A and B can do a piece of work together in 9 days while A alone can do it in 15 days. They start working together but B leaves 3 days before the completion of the work. For how many days did A and B work together?
A)`7.2`
B)`7.5`
C)`8.1`
D)8

Options

A. `7.2`
B. `7.5`
C. `8.1`
D. 8

Answer

To solve the problem step by step, we will first determine the work rates of A and B, then calculate how long they worked together before B left.

### Step 1: Determine the work rates of A and B
- A can complete the work in 15 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{15} \text{ (work per day)} \]

- A and B together can complete the work in 9 days. Therefore, their combined work rate is:
\[ \text{Work rate of A and B} = \frac{1}{9} \text{ (work per day)} \]

### Step 2: Calculate B's work rate
- Since A's work rate is \(\frac{1}{15}\) and A and B's combined work rate is \(\frac{1}{9}\), we can find B's work rate by subtracting A's work rate from their combined work rate:
\[ \text{Work rate of B} = \text{Work rate of A and B} - \text{Work rate of A} \]
\[ \text{Work rate of B} = \frac{1}{9} - \frac{1}{15} \]

- To perform this subtraction, we need a common denominator, which is 45:
\[ \frac{1}{9} = \frac{5}{45}, \quad \frac{1}{15} = \frac{3}{45} \]
\[ \text{Work rate of B} = \frac{5}{45} - \frac{3}{45} = \frac{2}{45} \text{ (work per day)} \]

### Step 3: Calculate the total work
- The total work can be calculated using A's work rate:
\[ \text{Total work} = \text{Work rate of A} \times \text{Total days A works alone} = 15 \text{ (days)} \times 1 \text{ (unit of work)} = 1 \text{ unit of work} \]

### Step 4: Determine how much work is left when B leaves
- B leaves 3 days before the work is completed. Let's denote the total time taken to complete the work as \( T \). Therefore, A and B worked together for \( T - 3 \) days.

- The total work done by A and B together in \( T - 3 \) days is:
\[ \text{Work done by A and B} = (T - 3) \times \left(\frac{1}{9}\right) \]

- The work done by A alone in the remaining 3 days is:
\[ \text{Work done by A} = 3 \times \left(\frac{1}{15}\right) \]

### Step 5: Set up the equation for total work
- The total work equation is:
\[ (T - 3) \times \left(\frac{1}{9}\right) + 3 \times \left(\frac{1}{15}\right) = 1 \]

### Step 6: Solve the equation
- Substituting the values:
\[ \frac{T - 3}{9} + \frac{3}{15} = 1 \]
- Simplifying \(\frac{3}{15}\) to \(\frac{1}{5}\):
\[ \frac{T - 3}{9} + \frac{1}{5} = 1 \]

- Finding a common denominator (45):
\[ \frac{5(T - 3)}{45} + \frac{9}{45} = 1 \]
\[ \frac{5T - 15 + 9}{45} = 1 \]
\[ 5T - 6 = 45 \]
\[ 5T = 51 \implies T = \frac{51}{5} = 10.2 \text{ days} \]

### Step 7: Calculate how many days A and B worked together
- Since B left 3 days before the work was completed:
\[ \text{Days A and B worked together} = T - 3 = 10.2 - 3 = 7.2 \text{ days} \]

### Final Answer
Thus, A and B worked together for **7.2 days**.