A can do a work in 8 days and B can do it in 12 days. If A, B and C can do it in 4 days together then in how many days C can do it alone?

Options

A. 1. 16
B. 2. 24
C. 3. 18
D. 4. 25

Answer

To solve the problem step by step, we need to find out how many days C can do the work alone, given the efficiencies of A, B, and C.

### Step 1: Determine the total work
We know that:
- A can complete the work in 8 days.
- B can complete the work in 12 days.
- A, B, and C together can complete the work in 4 days.

To find the total work, we can take the Least Common Multiple (LCM) of the days taken by A, B, and the time they take together.

The LCM of 8, 12, and 4 is 24. Thus, we can consider the total work to be 24 units.

**Hint:** Use the LCM to find a common measure of work that can be divided by the days taken by A, B, and C.

### Step 2: Calculate the efficiency of A and B
Next, we calculate how much work A and B can do in one day:
- A's efficiency = Total work / Days taken by A = 24 units / 8 days = 3 units/day.
- B's efficiency = Total work / Days taken by B = 24 units / 12 days = 2 units/day.

**Hint:** Efficiency is calculated as total work divided by the number of days taken to complete that work.

### Step 3: Calculate the combined efficiency of A and B
Now, we add the efficiencies of A and B:
- Combined efficiency of A and B = A's efficiency + B's efficiency = 3 units/day + 2 units/day = 5 units/day.

**Hint:** When working together, simply add the efficiencies of the individuals.

### Step 4: Calculate the efficiency of A, B, and C together
Since A, B, and C together can complete the work in 4 days, their combined efficiency is:
- Combined efficiency of A, B, and C = Total work / Days taken by A, B, and C together = 24 units / 4 days = 6 units/day.

**Hint:** The combined efficiency of a group is the total work divided by the time they take together.

### Step 5: Find C's efficiency
Now, we can find C's efficiency by subtracting the combined efficiency of A and B from the combined efficiency of A, B, and C:
- C's efficiency = Combined efficiency of A, B, and C - Combined efficiency of A and B = 6 units/day - 5 units/day = 1 unit/day.

**Hint:** To find the efficiency of one individual in a group, subtract the combined efficiency of the others from the total combined efficiency.

### Step 6: Calculate the number of days C can do the work alone
Now that we know C's efficiency, we can find out how many days C would take to complete the entire work alone:
- Days taken by C = Total work / C's efficiency = 24 units / 1 unit/day = 24 days.

**Hint:** To find the time taken to complete the work alone, divide the total work by the individual's efficiency.

### Final Answer
C can complete the work alone in **24 days**.