A and B can do a job in 15 days and 10 days, respectively. They began the work together but A leaves after some days and B finished the remaining job in 5 days. After how many days did A leave?

Options

A. 2 days
B. 3 days
C. 1 day
D. None of these

Answer

To solve the problem step by step, let's break it down:

### Step 1: Determine the work done by A and B in one day.
- A can complete the job in 15 days. Therefore, the work done by A in one day = \( \frac{1}{15} \) of the job.
- B can complete the job in 10 days. Therefore, the work done by B in one day = \( \frac{1}{10} \) of the job.

### Step 2: Calculate the combined work done by A and B in one day.
- Combined work done by A and B in one day = \( \frac{1}{15} + \frac{1}{10} \).
- To add these fractions, we find a common denominator, which is 30.
- \( \frac{1}{15} = \frac{2}{30} \)
- \( \frac{1}{10} = \frac{3}{30} \)
- Therefore, combined work = \( \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6} \) of the job.

### Step 3: Determine how much work B does alone in 5 days.
- Since B finishes the remaining work in 5 days, the work done by B in 5 days = \( 5 \times \frac{1}{10} = \frac{5}{10} = \frac{1}{2} \) of the job.

### Step 4: Calculate the total work completed before B finishes the job.
- If B completed \( \frac{1}{2} \) of the job in 5 days, then the remaining work before B worked alone was also \( \frac{1}{2} \) of the job.

### Step 5: Determine how many days A and B worked together to complete half the job.
- Since A and B together can complete \( \frac{1}{6} \) of the job in one day, we need to find out how many days (let's call it \( x \)) they worked together to complete \( \frac{1}{2} \) of the job.
- The equation will be:
\[ x \times \frac{1}{6} = \frac{1}{2} \]
- Solving for \( x \):
\[ x = \frac{1/2}{1/6} = \frac{1}{2} \times \frac{6}{1} = 3 \]

### Conclusion:
- A and B worked together for 3 days before A left.

### Final Answer:
A left after 3 days.

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