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?
11 days
Understanding the Work and Time Problem
This problem involves calculating the time taken by individuals and a team to complete a certain amount of work. We are given the time A and B take individually to complete the entire work and the period they work together before A leaves. We need to find how long B takes to finish the remaining part of the work.
Calculating Individual Work Rates
In work and time problems, the work rate is the amount of work done by a person in one unit of time (usually one day). If a person can complete a work in 'd' days, their daily work rate is $\frac{1}{d}$ of the total work.
- A alone completes the work in 14 days.
- So, A's daily work rate is $\frac{1}{14}$ of the work per day.
- B alone completes the same work in 21 days.
- So, B's daily work rate is $\frac{1}{21}$ of the work per day.
Calculating Combined Work Rate
When A and B work together, their combined daily work rate is the sum of their individual daily work rates.
Combined daily work rate of A and B = A's daily rate + B's daily rate
Combined daily work rate = $\frac{1}{14} + \frac{1}{21}$
To add these fractions, we find a common denominator. The least common multiple (LCM) of 14 and 21 is 42.
$\frac{1}{14} = \frac{1 \times 3}{14 \times 3} = \frac{3}{42}$
$\frac{1}{21} = \frac{1 \times 2}{21 \times 2} = \frac{2}{42}$
Combined daily work rate = $\frac{3}{42} + \frac{2}{42} = \frac{3+2}{42} = \frac{5}{42}$ of the work per day.
Work Done by A and B Together
A and B start the work together and work for 4 days before A leaves. The amount of work done in these 4 days is calculated by multiplying their combined daily work rate by the number of days they worked together.
Work done by A and B in 4 days = Combined daily work rate $\times$ Number of days worked together
Work done in 4 days = $\frac{5}{42} \times 4 = \frac{20}{42}$
This fraction can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 2.
Work done in 4 days = $\frac{20 \div 2}{42 \div 2} = \frac{10}{21}$ of the total work.
Calculating Remaining Work
The total work is considered as 1 unit. The remaining work is the total work minus the work done by A and B together.
Remaining work = Total work - Work done in 4 days
Remaining work = $1 - \frac{10}{21}$
To subtract, we write 1 as $\frac{21}{21}$.
Remaining work = $\frac{21}{21} - \frac{10}{21} = \frac{21-10}{21} = \frac{11}{21}$ of the total work.
Time Taken by B to Complete the Remaining Work
After A leaves, only B is left to complete the remaining work. We know B's daily work rate is $\frac{1}{21}$ of the work per day. To find the time B takes to complete the remaining $\frac{11}{21}$ of the work, we divide the remaining work by B's daily work rate.
Time taken by B = $\frac{\text{Remaining work}}{\text{B's daily work rate}}$
Time taken by B = $\frac{11/21}{1/21}$
Dividing by a fraction is the same as multiplying by its reciprocal.
Time taken by B = $\frac{11}{21} \times \frac{21}{1} = \frac{11 \times 21}{21 \times 1} = \frac{231}{21}$
Dividing 231 by 21 gives 11.
Time taken by B = 11 days.
So, B will complete the remaining work in 11 days.
Summary of Steps
| Step | Description | Calculation |
|---|---|---|
| 1 | A's daily work rate | $\frac{1}{14}$ |
| 2 | B's daily work rate | $\frac{1}{21}$ |
| 3 | Combined daily work rate (A + B) | $\frac{1}{14} + \frac{1}{21} = \frac{5}{42}$ |
| 4 | Work done by A & B in 4 days | $\frac{5}{42} \times 4 = \frac{10}{21}$ |
| 5 | Remaining work | $1 - \frac{10}{21} = \frac{11}{21}$ |
| 6 | Time taken by B for remaining work | $\frac{11/21}{1/21} = 11$ days |
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula |
|---|---|---|
| Individual Work Rate | Amount of work done by one person in one unit of time. | If a person finishes work in 'd' days, rate = $\frac{1}{d}$ per day. |
| Total Work | The entire task to be completed, usually represented as 1 unit. | $-$ |
| Work Done | Rate $\times$ Time | Work = Rate $\times$ Time |
| Time Taken | $\frac{\text{Total Work}}{\text{Rate}}$ or $\frac{\text{Amount of Work}}{\text{Rate}}$ | Time = $\frac{\text{Work}}{\text{Rate}}$ |
| Combined Rate | Sum of individual rates when multiple people work together. | Rate(A+B) = Rate(A) + Rate(B) |
Additional Information on Work and Time Problems
Work and time problems are a common topic in quantitative aptitude. They often involve scenarios where multiple people work on a task, sometimes starting or leaving at different times. Understanding the concept of work rate is key to solving these problems.
- The total work is usually treated as a single unit (1).
- If a person works for 't' days at a rate of 'r' per day, the work done is 'r $\times$ t'.
- If multiple people work together, their rates add up to find the combined rate.
- Problems might involve efficiency (how fast someone works), which is directly related to their work rate. Higher efficiency means a higher work rate and less time taken.
- Sometimes, the work is measured in units (like pages typed, articles made), but often it's just considered a total task to be completed.
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
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?
X is twice as good as workman as Y. Together, they finish the work in 18 days. In how many days can it be done by each separately?
