A can finish 25% of a task in 3 days and B can finish half of the task in 18 days. If they work on it together, in how many days can they finish the task?
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Solving Work and Time Problems: A Step-by-Step Guide
This problem asks us to find the time it takes for two individuals, A and B, to complete a task when working together, given their individual rates of work. Work and time problems require us to first determine the efficiency or rate at which each person works individually.
Understanding Individual Work Rates
First, let's figure out how long each person takes to complete the entire task on their own.
- A finishes 25% of the task in 3 days. Since 25% is equivalent to \(\frac{1}{4}\) of the task, A takes 4 times as long to complete the whole task.
- Time taken by A to complete the whole task = \(3 \text{ days} \times 4 = 12 \text{ days}\).
- B finishes half of the task in 18 days. Since half is 50% or \(\frac{1}{2}\) of the task, B takes 2 times as long to complete the whole task.
- Time taken by B to complete the whole task = \(18 \text{ days} \times 2 = 36 \text{ days}\).
Calculating Daily Work Efficiency
Now that we know the total time each person takes, we can calculate the fraction of the task they complete in one day. This is their daily work rate or efficiency.
- A's daily work rate = \(\frac{1}{\text{Time taken by A}} = \frac{1}{12}\) of the task per day.
- B's daily work rate = \(\frac{1}{\text{Time taken by B}} = \frac{1}{36}\) of the task per day.
Calculating Combined Work Rate
When A and B work together, their work rates add up. To find their combined daily work rate, we sum their individual daily rates.
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, we find a common denominator, which is 36.
\(\frac{1}{12} = \frac{1 \times 3}{12 \times 3} = \frac{3}{36}\)
Combined daily work rate = \(\frac{3}{36} + \frac{1}{36} = \frac{3 + 1}{36} = \frac{4}{36}\)
Simplifying the fraction:
Combined daily work rate = \(\frac{4}{36} = \frac{1}{9}\) of the task per day.
Time Taken to Finish the Task Together
If A and B together complete \(\frac{1}{9}\) of the task in one day, the total time taken to complete the entire task (which is 1 whole task) is the reciprocal of their combined daily work rate.
Time taken together = \(\frac{1}{\text{Combined daily work rate}}\)
Time taken together = \(\frac{1}{\frac{1}{9}} = 1 \times 9 = 9\) days.
Therefore, if A and B work on the task together, they can finish it in 9 days.
| Individual | Fraction of Task Done | Time Taken | Time for Full Task | Daily Work Rate |
|---|---|---|---|---|
| A | 25% or \(\frac{1}{4}\) | 3 days | \(3 \times 4 = 12\) days | \(\frac{1}{12}\) |
| B | 50% or \(\frac{1}{2}\) | 18 days | \(18 \times 2 = 36\) days | \(\frac{1}{36}\) |
| A & B Together | 1 (whole task) | ? | ? | \(\frac{1}{12} + \frac{1}{36} = \frac{4}{36} = \frac{1}{9}\) |
Revision Table: Key Concepts in Work and Time
| Concept | Explanation | Formula |
|---|---|---|
| Work Rate | The amount of work done per unit of time. | Work Rate = \(\frac{\text{Work Done}}{\text{Time Taken}}\) |
| Time Taken | The total duration to complete a specific amount of work. | Time Taken = \(\frac{\text{Total Work}}{\text{Work Rate}}\) |
| Individual Time for Full Task | If a fraction of work is done in 'x' days, the time for the full task is 'x' divided by the fraction. | Time (Full Task) = \(\frac{\text{Time Taken for Fraction}}{\text{Fraction of Work Done}}\) |
| Combined Work Rate | When multiple people work together, their individual work rates add up. | Combined Rate = Rate\(_1\) + Rate\(_2\) + ... |
Additional Information on Solving Work Problems
Work and time problems often involve calculating efficiency and combining efforts. A key idea is to consider the total work as '1 unit' or equivalent to the Least Common Multiple (LCM) of the individual times if dealing with discrete units.
- Efficiency Method: Consider the total work as the LCM of the days taken by individuals. This helps avoid fractions. For example, LCM of 12 and 36 is 36. Assume total work is 36 units. A does 36/12 = 3 units/day. B does 36/36 = 1 unit/day. Together they do 3+1 = 4 units/day. Time taken together = Total work / Combined rate = 36 / 4 = 9 days. This method is useful for more complex problems.
- Understanding Proportionality: Work is directly proportional to time and number of workers (assuming constant efficiency). If more people work, less time is needed. If efficiency increases, less time is needed.
- Consistent Units: Always ensure the units of time (days, hours, minutes) are consistent throughout the problem calculation.
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