Work and Time Calculation
This problem involves calculating the combined time taken by two individuals, Narendra and Amit, to complete a job, given their individual completion times.
Work Rate Analysis
- Narendra's time to complete the job = 10 days.
- Narendra's work rate per day = $\frac{1}{10}$ of the job.
- Amit's time to complete the job = 12 days.
- Amit's work rate per day = $\frac{1}{12}$ of the job.
Combined Work Rate
To find the time taken when working together, we first calculate their combined work rate per day.
- Combined work rate = Narendra's rate + Amit's rate
- Combined work rate = $\frac{1}{10} + \frac{1}{12}$
- Find a common denominator (LCM of 10 and 12 is 60):
- Combined work rate = $\frac{6}{60} + \frac{5}{60} = \frac{11}{60}$ of the job per day.
Time Taken Together
The total time required to complete the job together is the reciprocal of their combined work rate.
- Time taken together = $\frac{1}{\text{Combined work rate}}$
- Time taken together = $\frac{1}{\frac{11}{60}}$
- Time taken together = $\frac{60}{11}$ days.
Final Answer Conversion
Convert the improper fraction to a mixed number.
- $\frac{60}{11}$ can be written as $5$ with a remainder of $5$ ($60 = 5 \times 11 + 5$).
- Therefore, the time taken is $5\frac{5}{11}$ days.
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