A and B together can complete a work in 45 days and 40 days, respectively. B alone completed the remaining work in 23 days. After how many days did A leave?

Question

Accepted Answer

### Step-by-Step Solution:

1. **Total Work Calculation:**  
   The total work is the least common multiple (LCM) of the time taken by A and B to complete the work individually:  
   $$ 	ext{Total Work} = 	ext{LCM}(45, 40) = 360 \, 	ext{units}. $$

2. **Efficiency of A and B:**  
   - A's efficiency = $$ \frac{	ext{Total Work}}{	ext{Time}} = \frac{360}{45} = 8 \, 	ext{units/day}. $$  
   - B's efficiency = $$ \frac{	ext{Total Work}}{	ext{Time}} = \frac{360}{40} = 9 \, 	ext{units/day}. $$

3. **Work Done by B in 23 Days:**  
   B alone worked for 23 days:  
   $$ 	ext{Work done by B} = 23 	imes 9 = 207 \, 	ext{units}. $$

4. **Remaining Work to be Done by A and B Together:**  
   Remaining work:  
   $$ 	ext{Remaining Work} = 	ext{Total Work} - 	ext{Work done by B} = 360 - 207 = 153 \, 	ext{units}. $$

5. **Days Taken by A and B Together to Complete Remaining Work:**  
   A and B together complete:  
   $$ (8 + 9) = 17 \, 	ext{units/day}. $$  
   Time taken to complete the remaining work:  
   $$ \frac{	ext{Remaining Work}}{	ext{Combined Efficiency}} = \frac{153}{17} = 9 \, 	ext{days}. $$

Thus, A worked for **9 days** before leaving.

### Final Answer:  
The correct answer is **(d) 9**.

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Answer

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