**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. B alone could do the work in how many days ?**

Let's solve this step by step.

### 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 is:

\[
\text{Work rate of A and B together} = \frac{1}{3} \text{ work per day}
\]

### Step 2: Calculate the work done by A and B in 2 days.
Since A and B work together for 2 days, the amount of work completed in these 2 days is:

\[
\text{Work done by A and B in 2 days} = 2 \times \frac{1}{3} = \frac{2}{3} \text{ of the work}
\]

### Step 3: Determine the remaining work.
After 2 days, \(\frac{2}{3}\) of the work is completed. Therefore, the remaining work is:

\[
\text{Remaining work} = 1 - \frac{2}{3} = \frac{1}{3}
\]

### Step 4: Determine the work rate of A alone.
Let's assume A can complete the work in \(a\) days. Therefore, A's work rate is:

\[
\text{Work rate of A} = \frac{1}{a} \text{ work per day}
\]

### Step 5: Calculate the time taken by A to complete the remaining work.
The remaining \(\frac{1}{3}\) of the work is completed by A in 2 days after B left. Therefore:

\[
\frac{1}{3} = 2 \times \frac{1}{a}
\]

Solving for \(a\):

\[
\frac{1}{3} = \frac{2}{a} \implies a = 6 \text{ days}
\]

So, A alone can complete the entire work in 6 days.

### Step 6: Determine B's work rate.
Let \(b\) be the number of days B alone would take to complete the work. The combined work rate of A and B is:

\[
\frac{1}{a} + \frac{1}{b} = \frac{1}{3}
\]

Substitute \(a = 6\) into the equation:

\[
\frac{1}{6} + \frac{1}{b} = \frac{1}{3}
\]

### Step 7: Solve for \(b\).
\[
\frac{1}{b} = \frac{1}{3} - \frac{1}{6}
\]
\[
\frac{1}{b} = \frac{2 - 1}{6} = \frac{1}{6}
\]
\[
b = 6 \text{ days}
\]

Therefore, B alone could do the work in **6 days**.

Would you like more details, or do you have any questions?

Here are 5 related questions you might find interesting:
1. How long would it take A alone to complete half of the work?
2. If B worked alone for 2 days, how much of the work would be left?
3. What if A and B started together, but B left after 1 day? How many more days would A need to complete the work?
4. If both A and B work on the task, but A is twice as efficient as B, how long would they take to complete the work together?
5. How would the result change if B left after only 1 day instead of 2 days?

**Tip:** When solving work problems, it's often helpful to express work rates as fractions of the total work per unit time (e.g., work per day) to make the calculations easier.

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, B alone could do the work in how many days?

Explore how to calculate the time B would take to complete a task alone after working together with A. This detailed solution covers concepts of work and time, using algebraic equations and the work-time theorem. Ideal for students in Grades 9-11.

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