Step 1: Understanding the Concept:
Time and Work problems are based on the inverse relationship between the time taken to complete a task and the efficiency of the worker.
To simplify these problems, we assume a "Total Work" unit. The most convenient number for Total Work is the Least Common Multiple (LCM) of the days given.
By converting the days into "Units per Day" (Efficiency), we can easily sum the work done by multiple people and calculate the remaining work after someone leaves.
Key Formula or Approach:
1. Total Work = \(LCM(\text{individual times})\)
2. Efficiency = \(\frac{\text{Total Work}}{\text{Time Taken}}\)
3. Time = \(\frac{\text{Remaining Work}}{\text{Efficiency}}\)
Step 2: Detailed Explanation:
Let's define the total work by taking the LCM of 12 and 15:
\[ \text{Total Work} = 60 \text{ units} \]
Now, find the individual efficiencies:
Efficiency of A = \(60 / 12 = 5\) units/day.
Efficiency of B = \(60 / 15 = 4\) units/day.
When working together, their combined efficiency is:
\[ \text{Combined Efficiency} = 5 + 4 = 9 \text{ units/day} \]
They work together for 5 days. We calculate the work completed in this period:
\[ \text{Work done in 5 days} = 9 \times 5 = 45 \text{ units} \]
Now, find the amount of work still remaining after A leaves the job:
\[ \text{Remaining Work} = 60 - 45 = 15 \text{ units} \]
B must finish these 15 units alone. Since B's efficiency is 4 units/day:
\[ \text{Time taken by B} = \frac{15}{4} \]
\[ 15 \div 4 = 3.75 \text{ days} \]
Step 3: Final Answer:
B will take 3.75 more days to complete the remaining work. Thus, Option (A) is the correct choice.