Ram can do a work in 20 days. Raj can do `(1/30)` of a piece of work in a day. In how many days can they do the same work, working together?

Answer

To solve the problem step by step, we will determine how many days Ram and Raj can complete the work together.

### Step 1: Determine the work done by Ram in one day.
Ram can complete the entire work in 20 days. Therefore, the work done by Ram in one day (let's denote this as \( R \)) is:
\[ R = \frac{W}{20} \]
where \( W \) is the total work.

### Step 2: Determine the work done by Raj in one day.
Raj can do \( \frac{1}{30} \) of the work in one day. Therefore, the work done by Raj in one day (let's denote this as \( J \)) is:
\[ J = \frac{W}{30} \]

### Step 3: Calculate the total work done by Ram and Raj together in one day.
When working together, the total work done in one day by both Ram and Raj is the sum of their individual work:
\[ \text{Total work done in one day} = R + J = \frac{W}{20} + \frac{W}{30} \]

### Step 4: Find a common denominator and simplify.
To add \( \frac{W}{20} \) and \( \frac{W}{30} \), we need a common denominator. The least common multiple of 20 and 30 is 60. Therefore:
\[ R = \frac{W}{20} = \frac{3W}{60} \]
\[ J = \frac{W}{30} = \frac{2W}{60} \]
Now, adding these:
\[ R + J = \frac{3W}{60} + \frac{2W}{60} = \frac{5W}{60} = \frac{W}{12} \]

### Step 5: Calculate the number of days to complete the work together.
If Ram and Raj together can complete \( \frac{W}{12} \) of the work in one day, then the total number of days \( T \) required to complete the entire work \( W \) is given by:
\[ T = \frac{W}{\frac{W}{12}} = 12 \text{ days} \]

### Final Answer:
Ram and Raj can complete the work together in **12 days**.

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