Pratima and Diksha can complete a typing work separately in 10 hours and 15 hours, respectively. After typing for 4 hours alone, Pratima leaves the work. In how many hours will Diksha complete the remaining typing work?
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Solving the Typing Work Problem: Pratima and Diksha
This problem involves understanding work and time concepts. We are given the time it takes two individuals, Pratima and Diksha, to complete a typing work separately. We need to calculate how long Diksha takes to finish the remaining work after Pratima has worked alone for a specific duration.
Understanding Work Rate
The work rate is the amount of work done per unit of time. If someone can complete a work in 'T' hours, their work rate is 1/T of the work per hour.
- Pratima completes the typing work in 10 hours.
- Pratima's work rate per hour = $\frac{1}{10}$ of the work.
- Diksha completes the typing work in 15 hours.
- Diksha's work rate per hour = $\frac{1}{15}$ of the work.
Calculating Work Done by Pratima
Pratima works alone for 4 hours. To find the amount of work she completes, we multiply her work rate by the time she worked:
Work done by Pratima = Pratima's rate $\times$ Time worked
Work done by Pratima = $\frac{1}{10} \times 4 = \frac{4}{10} = \frac{2}{5}$ of the typing work.
So, Pratima completed $\frac{2}{5}$ of the total typing work before leaving.
Calculating Remaining Work
The total work is considered as 1 unit. After Pratima leaves, the remaining work is:
Remaining work = Total work - Work done by Pratima
Remaining work = $1 - \frac{2}{5} = \frac{5}{5} - \frac{2}{5} = \frac{3}{5}$ of the typing work.
Thus, $\frac{3}{5}$ of the typing work still needs to be completed.
Calculating Time Taken by Diksha for Remaining Work
Diksha will complete the remaining $\frac{3}{5}$ of the work at her own work rate, which is $\frac{1}{15}$ of the work per hour. The time taken by Diksha is the remaining work divided by Diksha's rate:
Time taken by Diksha = $\frac{\text{Remaining Work}}{\text{Diksha's Rate}}$
Time taken by Diksha = $\frac{\frac{3}{5}}{\frac{1}{15}}$
To divide by a fraction, we multiply by its reciprocal:
Time taken by Diksha = $\frac{3}{5} \times \frac{15}{1}$
Time taken by Diksha = $\frac{3 \times 15}{5 \times 1} = \frac{45}{5} = 9$ hours.
Therefore, Diksha will take 9 hours to complete the remaining typing work.
Summary of Calculations
| Person | Time to Complete (hours) | Work Rate (work/hour) |
|---|---|---|
| Pratima | 10 | $\frac{1}{10}$ |
| Diksha | 15 | $\frac{1}{15}$ |
| Step | Calculation | Result |
|---|---|---|
| Work done by Pratima in 4 hours | $\frac{1}{10} \times 4$ | $\frac{2}{5}$ of the work |
| Remaining Work | $1 - \frac{2}{5}$ | $\frac{3}{5}$ of the work |
| Time taken by Diksha for remaining work | $\frac{3/5}{1/15} = \frac{3}{5} \times 15$ | 9 hours |
The remaining typing work will be completed by Diksha in 9 hours.
Revision Table: Typing Work Problem Concepts
| Concept | Explanation |
|---|---|
| Work Rate | The fraction of work done per unit of time. If time taken is T, rate is $1/T$. |
| Work Done | Work Rate $\times$ Time Worked. |
| Total Work | Usually considered as 1 unit. |
| Remaining Work | Total Work - Work Done. |
| Time to Complete Remaining Work | Remaining Work / Person's Work Rate. |
Additional Information: Work and Time Problems
Work and time problems are common in quantitative aptitude. They are based on the principle that the amount of work done is directly proportional to the time taken and the rate at which the work is done. A higher work rate means less time is needed to complete the same amount of work.
Key points to remember:
- If multiple people work together, their individual work rates are added to find the combined work rate.
- If the efficiency of a person changes, their work rate changes accordingly.
- Always express work rate as 'work per unit time' (e.g., work per hour, work per day).
- Ensure units of time are consistent throughout the problem.
These problems often involve scenarios where individuals work alone for a time, leave, or join others, requiring calculation of work done in parts and the time taken for the remaining work.
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