Explanation: In work and time problems, the efficiency of individuals or groups is often expressed as the fraction of work done per unit time. If A and B together can complete a piece of work in 8 days, their combined efficiency is 1/8 of the work per day. This means that together, A and B can complete 1/8 of the work in one day.
If B alone can complete the work in 12 days, B's efficiency is 1/12 of the work per day. This means that B can complete 1/12 of the work in one day.
To find A's efficiency, we subtract B's efficiency from the combined efficiency:
\[ \text{A's efficiency} = \text{Combined efficiency} - \text{B's efficiency} \]
\[ \text{A's efficiency} = \frac{1}{8} - \frac{1}{12} \]
To perform the subtraction, we need a common denominator. The least common multiple of 8 and 12 is 24. Therefore, we convert the fractions:
\[ \frac{1}{8} = \frac{3}{24} \]
\[ \frac{1}{12} = \frac{2}{24} \]
Now we can subtract the fractions:
\[ \text{A's efficiency} = \frac{3}{24} - \frac{2}{24} = \frac{1}{24} \]
This means A's efficiency is 1/24 of the work per day. Therefore, A alone can complete the work in 24 days.
The correct answer is (A) 24 days.
This problem tests the understanding of work and time concepts, specifically the calculation of individual efficiencies based on combined efficiencies. It is important to correctly apply the subtraction of efficiencies and convert the efficiency into the number of days required to complete the work. Common mistakes include incorrect addition of efficiencies or confusion between combined and individual efficiencies.