A and B can do a job in 12 days, B and C in 15 days and C and A in 20 days. How long would A take to do that work?
A तथा B एक कार्य 12 दिनों में कर सकते हैं। वही कार्य B तथा C 15 दिनों में और C तथा A 20 दिनों में कर सकते हैं। तदनुसार A अकेला वही कार्य कितने दिनों में कर सकता है?

Options

A. A. 20 day / 20 दिन
B. B. 60 day/ 60 दिन
C. C. 30 day/ 30 दिन
D. D. 40 day/ 40 दिन

Explanation

To solve the problem step by step, we will start by defining the work done by each pair of workers and then derive the time taken by A to complete the work alone.

### Step 1: Define the work done by each pair
Let:
- Work done by A and B in one day = \( \frac{1}{12} \)
- Work done by B and C in one day = \( \frac{1}{15} \)
- Work done by C and A in one day = \( \frac{1}{20} \)

### Step 2: Set up equations
From the work done by pairs, we can set up the following equations:
1. \( \frac{1}{A} + \frac{1}{B} = \frac{1}{12} \) (Equation 1)
2. \( \frac{1}{B} + \frac{1}{C} = \frac{1}{15} \) (Equation 2)
3. \( \frac{1}{C} + \frac{1}{A} = \frac{1}{20} \) (Equation 3)

### Step 3: Express \( \frac{1}{B} \) and \( \frac{1}{C} \) in terms of \( \frac{1}{A} \)
From Equation 1:
\[ \frac{1}{B} = \frac{1}{12} - \frac{1}{A} \]

From Equation 3:
\[ \frac{1}{C} = \frac{1}{20} - \frac{1}{A} \]

### Step 4: Substitute \( \frac{1}{B} \) and \( \frac{1}{C} \) into Equation 2
Substituting these expressions into Equation 2:
\[ \left( \frac{1}{12} - \frac{1}{A} \right) + \left( \frac{1}{20} - \frac{1}{A} \right) = \frac{1}{15} \]

### Step 5: Simplify the equation
Combine the terms:
\[ \frac{1}{12} + \frac{1}{20} - \frac{2}{A} = \frac{1}{15} \]

### Step 6: Find a common denominator
The common denominator for 12, 20, and 15 is 60. Convert each fraction:
\[ \frac{5}{60} + \frac{3}{60} - \frac{2}{A} = \frac{4}{60} \]

### Step 7: Combine and solve for \( \frac{2}{A} \)
Combine the fractions:
\[ \frac{8}{60} - \frac{2}{A} = \frac{4}{60} \]

Now, isolate \( \frac{2}{A} \):
\[ \frac{2}{A} = \frac{8}{60} - \frac{4}{60} = \frac{4}{60} = \frac{1}{15} \]

### Step 8: Solve for \( A \)
Taking the reciprocal gives:
\[ A = 2 \times 15 = 30 \]

### Conclusion
Thus, A alone can complete the work in **30 days**.