Explanation: In work and time problems, the efficiency of an individual is the reciprocal of the time they take to complete a task. For example, if A can complete a piece of work in 10 days, A's efficiency is 1/10 of the work per day. Similarly, if B can complete the same piece of work in 30 days, B's efficiency is 1/30 of the work per day.
When A and B work together, their combined efficiency is the sum of their individual efficiencies. Therefore, the combined efficiency of A and B is:
\[ \text{Combined Efficiency} = \frac{1}{10} + \frac{1}{30} \]
To add these fractions, we need a common denominator. The least common multiple of 10 and 30 is 30. So, we convert the fractions:
\[ \frac{1}{10} = \frac{3}{30} \]
\[ \frac{1}{30} = \frac{1}{30} \]
Adding these fractions gives:
\[ \text{Combined Efficiency} = \frac{3}{30} + \frac{1}{30} = \frac{4}{30} = \frac{2}{15} \]
The combined efficiency of A and B is 2/15 of the work per day. To find the time taken to complete the work together, we take the reciprocal of the combined efficiency:
\[ \text{Time Taken} = \frac{1}{\text{Combined Efficiency}} = \frac{1}{\frac{2}{15}} = \frac{15}{2} = 7.5 \text{ days} \]
Therefore, A and B together can complete the work in 7.5 days. This is the correct answer.
It is important to note that the combined efficiency is not the sum of the times taken by A and B individually. Instead, it is the sum of their efficiencies, which are the reciprocals of their individual times. This is a common misconception that students often make, leading to incorrect answers.
In summary, the key concepts to remember are:
1. Efficiency is the reciprocal of the time taken to complete a task.
2. Combined efficiency is the sum of individual efficiencies.
3. The time taken to complete the work together is the reciprocal of the combined efficiency.
By understanding these concepts, students can solve similar work and time problems accurately and efficiently.