Step 1: Conceptualization:
This problem involves work and time. The objective is to determine the total job completion time and subsequently the remaining time.
Step 2: Core Principle:
If a portion of work is accomplished within a specific timeframe, the time required for the remaining work can be calculated.
Step 3: Detailed Calculation:
Method 1: Total Time Calculation
The individual completes \( \frac{5}{8} \) of the task in 10 days.
The time to complete the entire job (1 whole job) is determined by the proportion:
\[ \text{Total Job Time} = \frac{\text{Days Invested}}{\text{Fraction of Work Completed}} = \frac{10}{\frac{5}{8}} = 10 \times \frac{8}{5} = 2 \times 8 = 16 \text{ days} \]The cumulative time for the job is 16 days.
Work already performed spans 10 days.
\[ \text{Additional Days Required} = \text{Total Job Time} - \text{Time Already Worked} = 16 - 10 = 6 \text{ days} \]
Method 2: Remaining Work Calculation
Fraction of work accomplished = \( \frac{5}{8} \)
Fraction of work remaining = \( 1 - \frac{5}{8} = \frac{3}{8} \)
It is established that \( \frac{5}{8} \) of the work requires 10 days.
The time required for \( \frac{1}{8} \) of the work is:
\[ \text{Time for } \frac{1}{8} \text{ Work Unit} = \frac{10 \text{ days}}{5} = 2 \text{ days} \]Subsequently, the time for the remaining \( \frac{3}{8} \) of the work is calculated:
\[ \text{Time for } \frac{3}{8} \text{ Work Remaining} = 3 \times (\text{Time for } \frac{1}{8} \text{ Work Unit}) = 3 \times 2 = 6 \text{ days} \]
Step 4: Conclusion:
The individual will require an additional 6 days to finalize the job.