Explanation: To solve this problem, we need to determine the work rates of A and B and then find the combined work rate to determine how long it would take for them to complete the work together.
First, let's calculate the work rate of A. Since A can complete the work in 30 days, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{30} \text{ of the work per day} \]
A works for 10 days, so the amount of work A completes is:
\[ \text{Work done by A in 10 days} = 10 \times \frac{1}{30} = \frac{10}{30} = \frac{1}{3} \]
This means that \(\frac{2}{3}\) of the work remains after A leaves. B completes the remaining \(\frac{2}{3}\) of the work in 16 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{\frac{2}{3}}{16} = \frac{2}{3 \times 16} = \frac{2}{48} = \frac{1}{24} \text{ of the work per day} \]
Now, we need to find the combined work rate of A and B. The combined work rate is the sum of their individual work rates:
\[ \text{Combined work rate} = \frac{1}{30} + \frac{1}{24} \]
To add these fractions, we need a common denominator. The least common multiple of 30 and 24 is 120. So, we convert the fractions:
\[ \frac{1}{30} = \frac{4}{120} \]
\[ \frac{1}{24} = \frac{5}{120} \]
Adding these fractions gives:
\[ \text{Combined work rate} = \frac{4}{120} + \frac{5}{120} = \frac{9}{120} = \frac{3}{40} \]
The combined work rate of A and B is \(\frac{3}{40}\) of the work per day. To find the total time taken to complete the work together, we take the reciprocal of the combined work rate:
\[ \text{Total time} = \frac{1}{\frac{3}{40}} = \frac{40}{3} = 13 \frac{1}{3} \text{ days} \]
Therefore, A and B together can complete the work in 13 \(\frac{1}{3}\) days, which is the correct answer.