Aarif, Arun and Abraham can do a work in 12, 20 and 24 days, respectively. They all begin together. Arun leaves the work 3 days and Abraham 6 days before its completion. In how many days is the work finished?
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Solving the Work and Time Problem
This question involves calculating the total time taken to complete a work when multiple individuals with different work rates start together, but some leave before the work is finished. We need to determine the total number of days until the work is fully completed.
Understanding the Work Rates
First, let's figure out how much work each person can do in a single day. This is their daily work rate.
- Aarif can do the work in 12 days. So, Aarif's daily work rate is \( \frac{1}{12} \) of the total work.
- Arun can do the work in 20 days. So, Arun's daily work rate is \( \frac{1}{20} \) of the total work.
- Abraham can do the work in 24 days. So, Abraham's daily work rate is \( \frac{1}{24} \) of the total work.
Setting Up the Problem
Let the total time taken to complete the work be T days.
We know the following:
- Arun leaves 3 days before the work is completed. This means Arun worked for \( (T - 3) \) days.
- Abraham leaves 6 days before the work is completed. This means Abraham worked for \( (T - 6) \) days.
- Aarif works for the entire duration until the work is finished, which is T days.
The total work done is the sum of the work done by each person. The total work is considered as 1 unit.
Work done by Aarif = \( \text{Aarif's daily rate} \times \text{Days Aarif worked} = \frac{1}{12} \times T = \frac{T}{12} \)
Work done by Arun = \( \text{Arun's daily rate} \times \text{Days Arun worked} = \frac{1}{20} \times (T-3) = \frac{T-3}{20} \)
Work done by Abraham = \( \text{Abraham's daily rate} \times \text{Days Abraham worked} = \frac{1}{24} \times (T-6) = \frac{T-6}{24} \)
The equation representing the total work done is:
\( \text{Work done by Aarif} + \text{Work done by Arun} + \text{Work done by Abraham} = 1 \)
\( \frac{T}{12} + \frac{T-3}{20} + \frac{T-6}{24} = 1 \)
Solving for Total Time (T)
To solve this equation, we find the least common multiple (LCM) of the denominators 12, 20, and 24. The LCM of 12, 20, and 24 is 120.
Multiply the entire equation by 120:
\( 120 \times \left( \frac{T}{12} \right) + 120 \times \left( \frac{T-3}{20} \right) + 120 \times \left( \frac{T-6}{24} \right) = 120 \times 1 \)
\( 10T + 6(T-3) + 5(T-6) = 120 \)
Now, distribute the numbers outside the parentheses:
\( 10T + 6T - 18 + 5T - 30 = 120 \)
Combine the terms with T and the constant terms:
\( (10T + 6T + 5T) + (-18 - 30) = 120 \)
\( 21T - 48 = 120 \)
Add 48 to both sides of the equation:
\( 21T = 120 + 48 \)
\( 21T = 168 \)
Divide by 21 to find the value of T:
\( T = \frac{168}{21} \)
\( T = 8 \)
So, the total time taken to finish the work is 8 days.
Detailed Calculation Steps
| Step | Calculation | Explanation |
|---|---|---|
| 1 | Individual Rates | Aarif: \(1/12\), Arun: \(1/20\), Abraham: \(1/24\) |
| 2 | Set up Equation | \( \frac{T}{12} + \frac{T-3}{20} + \frac{T-6}{24} = 1 \) |
| 3 | Find LCM | LCM(12, 20, 24) = 120 |
| 4 | Multiply by LCM | \( 10T + 6(T-3) + 5(T-6) = 120 \) |
| 5 | Simplify | \( 10T + 6T - 18 + 5T - 30 = 120 \) |
| 6 | Combine Terms | \( 21T - 48 = 120 \) |
| 7 | Isolate T | \( 21T = 168 \) |
| 8 | Solve for T | \( T = \frac{168}{21} = 8 \) |
Conclusion
Based on our calculations, the work is finished in 8 days.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula |
|---|---|---|
| Work Rate | Amount of work done by a person/group in one unit of time (e.g., 1 day). | If a person takes N days to complete a work, their daily rate is \( \frac{1}{N} \). |
| Total Work | Usually considered as 1 unit, or the LCM of the individual days to simplify calculations. | Total Work = Sum of Work done by individuals/groups. |
| Work Done | The portion of work completed by a person/group. | Work Done = Rate \(\times\) Time |
| Combined Rate | The sum of individual rates when people work together. | If rates are \(R_1, R_2, \dots \), combined rate is \( R_1 + R_2 + \dots \). |
Additional Information: Solving Time and Work Problems
Time and work problems often involve scenarios with multiple individuals working together, sometimes with people joining or leaving. A common approach is to use the concept of work rate.
- Work Rate Method: Assign a rate (\(1/\text{days}\)) to each person. If people work together, their rates add up. If someone leaves, their contribution stops for the remaining time.
- LCM Method (Total Work Unit): Assume the total work unit is the LCM of the days taken by each person. This converts fractions into whole numbers, making calculation easier. Calculate the number of units of work done by each person per day.
- Segmentation Method: Divide the total time into segments based on who is working during that period. Calculate the work done in each segment and sum it up to equal the total work.
In this specific problem, since the leaving times are given relative to the completion time, using a variable for total time (T) and setting up an equation based on total work (1) is an effective method.
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