Work and Time Calculation: A's Solo Effort
This problem involves calculating the time taken by an individual worker based on relative efficiency and combined work time.
Step-by-Step Solution
- Define Work Rates: Let the rate at which B works be $r_B$ (units of work per day). Since A is twice as good a workman as B, A's rate of work is $r_A = 2r_B$.
- Calculate Combined Rate: When working together, their combined rate is $r_A + r_B = 2r_B + r_B = 3r_B$.
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Determine Total Work: They finish the work in 10 days. The total amount of work (W) can be calculated as:
$W = (\text{Combined Rate}) \times (\text{Time Taken Together}) $
$W = (3r_B) \times 10 \text{ days} = 30r_B \text{ units of work} $
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Calculate A's Time Alone: To find how many days A will finish the work alone, divide the total work by A's individual rate:
$ \text{Time for A} = \frac{\text{Total Work}}{r_A} $
$ \text{Time for A} = \frac{30r_B}{2r_B} $
$ \text{Time for A} = 15 \text{ days} $
Therefore, A will finish the work alone in 15 days.
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