Explanations:In work and time problems, the work rate of an individual is the fraction of the work they can complete in one day. The combined work rate of multiple individuals is the sum of their individual work rates. The total time taken to complete the work together is the reciprocal of the combined work rate.
Let's break down the problem step-by-step:
1. **Determine the work rate of A:**
- A can complete the work in 30 days.
- Therefore, A's work rate is \( \frac{1}{30} \) of the work per day.
2. **Calculate the work done by A in 10 days:**
- In 10 days, A completes \( 10 \times \frac{1}{30} = \frac{10}{30} = \frac{1}{3} \) of the work.
3. **Determine the remaining work:**
- The remaining work is \( 1 - \frac{1}{3} = \frac{2}{3} \) of the work.
4. **Determine the work rate of B:**
- B completes the remaining \( \frac{2}{3} \) of the work in 16 days.
- Therefore, B's work rate is \( \frac{\frac{2}{3}}{16} = \frac{2}{3 \times 16} = \frac{2}{48} = \frac{1}{24} \) of the work per day.
5. **Calculate the combined work rate of A and B:**
- The combined work rate is \( \frac{1}{30} + \frac{1}{24} \).
- To add these fractions, find a common denominator, which is 120.
- \( \frac{1}{30} = \frac{4}{120} \) and \( \frac{1}{24} = \frac{5}{120} \).
- Therefore, the combined work rate is \( \frac{4}{120} + \frac{5}{120} = \frac{9}{120} = \frac{3}{40} \) of the work per day.
6. **Determine the total time taken to complete the work together:**
- The total time taken is the reciprocal of the combined work rate.
- \( \frac{1}{\frac{3}{40}} = \frac{40}{3} = 13 \frac{1}{3} \) days.
Thus, A and B together can complete the work in \( 13 \frac{1}{3} \) days, which corresponds to option B.