A and B can do a piece of work in 20 and 25 days respectively. They began to work together but A leaves after sometime and B completed 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. `3(5)/(9)` days

Answer

To solve the problem, we need to find out how many days A worked before leaving, given that A and B can complete a piece of work in 20 and 25 days respectively, and B completed the remaining work in 17 days after A left.

### Step-by-Step Solution:

1. **Calculate the work done by A and B in one day:**
- A's work rate = 1/20 (A can complete 1/20 of the work in one day)
- B's work rate = 1/25 (B can complete 1/25 of the work in one day)

2. **Find the combined work rate of A and B:**
- Combined work rate = A's work rate + B's work rate
- Combined work rate = 1/20 + 1/25
- To add these fractions, we need a common denominator. The least common multiple of 20 and 25 is 100.
- Convert the fractions:
- 1/20 = 5/100
- 1/25 = 4/100
- Combined work rate = 5/100 + 4/100 = 9/100

3. **Determine the total work done by B in 17 days:**
- Work done by B in 17 days = B's work rate × 17
- Work done by B = (1/25) × 17 = 17/25

4. **Calculate the total work completed:**
- Since the total work is considered as 1 (or 100%), the work completed by A and B together before A left can be calculated as:
- Total work = Work completed by A and B together + Work completed by B alone
- Let the work completed by A and B together before A left be W.
- Therefore, W + 17/25 = 1
- W = 1 - 17/25 = 8/25

5. **Calculate the time A and B worked together:**
- Let the number of days A and B worked together be x.
- The work done by A and B together in x days = (Combined work rate) × x
- (9/100) × x = 8/25
- To solve for x, we can multiply both sides by 100 to eliminate the fraction:
- 9x = (8/25) × 100
- 9x = 32
- x = 32/9
- x ≈ 3.56 days

### Conclusion:
A worked together with B for approximately 3.56 days before leaving.