Ravi, Rohan and Rajesh alone can complete a work in 10, 12 and 15 days respectively. In how many days can the work be completed, if all three work together?
A. 4 days
B. 5 days
C. 3 days
D. 8 days

Options

A. B
B. C
C. D
D. A

Answer

To solve the problem of how many days Ravi, Rohan, and Rajesh can complete the work together, we will follow these steps:

### Step 1: Determine the work done by each person in one day
- Ravi can complete the work in 10 days, so his work rate is \( \frac{1}{10} \) of the work per day.
- Rohan can complete the work in 12 days, so his work rate is \( \frac{1}{12} \) of the work per day.
- Rajesh can complete the work in 15 days, so his work rate is \( \frac{1}{15} \) of the work per day.

### Step 2: Calculate the combined work rate of all three
To find the total work rate when all three work together, we add their individual work rates:
\[ \text{Combined Work Rate} = \frac{1}{10} + \frac{1}{12} + \frac{1}{15} \]

### 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:
\[ \text{Combined Work Rate} = \frac{6}{60} + \frac{5}{60} + \frac{4}{60} = \frac{15}{60} \]

### Step 5: Simplify the combined work rate
\[ \frac{15}{60} = \frac{1}{4} \]
This means that together, they can complete \( \frac{1}{4} \) of the work in one day.

### Step 6: Calculate the total time to complete the work
If they complete \( \frac{1}{4} \) of the work in one day, the total time to complete the entire work is:
\[ \text{Total Time} = \frac{1}{\frac{1}{4}} = 4 \text{ days} \]

### Final Answer
Thus, if all three work together, they can complete the work in **4 days**.