`A`, `B` and `C` can do a piece of work in `10` days, `12` days and `15` days respectively. How long will they take to finish it if they work together?

Options

A. `2days`
B. `3 days`
C. `5 days`
D. `4 days`

Answer

To solve the problem of how long A, B, and C will take to finish the work together, we will 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 does:
\[ \text{Work done by A in 1 day} = \frac{1}{10} \text{ units} \]

- **B** can complete the work in 12 days, so in one day, B does:
\[ \text{Work done by B in 1 day} = \frac{1}{12} \text{ units} \]

- **C** can complete the work in 15 days, so in one day, C does:
\[ \text{Work done by C in 1 day} = \frac{1}{15} \text{ units} \]

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

### 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:
- For A:
\[ \frac{1}{10} = \frac{6}{60} \]
- For B:
\[ \frac{1}{12} = \frac{5}{60} \]
- For C:
\[ \frac{1}{15} = \frac{4}{60} \]

### Step 4: Add the fractions.
Now we can add the fractions:
\[ \text{Total work done in 1 day} = \frac{6}{60} + \frac{5}{60} + \frac{4}{60} = \frac{15}{60} \]
This simplifies to:
\[ \frac{15}{60} = \frac{1}{4} \text{ units} \]

### Step 5: Calculate the total time taken to complete the work.
Since A, B, and C together complete \(\frac{1}{4}\) of the work in one day, we need to find out how many days it will take to complete 1 unit of work:
\[ \text{Time taken} = \frac{1 \text{ unit}}{\frac{1}{4} \text{ units/day}} = 4 \text{ days} \]

### Final Answer:
Thus, A, B, and C will take **4 days** to finish the work if they work together.

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