A piece of work can be done by Ram and Shyam in 12 days, by Shyam and Hari in 15 days and by Hari and Ram in 20 days. Ram alone will complete the work in.

राम तथा श्याम एक कार्य 12 दिनों में, श्याम तथा हरि 15 दिनों में, और हरि तथा राम 20 दिनों में कर सकते हैं। तदनुसार, राम अकेला 2 वह कार्य कितने दिनों में पूरा कर सकता है?

Options

A. A )30 days
B. B) 32 days
C. C) 36 days
D. D) 42 days

Explanation

To find out how many days Ram alone will take to complete the work, we can follow these steps:

### Step 1: Define the work done by each pair
Let:
- The work done by Ram and Shyam in one day = \( R + S \)
- The work done by Shyam and Hari in one day = \( S + H \)
- The work done by Hari and Ram in one day = \( H + R \)

From the problem, we know:
- \( R + S = \frac{1}{12} \) (since they complete the work in 12 days)
- \( S + H = \frac{1}{15} \) (since they complete the work in 15 days)
- \( H + R = \frac{1}{20} \) (since they complete the work in 20 days)

### Step 2: Set up the equations
We can rewrite the equations as:
1. \( R + S = \frac{1}{12} \) (1)
2. \( S + H = \frac{1}{15} \) (2)
3. \( H + R = \frac{1}{20} \) (3)

### Step 3: Add all equations
Adding all three equations:
\[ (R + S) + (S + H) + (H + R) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]
This simplifies to:
\[ 2R + 2S + 2H = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]

### Step 4: Find a common denominator
The least common multiple (LCM) of 12, 15, and 20 is 60. Now we convert each fraction:
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60} \]
Thus,
\[ \frac{1}{12} + \frac{1}{15} + \frac{1}{20} = \frac{5 + 4 + 3}{60} = \frac{12}{60} = \frac{1}{5} \]

### Step 5: Substitute back into the equation
Now we have:
\[ 2R + 2S + 2H = \frac{1}{5} \]
Dividing everything by 2:
\[ R + S + H = \frac{1}{10} \]

### Step 6: Solve for individual work rates
Now we can express \( H \) in terms of \( R \) and \( S \):
From equation (1):
\[ S = \frac{1}{12} - R \]
Substituting \( S \) into equation (2):
\[ \left(\frac{1}{12} - R\right) + H = \frac{1}{15} \]
Rearranging gives:
\[ H = \frac{1}{15} - \left(\frac{1}{12} - R\right) = R - \frac{1}{12} + \frac{1}{15} \]
Finding a common denominator for \( \frac{1}{12} \) and \( \frac{1}{15} \):
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60} \]
Thus,
\[ H = R - \left(\frac{5}{60} - \frac{4}{60}\right) = R - \frac{1}{60} \]

### Step 7: Substitute \( H \) back into equation (3)
From equation (3):
\[ H + R = \frac{1}{20} \]
Substituting \( H \):
\[ \left(R - \frac{1}{60}\right) + R = \frac{1}{20} \]
This simplifies to:
\[ 2R - \frac{1}{60} = \frac{1}{20} \]
Finding a common denominator for \( \frac{1}{20} \):
\[ \frac{1}{20} = \frac{3}{60} \]
Thus:
\[ 2R - \frac{1}{60} = \frac{3}{60} \]
Adding \( \frac{1}{60} \) to both sides:
\[ 2R = \frac{3}{60} + \frac{1}{60} = \frac{4}{60} = \frac{1}{15} \]
Dividing by 2:
\[ R = \frac{1}{30} \]

### Step 8: Find the time taken by Ram alone
Since \( R = \frac{1}{30} \), it means Ram can complete the work alone in 30 days.

### Final Answer:
Ram alone will complete the work in **30 days**.

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