Ravi , Mohan and Govind can complete a task in 12 days , 10 days and 15 days , respectively . In how many days can Ravi , Mohan and Govind together complete the same task ?

Options

A. 6
B. 4
C. 8
D. 2

Answer

To solve the problem of how many days Ravi, Mohan, and Govind can complete a task together, we can follow these steps:

### Step 1: Determine Individual Work Rates
- Ravi can complete the task in 12 days. Therefore, his work rate is:
\[ \text{Ravi's work rate} = \frac{1}{12} \text{ (tasks per day)} \]

- Mohan can complete the task in 10 days. Therefore, his work rate is:
\[ \text{Mohan's work rate} = \frac{1}{10} \text{ (tasks per day)} \]

- Govind can complete the task in 15 days. Therefore, his work rate is:
\[ \text{Govind's work rate} = \frac{1}{15} \text{ (tasks per day)} \]

### Step 2: Calculate Combined Work Rate
Now, we need to find the combined work rate of Ravi, Mohan, and Govind by adding their individual work rates:
\[ \text{Combined work rate} = \frac{1}{12} + \frac{1}{10} + \frac{1}{15} \]

### Step 3: Find a Common Denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of 12, 10, and 15 is 60. We will convert each fraction:
- For \(\frac{1}{12}\):
\[ \frac{1}{12} = \frac{5}{60} \]

- For \(\frac{1}{10}\):
\[ \frac{1}{10} = \frac{6}{60} \]

- For \(\frac{1}{15}\):
\[ \frac{1}{15} = \frac{4}{60} \]

### Step 4: Add the Converted Fractions
Now, we can add the fractions:
\[ \text{Combined work rate} = \frac{5}{60} + \frac{6}{60} + \frac{4}{60} = \frac{15}{60} \]

### Step 5: Simplify the Combined Work Rate
Simplifying \(\frac{15}{60}\):
\[ \frac{15}{60} = \frac{1}{4} \]

### Step 6: Calculate the Time Taken to Complete the Task
The combined work rate of \(\frac{1}{4}\) means that together, they complete \(\frac{1}{4}\) of the task in one day. Therefore, the time taken to complete the entire task is the reciprocal of the combined work rate:
\[ \text{Time taken} = \frac{1}{\frac{1}{4}} = 4 \text{ days} \]

### Final Answer
Thus, Ravi, Mohan, and Govind together can complete the task in **4 days**.

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