A can do a work in 45 days and B can do it in 40 days. Both starts the work together and after some days Aleaves the work. If the work is completed in 32 days, then for how many days A worked?

Options

A. 9 days
B. 8 days
C. 10 days
D. 12 days

Answer

To solve the problem step by step, we will follow the given information about A and B's work rates and the total time taken to complete the work.

### Step 1: Determine the work rates of A and B
- A can complete the work in 45 days, so A's work rate (efficiency) is:
\[ \text{Efficiency of A} = \frac{1 \text{ work}}{45 \text{ days}} = \frac{1}{45} \text{ work/day} \]

- B can complete the work in 40 days, so B's work rate (efficiency) is:
\[ \text{Efficiency of B} = \frac{1 \text{ work}}{40 \text{ days}} = \frac{1}{40} \text{ work/day} \]

### Step 2: Find the LCM of the days to determine the total work
- The LCM of 45 and 40 is 360. We can consider the total work as 360 units.

### Step 3: Calculate the efficiencies in terms of work units
- A's efficiency in units of work is:
\[ \text{Efficiency of A} = \frac{360 \text{ units}}{45 \text{ days}} = 8 \text{ units/day} \]
- B's efficiency in units of work is:
\[ \text{Efficiency of B} = \frac{360 \text{ units}}{40 \text{ days}} = 9 \text{ units/day} \]

### Step 4: Calculate the total work done by B in 32 days
- Since both A and B start the work together and the total work is completed in 32 days, we can calculate the work done by B:
\[ \text{Work done by B} = \text{Efficiency of B} \times \text{Time} = 9 \text{ units/day} \times 32 \text{ days} = 288 \text{ units} \]

### Step 5: Determine the remaining work
- The total work is 360 units, so the remaining work that A must complete is:
\[ \text{Remaining work} = 360 \text{ units} - 288 \text{ units} = 72 \text{ units} \]

### Step 6: Calculate the time taken by A to complete the remaining work
- A's efficiency is 8 units/day, so the time taken by A to complete the remaining 72 units of work is:
\[ \text{Time taken by A} = \frac{\text{Remaining work}}{\text{Efficiency of A}} = \frac{72 \text{ units}}{8 \text{ units/day}} = 9 \text{ days} \]

### Step 7: Conclusion
- Since A worked for 9 days before leaving the work, we conclude that A worked for **9 days**.

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