A started a work and left after working for 4 days, then B was called and he finished the work in 18 days. If A left the work after working for 6 days then B would have finished the remaining work in 12 days. In how many days can each of them, working alone finish the whole work?

Answer

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

### Step 1: Set up the equations based on the given information.

1. **First Scenario**: A works for 4 days and then B finishes the remaining work in 18 days.
- Work done by A in 4 days = \( 4A \)
- Work done by B in 18 days = \( 18B \)
- Total work = \( 4A + 18B \) (Equation 1)

2. **Second Scenario**: A works for 6 days and then B finishes the remaining work in 12 days.
- Work done by A in 6 days = \( 6A \)
- Work done by B in 12 days = \( 12B \)
- Total work = \( 6A + 12B \) (Equation 2)

### Step 2: Equate the total work from both scenarios.

Since the total work is the same in both scenarios, we can set Equation 1 equal to Equation 2:

\[ 4A + 18B = 6A + 12B \]

### Step 3: Simplify the equation.

Rearranging the equation gives:

\[ 4A + 18B - 12B = 6A \]
\[ 4A + 6B = 6A \]
\[ 6B = 6A - 4A \]
\[ 6B = 2A \]
\[ \frac{A}{B} = 3 \quad \text{(Equation 3)} \]

### Step 4: Substitute \( A \) in terms of \( B \).

From Equation 3, we can express \( A \) as:

\[ A = 3B \]

### Step 5: Substitute \( A \) back into one of the total work equations.

Let's substitute \( A \) into Equation 1:

\[ 4(3B) + 18B = \text{Total Work} \]
\[ 12B + 18B = \text{Total Work} \]
\[ 30B = \text{Total Work} \]

### Step 6: Find the total work.

Now we know that the total work is \( 30B \).

### Step 7: Calculate the number of days each person takes to finish the work alone.

1. **For A**:
\[ \text{Days taken by A} = \frac{\text{Total Work}}{A} = \frac{30B}{3B} = 10 \text{ days} \]

2. **For B**:
\[ \text{Days taken by B} = \frac{\text{Total Work}}{B} = \frac{30B}{B} = 30 \text{ days} \]

### Final Answer:
- A can finish the work alone in **10 days**.
- B can finish the work alone in **30 days**.