A can do a piece of work in 10 days, B can do it in 12 days and C can do it in 15 days. In how many days will A, B and C finish it, working all together?

Options

A. 6 days
B. `5 1/4 ` days
C. `4 (4)/(11)` days
D. 4 days

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 in one day, A can do \( \frac{1}{10} \) of the work.
- B can complete the work in 12 days, so in one day, B can do \( \frac{1}{12} \) of the work.
- C can complete the work in 15 days, so in one day, C can do \( \frac{1}{15} \) of the work.

### Step 2: Calculate the total work done by A, B, and C in one day.
To find the total work done by A, B, and C together in one day, we add their individual contributions:
\[ \text{Total work in one day} = \frac{1}{10} + \frac{1}{12} + \frac{1}{15} \]

### Step 3: Find a common denominator.
The least common multiple (LCM) of 10, 12, and 15 is 60. We will convert each fraction to have a denominator of 60:
- \( \frac{1}{10} = \frac{6}{60} \)
- \( \frac{1}{12} = \frac{5}{60} \)
- \( \frac{1}{15} = \frac{4}{60} \)

### Step 4: Add the fractions.
Now we can add the fractions:
\[ \frac{6}{60} + \frac{5}{60} + \frac{4}{60} = \frac{6 + 5 + 4}{60} = \frac{15}{60} \]

### Step 5: Simplify the total work done in one day.
\[ \frac{15}{60} = \frac{1}{4} \]
This means A, B, and C together can complete \( \frac{1}{4} \) of the work in one day.

### Step 6: Calculate the total time taken to complete the work.
If they can complete \( \frac{1}{4} \) of the work in one day, then to complete the entire work (1 whole), it will take:
\[ \text{Total days} = \frac{1}{\frac{1}{4}} = 4 \text{ days} \]

### Final Answer:
A, B, and C together will finish the work in **4 days**.

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