Step 1: Concept Introduction:
This is a work and time problem. The objective is to determine the total time for the job and then the remaining time.
Step 2: Fundamental Principle:
Given a portion of work completed within a specific timeframe, we can ascertain the time required for the outstanding portion.
Step 3: Detailed Analysis:
Method 1: Calculate Total Time First
\( \frac{5}{8} \) of the job is completed in 10 days.
To calculate the time for the entire job (1 unit of work), we apply the formula:
\[ \text{Total time} = \frac{\text{Duration}}{\text{Fraction of work}} = \frac{10 \text{ days}}{\frac{5}{8}} = 10 \times \frac{8}{5} = 16 \text{ days} \]The total duration for the job is 16 days.
Time already spent is 10 days.
\[ \text{Remaining time} = \text{Total time} - \text{Time spent} = 16 - 10 = 6 \text{ days} \]Method 2: Calculate Remaining Work Directly
Fraction of work completed = \( \frac{5}{8} \)
Fraction of work remaining = \( 1 - \frac{5}{8} = \frac{3}{8} \)
It is known that \( \frac{5}{8} \) of the work takes 10 days.
We calculate the time for \( \frac{1}{8} \) of the work:
\[ \text{Time per } \frac{1}{8} \text{ work} = \frac{10 \text{ days}}{5} = 2 \text{ days} \]Subsequently, we calculate the time for the remaining \( \frac{3}{8} \) of the work:
\[ \text{Time for } \frac{3}{8} \text{ work} = 3 \times (\text{Time per } \frac{1}{8} \text{ work}) = 3 \times 2 = 6 \text{ days} \]Step 4: Conclusion:
The remaining duration to complete the job is 6 days.