A and B can do a piece of work in 12 days, B and C in 15 days and A and C in 20 days. In how many days can each alone do the work?

Options

A. 30,20,50
B. 30,45,60
C. 30,20,60
D. 20,30,50

Explanation

To solve the problem step by step, let's denote the work done by A, B, and C in one day as \( A \), \( B \), and \( C \) respectively.

### Step 1: Set up the equations based on the given information
1. A and B can complete the work in 12 days:
\[ A + B = \frac{1}{12} \quad \text{(Equation 1)} \]
2. B and C can complete the work in 15 days:
\[ B + C = \frac{1}{15} \quad \text{(Equation 2)} \]
3. A and C can complete the work in 20 days:
\[ A + C = \frac{1}{20} \quad \text{(Equation 3)} \]

### Step 2: Combine the equations
Now, we will add all three equations together:
\[ (A + B) + (B + C) + (A + C) = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]
This simplifies to:
\[ 2A + 2B + 2C = \frac{1}{12} + \frac{1}{15} + \frac{1}{20} \]

### Step 3: Calculate the right-hand side
To add the fractions, we need a common denominator. The least common multiple (LCM) of 12, 15, and 20 is 60. Now we convert each fraction:
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60}, \quad \frac{1}{20} = \frac{3}{60} \]
Adding these gives:
\[ \frac{5}{60} + \frac{4}{60} + \frac{3}{60} = \frac{12}{60} = \frac{1}{5} \]

### Step 4: Simplify the equation
Now we have:
\[ 2A + 2B + 2C = \frac{1}{5} \]
Dividing the entire equation by 2:
\[ A + B + C = \frac{1}{10} \quad \text{(Equation 4)} \]

### Step 5: Find individual work rates
Now we can find the individual work rates by subtracting the equations:

1. To find \( C \):
\[ C = (A + B + C) - (A + B) = \frac{1}{10} - \frac{1}{12} \]
Finding a common denominator (60):
\[ C = \frac{6}{60} - \frac{5}{60} = \frac{1}{60} \]
So, \( C \) can complete the work alone in 60 days.

2. To find \( B \):
\[ B = (A + B + C) - (A + C) = \frac{1}{10} - \frac{1}{20} \]
Finding a common denominator (20):
\[ B = \frac{2}{20} - \frac{1}{20} = \frac{1}{20} \]
So, \( B \) can complete the work alone in 20 days.

3. To find \( A \):
\[ A = (A + B + C) - (B + C) = \frac{1}{10} - \frac{1}{15} \]
Finding a common denominator (30):
\[ A = \frac{3}{30} - \frac{2}{30} = \frac{1}{30} \]
So, \( A \) can complete the work alone in 30 days.

### Final Answer
- A can do the work alone in **30 days**.
- B can do the work alone in **20 days**.
- C can do the work alone in **60 days**.