Step 1: Understanding the Concept:
This is a standard Time and Work problem. The fundamental principle is that the amount of work done is the product of time and the rate of work (efficiency).
The time taken to complete a job is inversely proportional to the efficiency.
When two people work together, their combined efficiency is the sum of their individual efficiencies.
Key Formula or Approach:
If A can do a work in \( x \) days and B in \( y \) days, the time taken for them to finish the work together is:
\[ \text{Time Taken} = \frac{x \times y}{x + y} \text{ days} \]
Alternatively, you can add their 1-day work capacities: \( \frac{1}{x} + \frac{1}{y} = \frac{1}{\text{Together Time}} \).
Step 2: Detailed Explanation:
Given:
Time taken by A = 12 days
Time taken by B = 18 days
Let's use the fractional approach (1-day work):
Work done by A in 1 day = \( \frac{1}{12} \) of the total work.
Work done by B in 1 day = \( \frac{1}{18} \) of the total work.
Total work done by (A + B) in 1 day = \( \frac{1}{12} + \frac{1}{18} \)
To add these fractions, find the LCM of 12 and 18, which is 36:
\[ \text{Combined daily work} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36} \]
Since they do \( \frac{5}{36} \) of the work in one day, the total time to complete the full work is the reciprocal of this value:
\[ \text{Total Time} = \frac{36}{5} \text{ days} \]
Performing the division:
\[ 36 \div 5 = 7.2 \]
So, they will complete the work in 7.2 days together.
Alternatively, using the direct formula:
\[ \text{Time} = \frac{12 \times 18}{12 + 18} = \frac{216}{30} = 7.2 \text{ days.} \]
Step 3: Final Answer:
Working together, A and B will complete the task in 7.2 days.
This corresponds to Option (B).