A and B can complete a task together in 15 days, while A alone can complete it in 18.75 days. They start working together but A leaves 12.5 days before the completion of the work. For how many days did A and B work together ?

Options

A. 10
B. 13.75
C. 12.5
D. 11.25

Answer

To solve the problem step by step, we need to determine how many days A and B worked together before A left 12.5 days before the work was completed.

### Step 1: Determine the work done by A and B together.
A and B can complete the task together in 15 days. Therefore, the work done by A and B in one day is:
\[ \text{Work rate of A and B together} = \frac{1}{15} \text{ of the work per day} \]

### Step 2: Determine the work done by A alone.
A can complete the task alone in 18.75 days. Therefore, the work done by A in one day is:
\[ \text{Work rate of A} = \frac{1}{18.75} = \frac{4}{75} \text{ of the work per day} \]

### Step 3: Determine the work done by B alone.
Let the work done by B in one day be \( b \). Since A and B together can complete the work in 15 days, we have:
\[ \frac{1}{15} = \frac{4}{75} + b \]
To find \( b \), we first convert \(\frac{1}{15}\) to a fraction with a common denominator:
\[ \frac{1}{15} = \frac{5}{75} \]
Now substituting:
\[ \frac{5}{75} = \frac{4}{75} + b \]
Subtracting \(\frac{4}{75}\) from both sides gives:
\[ b = \frac{5}{75} - \frac{4}{75} = \frac{1}{75} \text{ of the work per day} \]

### Step 4: Determine the total work.
The total work can be considered as 1 unit of work.

### Step 5: Determine how much work is left when A leaves.
A leaves 12.5 days before the work is completed. Let \( x \) be the total number of days they worked together. Therefore, the total time to complete the work is \( x + 12.5 \) days. The work done in the last 12.5 days by B alone is:
\[ \text{Work done by B in 12.5 days} = 12.5 \times \frac{1}{75} = \frac{12.5}{75} = \frac{1}{6} \text{ of the work} \]

### Step 6: Calculate the work done together.
If B completed \(\frac{1}{6}\) of the work alone, then the work done together by A and B is:
\[ \text{Work done together} = 1 - \frac{1}{6} = \frac{5}{6} \text{ of the work} \]

### Step 7: Calculate the number of days they worked together.
Since A and B together can complete \(\frac{1}{15}\) of the work per day, we can find the number of days they worked together:
\[ \text{Days worked together} = \frac{\frac{5}{6}}{\frac{1}{15}} = \frac{5}{6} \times 15 = \frac{75}{6} = 12.5 \text{ days} \]

### Conclusion:
A and B worked together for **12.5 days**.