A and B together can complete a work in 3 days. They started together but after 2 days, B left the work. If the work is completed after 2 more days, Balone could do the work in how many days?

Options

A. 5
B. 6
C. 7
D. 10

Answer

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

### Step 1: Determine the combined work rate of A and B
A and B together can complete the work in 3 days. Therefore, their combined work rate (efficiency) is:
\[ \text{Efficiency of A and B (E_{AB})} = \frac{1}{3} \text{ (work per day)} \]

**Hint:** Remember that if they can complete the work in \( n \) days, their combined efficiency is \( \frac{1}{n} \).

### Step 2: Calculate the work done by A and B in 2 days
In 2 days, the amount of work done by A and B together is:
\[ \text{Work done in 2 days} = 2 \times E_{AB} = 2 \times \frac{1}{3} = \frac{2}{3} \]

**Hint:** Multiply the daily work rate by the number of days worked to find the total work done.

### Step 3: Determine the remaining work
The total work is considered as 1 unit. After 2 days of work, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{2}{3} = \frac{1}{3} \]

**Hint:** Subtract the work done from the total work to find the remaining work.

### Step 4: Find the work done by A alone in the next 2 days
After B leaves, A continues to work alone for 2 more days to complete the remaining work. Since the remaining work is \( \frac{1}{3} \), we can express A's efficiency as:
\[ \text{Let A's efficiency (E_A)} = \frac{1}{x} \text{ (where x is the number of days A takes to complete the work alone)} \]
In 2 days, A does:
\[ \text{Work done by A in 2 days} = 2 \times E_A \]
Setting this equal to the remaining work:
\[ 2 \times E_A = \frac{1}{3} \]
Thus,
\[ E_A = \frac{1}{6} \text{ (work per day)} \]

**Hint:** To find A's efficiency, divide the remaining work by the time taken to complete it.

### Step 5: Calculate B's efficiency
Now we know:
\[ E_{AB} = E_A + E_B \]
Substituting the known values:
\[ \frac{1}{3} = \frac{1}{6} + E_B \]
To find \( E_B \):
\[ E_B = \frac{1}{3} - \frac{1}{6} \]
Finding a common denominator (6):
\[ E_B = \frac{2}{6} - \frac{1}{6} = \frac{1}{6} \]

**Hint:** When subtracting fractions, ensure you have a common denominator.

### Step 6: Calculate the time taken by B to complete the work alone
Since \( E_B = \frac{1}{6} \), the time taken by B to complete the work alone is:
\[ \text{Time taken by B} = \frac{1}{E_B} = \frac{1}{\frac{1}{6}} = 6 \text{ days} \]

**Hint:** The time taken to complete the work alone is the reciprocal of the efficiency.

### Final Answer
B alone could complete the work in **6 days**.

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