A and B can do a piece of work in 45 and 40 days, respectively. They began the work together but A left after some days and B finished the remaining work in 23 days. After how many days did A leave?

Correct option is C

Given:

A can do the work in 45 days => A's 1 day work = $$\frac{1}{45}$$​

B can do the work in 40 days => B's 1 day work = $$\frac{1}{40}$$​​
They worked together for some days. After that, B alone worked for 23 days.
We are to find the number of days after which A left.

Formula Used:

Work done by two persons working together:

​$$\text{Combined work per day} = \left(\frac{1}{\text{A's time}} + \frac{1}{\text{B's time}}\right)$$​​

Solution:

​$$\text{Work by A and B in } x \text{ days} = x\left(\frac{1}{45} + \frac{1}{40}\right)$$​​

​$$\text{Work by B in 23 days} = 23 \times \frac{1}{40}$$​​

​$$\text{Total work} = x\left(\frac{1}{45} + \frac{1}{40}\right) + \frac{23}{40} = 1$$​​

​$$\frac{1}{45} + \frac{1}{40} = \frac{8 + 9}{360} = \frac{17}{360}$$​​

Let A worked for x days. Then,

​$$x\cdot \frac{17}{360} + \frac{23}{40} = 1$$​​

$$x\cdot 17 + \frac{23}{40} \cdot 360 = 360 \\ \ \\ 17 x + 207 =360\\ \ \\x= \frac{153}{17} = 9$$​​

Thus, the correct option is (c) 9 days

Alternate Method:

Total time = $$\frac{\text{Total work}}{\text{Work Done Per day}}$$   

LCM (45, 40) = 360

Then Total work = 360

A's Efficiency = $$\frac{360}{45}$$ = 8 Units

B's Efficiency = $$\frac{360}{40}$$ = 9 Units

Let, A left after t days and B finished in remaining days:

8t + 9 (t + 23) =360

8 t + 9 t + 207 = 360

17 t  = 153

t = $$\frac{153}{17}$$

t = 9 days