Calculating Work Completion Time

This problem involves calculating the time taken to complete a work based on individual efficiencies and changes in workforce.

Individual Work Rates

First, determine the rate at which each person works.

• A's efficiency: If A completes the work in 12 days, A completes $ \frac{1}{12} $ of the work per day.
• B's efficiency: If B completes the work in 18 days, B completes $ \frac{1}{18} $ of the work per day.

Combined Work Rate

Calculate their combined work rate when they work together.

• Combined rate = A's rate + B's rate = $ \frac{1}{12} + \frac{1}{18} $
• Find a common denominator (36): $ \frac{3}{36} + \frac{2}{36} = \frac{5}{36} $
• Together, they complete $ \frac{5}{36} $ of the work per day.

Work Done Together

Calculate the amount of work completed before A left.

• They worked together for 3 days.
• Work done in 3 days = Combined rate $ \times $ time = $ \frac{5}{36} \times 3 = \frac{15}{36} $
• Simplify the fraction: $ \frac{15}{36} = \frac{5}{12} $ of the work was completed.

Remaining Work Calculation

Determine the portion of the work left to be done.

• Total work = 1 (representing the whole work)
• Remaining work = Total work - Work done = $ 1 - \frac{5}{12} = \frac{12}{12} - \frac{5}{12} = \frac{7}{12} $
• $ \frac{7}{12} $ of the work remains.

Time for B to Complete Remaining Work

Calculate how many days B needs to finish the remaining work alone.

• B's rate = $ \frac{1}{18} $ of the work per day.
• Time = Remaining work / B's rate
• Time = $ \frac{7/12}{1/18} $
• Time = $ \frac{7}{12} \times 18 $
• Time = $ \frac{7 \times 3}{2} = \frac{21}{2} = 10.5 $ days.

B will take 10.5 days to complete the remaining work.