Raj can do a piece of work in 20 days and Rohan can do it in 12 days. How long will they take, if both work together?

Options

A. 5 days
B. `6 1/2` days
C. `7 1/2` days
D. 8days

Answer

To solve the problem of how long Raj and Rohan will take to complete the work together, we can follow these steps:

### Step 1: Determine the work done by Raj in one day.
Raj can complete the work in 20 days. Therefore, the fraction of work he does in one day is:
\[ \text{Work done by Raj in one day} = \frac{1}{20} \]

### Step 2: Determine the work done by Rohan in one day.
Rohan can complete the work in 12 days. Therefore, the fraction of work he does in one day is:
\[ \text{Work done by Rohan in one day} = \frac{1}{12} \]

### Step 3: Calculate the combined work done by Raj and Rohan in one day.
When both work together, we add the fractions of work they can do in one day:
\[ \text{Combined work in one day} = \frac{1}{20} + \frac{1}{12} \]

### Step 4: Find a common denominator to add the fractions.
The least common multiple (LCM) of 20 and 12 is 60. We convert both fractions:
\[ \frac{1}{20} = \frac{3}{60} \quad \text{and} \quad \frac{1}{12} = \frac{5}{60} \]
Now, we can add them:
\[ \text{Combined work in one day} = \frac{3}{60} + \frac{5}{60} = \frac{8}{60} = \frac{2}{15} \]

### Step 5: Determine how many days it will take to complete the work.
If they together complete \(\frac{2}{15}\) of the work in one day, to find out how many days it will take to complete the whole work (1 unit), we set up the equation:
\[ \text{Days taken} = \frac{1}{\text{Combined work in one day}} = \frac{1}{\frac{2}{15}} = \frac{15}{2} = 7.5 \text{ days} \]

### Final Answer:
Therefore, Raj and Rohan will take **7.5 days** to complete the work together.

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