If Mohit can complete \(\frac{2}{3}\)rd of a work in 24 days, then in how many days can \(\rm\frac{1}{9}^{th}\) of the work be complete by him?
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Calculating Work Completion Time
This problem involves understanding the relationship between the fraction of work done and the time taken. We are given how long it takes Mohit to complete a certain fraction of a task, and we need to find out how long it takes him to complete another fraction of the same task.
Understanding the Given Information
- Mohit completes \(\frac{2}{3}\)rd of the work.
- Time taken for \(\frac{2}{3}\)rd of the work is 24 days.
We need to find the time taken to complete \(\frac{1}{9}\)th of the work.
Step-by-Step Work Time Calculation
Step 1: Calculate the time taken to complete the entire work
If Mohit completes \(\frac{2}{3}\) of the work in 24 days, we can find the time taken to complete the full work (which is 1 whole). We can set up a proportion or simply divide the time taken by the fraction of work done:
Time for full work = \(\frac{\text{Time for partial work}}{\text{Fraction of work}}\)
Time for full work = \(\frac{24 \text{ days}}{\frac{2}{3}}\)
To divide by a fraction, we multiply by its reciprocal:
Time for full work = \(24 \times \frac{3}{2} \text{ days}\)
Time for full work = \(\frac{24 \times 3}{2} \text{ days}\)
Time for full work = \(12 \times 3 \text{ days}\)
Time for full work = \(36 \text{ days}\)
So, Mohit can complete the entire work in 36 days.
Step 2: Calculate the time taken to complete \(\frac{1}{9}\)th of the work
Now that we know the time taken for the full work, we can calculate the time taken for any fraction of the work. We need to find the time taken for \(\frac{1}{9}\)th of the work.
Time for \(\frac{1}{9}\)th work = \(\text{Time for full work} \times \text{Fraction of work}\)
Time for \(\frac{1}{9}\)th work = \(36 \text{ days} \times \frac{1}{9}\)
Time for \(\frac{1}{9}\)th work = \(\frac{36}{9} \text{ days}\)
Time for \(\frac{1}{9}\)th work = \(4 \text{ days}\)
Thus, Mohit can complete \(\frac{1}{9}\)th of the work in 4 days.
Final Answer Summary
Given that Mohit completes \(\frac{2}{3}\) of the work in 24 days, the time taken for the entire work is 36 days. Consequently, the time required to complete \(\frac{1}{9}\) of the work is 4 days.
| Work Done | Time Taken (Days) |
|---|---|
| \(\frac{2}{3}\) | 24 |
| 1 (Full Work) | \(24 \times \frac{3}{2} = 36\) |
| \(\frac{1}{9}\) | \(36 \times \frac{1}{9} = 4\) |
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula Example |
|---|---|---|
| Direct Proportion | Work done is directly proportional to the time taken (assuming constant work rate). More work means more time. | If work \(W_1\) takes time \(T_1\), then work \(W_2\) takes time \(T_2\), where \(\frac{W_1}{T_1} = \frac{W_2}{T_2}\). |
| Finding Time for Full Work | If a fraction of work is done in a certain time, time for full work is (Time for fraction) / (Fraction of work). | If \(\frac{a}{b}\) work is done in \(T\) days, full work takes \(T \times \frac{b}{a}\) days. |
| Finding Time for a Fraction | Time for a fraction of work is (Time for full work) \(\times\) (Fraction of work). | If full work takes \(T\) days, \(\frac{c}{d}\) work takes \(T \times \frac{c}{d}\) days. |
Additional Information: Work Rate
Another way to solve work and time problems is by using the concept of work rate. Work rate is the amount of work done per unit of time.
- If Mohit completes \(\frac{2}{3}\) of the work in 24 days, his work rate is \(\frac{\frac{2}{3} \text{ work}}{24 \text{ days}}\).
- Work rate = \(\frac{2}{3} \times \frac{1}{24} = \frac{2}{72} = \frac{1}{36}\) work per day.
- This means Mohit completes \(\frac{1}{36}\)th of the work each day.
- To find the time taken to complete \(\frac{1}{9}\)th of the work, we divide the amount of work by the work rate:
- Time = \(\frac{\text{Amount of work}}{\text{Work rate}}\)
- Time = \(\frac{\frac{1}{9}}{\frac{1}{36}}\) days
- Time = \(\frac{1}{9} \times \frac{36}{1}\) days
- Time = \(\frac{36}{9}\) days
- Time = 4 days
This method using work rate gives the same result and is a fundamental concept in work and time problems.
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