A, B and C can do a work in 10, 12 & 18 days respectively. In how many days all of them together will finish the work?

Answer

To solve the problem of how many days A, B, and C will take to finish the work together, we can follow these steps:

### Step 1: Determine the work done by each person in one day.
- A can complete the work in 10 days, so A's work in one day = \( \frac{1}{10} \) of the work.
- B can complete the work in 12 days, so B's work in one day = \( \frac{1}{12} \) of the work.
- C can complete the work in 18 days, so C's work in one day = \( \frac{1}{18} \) of the work.

### Step 2: Find the combined work done by A, B, and C in one day.
To find the combined efficiency (work done in one day) of A, B, and C, we add their individual work rates:
\[ \text{Combined work in one day} = \frac{1}{10} + \frac{1}{12} + \frac{1}{18} \]

### Step 3: Calculate the least common multiple (LCM) of the denominators.
The denominators are 10, 12, and 18. The LCM of these numbers is 180.

### Step 4: Convert each fraction to have a common denominator.
- A's work: \( \frac{1}{10} = \frac{18}{180} \)
- B's work: \( \frac{1}{12} = \frac{15}{180} \)
- C's work: \( \frac{1}{18} = \frac{10}{180} \)

### Step 5: Add the fractions.
Now, we can add the fractions:
\[ \text{Combined work in one day} = \frac{18}{180} + \frac{15}{180} + \frac{10}{180} = \frac{43}{180} \]

### Step 6: Calculate the total time taken to finish the work together.
To find the total time taken for A, B, and C to complete the work together, we take the total work (which we can consider as 1 unit of work) and divide it by their combined work rate:
\[ \text{Total time} = \frac{1 \text{ unit of work}}{\frac{43}{180}} = \frac{180}{43} \text{ days} \]

### Conclusion
Thus, A, B, and C together will finish the work in \( \frac{180}{43} \) days, which is approximately 4.19 days.

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