If Rami and Raman can do a job in 10 days and 15 days independently, how many days would they take to complete the same job working simultaneously?

Answer

To solve the problem of how many days Rami and Raman would take to complete the job working simultaneously, we can follow these steps:

### Step 1: Determine the work done by each person in one day.

- Rami can complete the job in 10 days.
- Therefore, Rami's work in one day = \( \frac{1}{10} \) of the job.

- Raman can complete the job in 15 days.
- Therefore, Raman's work in one day = \( \frac{1}{15} \) of the job.

### Step 2: Add the work done by both Rami and Raman in one day.

To find out how much work they can do together in one day, we add their individual work rates:

\[ \text{Work done by Rami in one day} + \text{Work done by Raman in one day} = \frac{1}{10} + \frac{1}{15} \]

### Step 3: Find a common denominator to add the fractions.

The least common multiple (LCM) of 10 and 15 is 30. We can convert both fractions:

\[ \frac{1}{10} = \frac{3}{30} \quad \text{and} \quad \frac{1}{15} = \frac{2}{30} \]

Now, we can add them:

\[ \frac{3}{30} + \frac{2}{30} = \frac{5}{30} \]

### Step 4: Simplify the result.

\[ \frac{5}{30} = \frac{1}{6} \]

This means that together, Rami and Raman can complete \( \frac{1}{6} \) of the job in one day.

### Step 5: Calculate the total time taken to complete the job.

If Rami and Raman complete \( \frac{1}{6} \) of the job in one day, then the total time taken to complete the entire job is the reciprocal of \( \frac{1}{6} \):

\[ \text{Total time} = \frac{1}{\frac{1}{6}} = 6 \text{ days} \]

### Final Answer:

Rami and Raman, working together, would take **6 days** to complete the job.

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