A and B can do a work in 30 days and 10 days respectively. If they work on alternate days beginning with A, in how many days will the work be completed?

Answer

To solve the problem step by step, we will first determine the work efficiencies of A and B, then calculate how much work they complete in a cycle of two days, and finally determine how many days it will take to complete the work.

### Step-by-Step Solution:

1. **Determine the Work Efficiency of A and B:**
- A can complete the work in 30 days. Therefore, A's work efficiency is:
\[ \text{Efficiency of A} = \frac{1 \text{ unit of work}}{30 \text{ days}} = \frac{1}{30} \text{ units per day} \]
- B can complete the work in 10 days. Therefore, B's work efficiency is:
\[ \text{Efficiency of B} = \frac{1 \text{ unit of work}}{10 \text{ days}} = \frac{1}{10} \text{ units per day} \]

2. **Calculate the Total Work:**
- We can assume the total work to be the least common multiple (LCM) of the days taken by A and B to complete the work. The LCM of 30 and 10 is 30 units of work.

3. **Calculate the Work Done in Two Days:**
- In the first day, A works and completes:
\[ \text{Work done by A in 1 day} = \frac{1}{30} \text{ units} \]
- In the second day, B works and completes:
\[ \text{Work done by B in 1 day} = \frac{1}{10} \text{ units} \]
- Therefore, in two days, the total work done is:
\[ \text{Total work in 2 days} = \frac{1}{30} + \frac{1}{10} = \frac{1}{30} + \frac{3}{30} = \frac{4}{30} = \frac{2}{15} \text{ units} \]

4. **Calculate the Number of Two-Day Cycles to Complete the Work:**
- To find out how many two-day cycles are needed to complete 30 units of work, we set up the equation:
\[ \text{Number of cycles} = \frac{30 \text{ units}}{\frac{2}{15} \text{ units per 2 days}} = 30 \times \frac{15}{2} = 225 \text{ cycles} \]
- This means it takes 225 two-day cycles to complete the work.

5. **Calculate the Total Days Taken:**
- Since each cycle takes 2 days, the total number of days taken to complete the work is:
\[ \text{Total days} = 225 \times 2 = 450 \text{ days} \]

6. **Final Adjustment for Remaining Work:**
- After 225 cycles, A will start again on the 451st day. Since they only need to complete 1 more unit of work, A will complete this in:
\[ \text{Days needed} = 1 \text{ unit} \times 30 \text{ days/unit} = \frac{1}{30} \text{ days} \]
- Therefore, the total time taken is:
\[ 450 + \frac{1}{30} \text{ days} \]

### Final Answer:
The total time taken to complete the work is approximately **450.03 days**.