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 analyze the work done by A and B, who work on alternate days, starting with A.

### Step 1: Determine the work done by A and B in one day.
- A can complete the work in 30 days. Therefore, A's work in one day is:
\[ \text{Work done by A in one day} = \frac{1}{30} \text{ of the work} \]

- B can complete the work in 10 days. Therefore, B's work in one day is:
\[ \text{Work done by B in one day} = \frac{1}{10} \text{ of the work} \]

### Step 2: Calculate the total work done in two days (one cycle of A and B).
- In the first day, A works:
\[ \text{Work done by A} = \frac{1}{30} \]

- In the second day, B works:
\[ \text{Work done by B} = \frac{1}{10} \]

- Therefore, the total work done in two days is:
\[ \text{Total work in 2 days} = \frac{1}{30} + \frac{1}{10} \]

To add these fractions, we need a common denominator (which is 30):
\[ \frac{1}{10} = \frac{3}{30} \]
So,
\[ \text{Total work in 2 days} = \frac{1}{30} + \frac{3}{30} = \frac{4}{30} = \frac{2}{15} \]

### Step 3: Determine how many complete cycles are needed to finish the work.
- The total work is considered as 1 unit. To find out how many cycles (2 days) are needed to complete the work, we can set up the equation:
\[ \text{Number of cycles} = \frac{1 \text{ unit of work}}{\frac{2}{15} \text{ unit of work per cycle}} = \frac{15}{2} = 7.5 \text{ cycles} \]

### Step 4: Calculate the total days taken.
- Each cycle takes 2 days, so:
\[ \text{Total days for 7 complete cycles} = 7 \times 2 = 14 \text{ days} \]

- In 14 days, the work done is:
\[ \text{Work done in 14 days} = 7 \times \frac{2}{15} = \frac{14}{15} \]

### Step 5: Calculate the remaining work and the last day.
- The remaining work after 14 days is:
\[ \text{Remaining work} = 1 - \frac{14}{15} = \frac{1}{15} \]

- On the 15th day, A will work again (since they alternate days). A can complete:
\[ \text{Work done by A on 15th day} = \frac{1}{30} \]
- However, A can only complete \(\frac{1}{15}\) of the work, which is less than \(\frac{1}{30}\). Thus, A will finish the remaining work.

### Step 6: Total days to complete the work.
- The total days taken to complete the work is:
\[ \text{Total days} = 14 + 1 = 15 \text{ days} \]

### Final Answer:
The work will be completed in **15 days**.

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