Explanations:In work rate problems, the key concept is to determine the rate at which each individual completes a task and then combine these rates to find the total rate when they work together. The work rate is typically expressed as a fraction of the work done per unit time. For example, if A can complete a task in 10 days, A's work rate is 1/10 of the work per day. Similarly, if B can complete the same task in 30 days, B's work rate is 1/30 of the work per day.
To find the combined work rate of A and B, we add their individual work rates. The combined work rate is:
\[
\text{Combined work rate} = \frac{1}{10} + \frac{1}{30}
\]
To add these fractions, we need a common denominator. The least common multiple of 10 and 30 is 30. Therefore, we can rewrite the fractions with a common denominator:
\[
\frac{1}{10} = \frac{3}{30} \quad \text{and} \quad \frac{1}{30} = \frac{1}{30}
\]
Adding these fractions, we get:
\[
\frac{3}{30} + \frac{1}{30} = \frac{4}{30} = \frac{2}{15}
\]
The combined work rate 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 work rate:
\[
\text{Time taken} = \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{2}{15}} = \frac{15}{2} = 7.5 \text{ days}
\]
Thus, 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 work rate is not simply the sum of the individual times taken by A and B. Instead, it is the sum of their work rates, which are fractions of the work done per unit time. The time taken to complete the work together is the reciprocal of the combined work rate.