A and B can do a piece of work in 28 and 35 days respectively. They began to work together but A leaves after some time and B completed the remaining work in 17 days. After how many days did A leave ?

Options

A. `14""2/5` days
B. `9 days `
C. `8 days `
D. `7""5/9` days

Answer

To solve the problem, we need to find out how many days A worked before leaving. Let's break down the solution step by step.

### Step 1: Determine the work rates of A and B
- A can complete the work in 28 days, so A's work rate is \( \frac{1}{28} \) of the work per day.
- B can complete the work in 35 days, so B's work rate is \( \frac{1}{35} \) of the work per day.

### Step 2: Calculate the combined work rate of A and B
When A and B work together, their combined work rate is:
\[ \text{Combined work rate} = \frac{1}{28} + \frac{1}{35} \]

To add these fractions, we need a common denominator. The least common multiple of 28 and 35 is 140.

Converting the fractions:
\[ \frac{1}{28} = \frac{5}{140} \quad \text{and} \quad \frac{1}{35} = \frac{4}{140} \]
Thus,
\[ \text{Combined work rate} = \frac{5}{140} + \frac{4}{140} = \frac{9}{140} \]

### Step 3: Set up the equation for the work done
Let \( x \) be the number of days A worked before leaving. The work done by A and B together in \( x \) days is:
\[ \text{Work done by A and B in } x \text{ days} = x \times \frac{9}{140} \]

### Step 4: Calculate the work done by B after A leaves
After A leaves, B works alone for 17 days. The work done by B in 17 days is:
\[ \text{Work done by B in 17 days} = 17 \times \frac{1}{35} = \frac{17}{35} \]

### Step 5: Total work equation
The total work is equal to 1 (the whole work). Therefore, we can set up the equation:
\[ x \times \frac{9}{140} + \frac{17}{35} = 1 \]

### Step 6: Solve for \( x \)
First, convert \( \frac{17}{35} \) to a fraction with a denominator of 140:
\[ \frac{17}{35} = \frac{68}{140} \]

Now, substitute this back into the equation:
\[ x \times \frac{9}{140} + \frac{68}{140} = 1 \]
Multiply through by 140 to eliminate the denominator:
\[ 9x + 68 = 140 \]
Now, solve for \( x \):
\[ 9x = 140 - 68 \]
\[ 9x = 72 \]
\[ x = \frac{72}{9} = 8 \]

### Conclusion
A worked for 8 days before leaving.