Rohan and Rohit together can do a certain job in 10 days while Rohan alone can do the same job in 15 days. How many days Rohit alone will do the same job ?

Options

A. 25 days
B. 32 days
C. 35 days
D. 30 days

Answer

To solve the problem, we need to find out how many days Rohit alone will take to complete the job. We can do this by using the concept of work rates.

### Step 1: Determine Rohan's work rate
Rohan can complete the job in 15 days. Therefore, his work rate is:
\[ \text{Rohan's work rate} = \frac{1 \text{ job}}{15 \text{ days}} = \frac{1}{15} \text{ jobs per day} \]

### Step 2: Determine the combined work rate of Rohan and Rohit
Rohan and Rohit together can complete the job in 10 days. Thus, their combined work rate is:
\[ \text{Combined work rate} = \frac{1 \text{ job}}{10 \text{ days}} = \frac{1}{10} \text{ jobs per day} \]

### Step 3: Calculate Rohit's work rate
Let Rohit's work rate be \( R \) (in jobs per day). The combined work rate of Rohan and Rohit can be expressed as:
\[ \text{Rohan's work rate} + \text{Rohit's work rate} = \text{Combined work rate} \]
Substituting the known values:
\[ \frac{1}{15} + R = \frac{1}{10} \]

### Step 4: Solve for Rohit's work rate
To find \( R \), we need to isolate it:
\[ R = \frac{1}{10} - \frac{1}{15} \]
To subtract these fractions, we need a common denominator. The least common multiple of 10 and 15 is 30. Therefore, we convert the fractions:
\[ \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30} \]
Now we can subtract:
\[ R = \frac{3}{30} - \frac{2}{30} = \frac{1}{30} \]

### Step 5: Determine the time Rohit takes to complete the job alone
Since Rohit's work rate is \( \frac{1}{30} \) jobs per day, it means he can complete the job in:
\[ \text{Time taken by Rohit} = \frac{1 \text{ job}}{R} = \frac{1}{\frac{1}{30}} = 30 \text{ days} \]

### Final Answer
Rohit alone will take **30 days** to complete the job.

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