A can finish a work in 24 days, B in 9 days and C in 12 days. B and C start the work but they are forced to leave after 3 days. The remaining work was done by A in

Options

A. 10 days
B. `10 (1)/(2) days`
C. `5 days`
D. ` 6 days`

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Determine the total work
We need to find the least common multiple (LCM) of the days taken by A, B, and C to finish the work.
- A can finish the work in 24 days.
- B can finish the work in 9 days.
- C can finish the work in 12 days.

**Calculation of LCM:**
- The LCM of 24, 9, and 12 is 72.

**Total work = 72 units.**

### Step 2: Calculate the efficiency of A, B, and C
Now we will calculate how much work each person can do in one day (their efficiency).

- **Efficiency of A:**
\[ \text{Efficiency of A} = \frac{72 \text{ units}}{24 \text{ days}} = 3 \text{ units/day} \]

- **Efficiency of B:**
\[ \text{Efficiency of B} = \frac{72 \text{ units}}{9 \text{ days}} = 8 \text{ units/day} \]

- **Efficiency of C:**
\[ \text{Efficiency of C} = \frac{72 \text{ units}}{12 \text{ days}} = 6 \text{ units/day} \]

### Step 3: Calculate the work done by B and C in 3 days
B and C work together for 3 days. We need to find the total work done by both in that time.

- **Total efficiency of B and C:**
\[ \text{Total efficiency} = \text{Efficiency of B} + \text{Efficiency of C} = 8 + 6 = 14 \text{ units/day} \]

- **Work done in 3 days:**
\[ \text{Work done} = \text{Total efficiency} \times \text{Time} = 14 \times 3 = 42 \text{ units} \]

### Step 4: Calculate the remaining work
Now we will find out how much work is left after B and C have worked for 3 days.

- **Remaining work:**
\[ \text{Remaining work} = \text{Total work} - \text{Work done by B and C} = 72 - 42 = 30 \text{ units} \]

### Step 5: Calculate the time taken by A to finish the remaining work
Now we need to determine how long it will take A to complete the remaining 30 units of work.

- **Time taken by A:**
\[ \text{Time} = \frac{\text{Remaining work}}{\text{Efficiency of A}} = \frac{30 \text{ units}}{3 \text{ units/day}} = 10 \text{ days} \]

### Final Answer
The remaining work was done by A in **10 days**.

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