Let's solve this step by step:

Step 1: Determine the work rates of A and B

• A can complete $\frac{1}{4}$ of the work in 10 days.

  • Work rate of A = $\frac{\frac{1}{4}}{10} = \frac{1}{40}$ (A completes $\frac{1}{40}$ of the work per day).

• B can complete $\frac{1}{3}$ of the work in 20 days.

  • Work rate of B = $\frac{\frac{1}{3}}{20} = \frac{1}{60}$ (B completes $\frac{1}{60}$ of the work per day).

Step 2: Combined work rate of A and B

• Combined work rate = $\frac{1}{40} + \frac{1}{60}$.
• To add these fractions, find the least common denominator (LCD):
  • LCD of 40 and 60 = 120.
  • $\frac{1}{40} = \frac{3}{120}$ and $\frac{1}{60} = \frac{2}{120}$.
  • Combined rate = $\frac{3}{120} + \frac{2}{120} = \frac{5}{120}$.

Step 3: Time taken to complete the entire work

• Combined rate = $\frac{5}{120} = \frac{1}{24}$ of the work per day.
• Time to complete 1 unit of work = $\frac{1}{\frac{1}{24}} = 24$ days.

Final Answer:

The total time taken to complete the work together is 24 days.

Let me know if you'd like further clarification or have additional questions!

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Related Questions:

1. What if A and B worked separately? How long would each take to finish the work?
2. If a third person, C, joined and completed $\frac{1}{5}$ of the work in 15 days, how would this change the total time?
3. How would the time change if A and B worked at double their rates?
4. What is the ratio of the work completed by A and B in one day?
5. How can this concept of combined work rates be applied to real-world scenarios?

Tip: Always calculate work rates individually before combining them. It simplifies the overall calculation!