7} \) of a work. With the help of B, they finish the whole work in 12 days. In how many days, B alone will complete the same work
A alone can complete a work in 14 days and B alone can complete the same work in 21 days. A and B start the work together but A leaves the work after 4 days of the starting of work. In how many days B will complete the remaining work?
For completing a certain work, A is 50% less efficient than B and B is 50% more efficient than C. Working together A, B and C can complete the work in 48 days. A alone can complete the same work in:
30 persons can do a piece of work in 24 days. How many more persons are required to complete the work in 20 days?
