A and B can finish a work in 15 days, B and C in 20 days, while C and A in 30 days, then A alone can finish the work in

Options

A. 40 days
B. 24 days
C. 120 days
D. 13 days

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Determine the total work
Given that:
- A and B can finish the work in 15 days
- B and C can finish the work in 20 days
- C and A can finish the work in 30 days

To find the total work, we can take the least common multiple (LCM) of the days taken by each pair:
- LCM of 15, 20, and 30 is 60.

So, we can consider the total work as 60 units.

### Step 2: Calculate the work done by each pair in one day
1. **A and B's work in one day**:
\[ \text{Work done by A and B in one day} = \frac{60}{15} = 4 \text{ units} \]

2. **B and C's work in one day**:
\[ \text{Work done by B and C in one day} = \frac{60}{20} = 3 \text{ units} \]

3. **C and A's work in one day**:
\[ \text{Work done by C and A in one day} = \frac{60}{30} = 2 \text{ units} \]

### Step 3: Set up equations for the efficiencies
Let:
- Efficiency of A = \( a \)
- Efficiency of B = \( b \)
- Efficiency of C = \( c \)

From the work done in one day, we can set up the following equations:
1. \( a + b = 4 \) (from A and B)
2. \( b + c = 3 \) (from B and C)
3. \( c + a = 2 \) (from C and A)

### Step 4: Solve the equations
We can add all three equations:
\[ (a + b) + (b + c) + (c + a) = 4 + 3 + 2 \]
This simplifies to:
\[ 2a + 2b + 2c = 9 \]
Dividing by 2:
\[ a + b + c = 4.5 \]

Now, we can find the individual efficiencies:
1. From \( a + b = 4 \):
\[ c = 4.5 - 4 = 0.5 \]

2. From \( b + c = 3 \):
\[ b + 0.5 = 3 \implies b = 3 - 0.5 = 2.5 \]

3. From \( c + a = 2 \):
\[ 0.5 + a = 2 \implies a = 2 - 0.5 = 1.5 \]

### Step 5: Calculate the time taken by A alone to finish the work
Now, we know the efficiency of A:
\[ \text{Efficiency of A} = a = 1.5 \text{ units/day} \]

To find out how many days A will take to complete the total work of 60 units:
\[ \text{Time taken by A} = \frac{\text{Total Work}}{\text{Efficiency of A}} = \frac{60}{1.5} = 40 \text{ days} \]

### Final Answer
A alone can finish the work in **40 days**.