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

Options

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

Answer

To solve the problem, we need to find out how many days Ram alone will take to complete the work. We are given the following information:

1. Ram and Shyam together can complete the work in 12 days.
2. Shyam and Harlin together can complete the work in 15 days.
3. Harlin and Ram together can complete the work in 20 days.

Let's denote:
- The work done by Ram in one day as \( R \).
- The work done by Shyam in one day as \( S \).
- The work done by Harlin in one day as \( H \).

From the information given, we can write the following equations based on the concept that Work = Rate × Time:

1. \( R + S = \frac{1}{12} \) (Equation 1)
2. \( S + H = \frac{1}{15} \) (Equation 2)
3. \( H + R = \frac{1}{20} \) (Equation 3)

### Step 1: Express each equation in terms of work done per day
From the equations, we can express the work done by each pair in one day.

### Step 2: Add all three equations
Now, we will add all three equations together:

\[ (R + S) + (S + H) + (H + R) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]

This simplifies to:

\[ 2R + 2S + 2H = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 12, 15, and 20 is 60. We convert each fraction:

\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60} \]

Adding these together:

\[ \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5} \]

### Step 4: Substitute back into the equation
Now we have:

\[ 2R + 2S + 2H = \frac{1}{5} \]

Dividing the entire equation by 2 gives:

\[ R + S + H = \frac{1}{10} \]

### Step 5: Find individual work rates
Now we can find the individual work rates. We can express \( H \) in terms of \( R \) and \( S \) using the equations we have:

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:
\[ H = \frac{1}{15} - \left(\frac{1}{12} - R\right) = R + \frac{1}{15} - \frac{1}{12} \]

Finding a common denominator for \( \frac{1}{15} - \frac{1}{12} \):
\[ \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{12} = \frac{5}{60} \quad \Rightarrow \quad H = R + \left(\frac{4}{60} - \frac{5}{60}\right) = R - \frac{1}{60} \]

### Step 6: Substitute \( H \) back into Equation 3
Now substitute \( H \) back into Equation 3:
\[ \left(R - \frac{1}{60}\right) + R = \frac{1}{20} \]
\[ 2R - \frac{1}{60} = \frac{1}{20} \]

Finding a common denominator for \( \frac{1}{20} \):
\[ \frac{1}{20} = \frac{3}{60} \]
So we have:
\[ 2R - \frac{1}{60} = \frac{3}{60} \]
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 7: Calculate the time taken by Ram alone
Since \( R = \frac{1}{30} \), this means Ram can complete the work alone in 30 days.

Thus, the final answer is:
**Ram alone will complete the work in 30 days.**