A can do a piece of work in 12 days. B can do that work in 36 days. After working together for 3 days, B left the job. In how many days can A complete the remaining work?
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Work and Time Problem Explained
This question involves a common type of quantitative aptitude problem dealing with work and time. We are given the time taken by two individuals, A and B, to complete a piece of work individually. They work together for a few days, and then one person leaves. We need to find the time taken by the remaining person to finish the rest of the work.
Step-by-Step Solution to the Work Problem
1. Calculate Individual Work Rates
The work rate of a person is the amount of work they can complete in one day. If a person completes a work in 'n' days, their daily work rate is \(\frac{1}{n}\) of the total work.
- A can do the work in 12 days.
- A's daily work rate = \( \frac{1}{12} \) of the total work.
- B can do the work in 36 days.
- B's daily work rate = \( \frac{1}{36} \) of the total work.
2. Calculate Combined Work Rate
When A and B work together, their work rates add up to find their combined daily work rate.
- Combined daily work rate of A and B = A's daily rate + B's daily rate
- Combined daily work rate = \( \frac{1}{12} + \frac{1}{36} \)
- To add these fractions, find a common denominator, which is 36.
- Combined daily work rate = \( \frac{3}{36} + \frac{1}{36} = \frac{3+1}{36} = \frac{4}{36} \)
- Simplifying the fraction, Combined daily work rate = \( \frac{1}{9} \) of the total work.
3. Calculate Work Done Together
A and B worked together for 3 days. We can calculate the total work completed by them in these 3 days using their combined daily work rate.
- Work done in 3 days = Combined daily work rate \(\times\) Number of days worked together
- Work done in 3 days = \( \frac{1}{9} \times 3 = \frac{3}{9} \)
- Simplifying the fraction, Work done in 3 days = \( \frac{1}{3} \) of the total work.
4. Calculate Remaining Work
After B leaves, A has to complete the remaining portion of the work. The remaining work is calculated by subtracting the work done from the total work (which is considered as 1 unit).
- Remaining work = Total work - Work done together
- Remaining work = \( 1 - \frac{1}{3} \)
- Remaining work = \( \frac{3}{3} - \frac{1}{3} = \frac{3-1}{3} = \frac{2}{3} \) of the total work.
5. Calculate Time for A to Complete Remaining Work
Now, only A is working on the remaining \( \frac{2}{3} \) of the work. We use A's daily work rate to find out how many days A will take.
- Time taken by A = Remaining work / A's daily work rate
- Time taken by A = \( \frac{2}{3} / \frac{1}{12} \)
- Dividing by a fraction is the same as multiplying by its reciprocal.
- Time taken by A = \( \frac{2}{3} \times \frac{12}{1} \)
- Time taken by A = \( \frac{2 \times 12}{3} = \frac{24}{3} \)
- Time taken by A = 8 days.
So, A can complete the remaining work in 8 days.
| Item | Value | Calculation |
|---|---|---|
| A's Daily Work | \( \frac{1}{12} \) | \( \frac{1}{\text{Days A takes}} \) |
| B's Daily Work | \( \frac{1}{36} \) | \( \frac{1}{\text{Days B takes}} \) |
| Combined Daily Work (A+B) | \( \frac{1}{9} \) | \( \frac{1}{12} + \frac{1}{36} \) |
| Work Done in 3 Days (A+B) | \( \frac{1}{3} \) | \( \frac{1}{9} \times 3 \) |
| Remaining Work | \( \frac{2}{3} \) | \( 1 - \frac{1}{3} \) |
| Time for A to finish remaining work | 8 days | \( \frac{2/3}{1/12} \) |
Work and Time Revision Table
| Concept | Formula | Explanation |
|---|---|---|
| Daily Work Rate | \( \frac{1}{\text{Total days to complete work}} \) | Amount of work done in one day. |
| Total Work | Work Rate \(\times\) Time | Usually considered as 1 unit or 100%. |
| Time Taken | \( \frac{\text{Total Work}}{\text{Work Rate}} \) | Number of days (or hours, etc.) to complete work. |
| Combined Work Rate | Sum of individual work rates | When multiple people work together. |
| Work Done | Combined Work Rate \(\times\) Days worked together | Amount of work completed by a group in a given time. |
| Remaining Work | Total Work - Work Done | Work left to be completed. |
Additional Information on Work and Time Concepts
Work and time problems often involve calculating rates and combining them. Understanding the inverse relationship between time and work rate is crucial: the more time a person takes, the lower their work rate.
- Efficiency: Sometimes problems refer to efficiency. Efficiency is directly proportional to the work rate. More efficient a person is, the more work they can do in a given time.
- Units: Ensure consistency in units (e.g., if rates are per day, time is in days).
- Part of Work: Problems often deal with completing a fraction or percentage of the work. The total work is always represented by 1 or 100%.
- LCM Method: Another common method to solve these problems is the LCM (Least Common Multiple) method. You can assume the total work to be the LCM of the number of days taken by individuals. For this problem, LCM of 12 and 36 is 36. Assume total work is 36 units. A's daily work = 36/12 = 3 units. B's daily work = 36/36 = 1 unit. Combined daily work = 3+1 = 4 units. Work done in 3 days = 4 * 3 = 12 units. Remaining work = 36 - 12 = 24 units. Time for A to complete remaining work = 24 units / 3 units per day = 8 days. This method can be simpler for some problems.
Important Questions from Work Efficiency
A man completes 7/8 of a job in 21 days. How many more days will it take him to finish the job if quantum of work further increased by 50%?
24 men and 12 women can do a piece of work in 30 days. In how many days can 12 men and 24 women do the same piece of work?
A and B together can do a piece of work in 4 days, B and C can do it in 6 days, A and C can do it in 8 days. Then A, B and C together can do the same work in :-
A can complete 50% of a work in 9 days and B can do 25% of the work in 9 days, if they work alone. If they work together then how much work (in percentage) can be completed in 6 days?
A is thrice as good a workman as B. C alone takes 48 days to paint a house. All three A, B and C working together take 16 days to paint the house. It will take how many days for A alone to paint the house?
