A can do a piece of work in 20 days while B can do it in 30 days. Both of them start the work together and work for some time, then B leaves. If A completes the remaining work in 10 days, then find the number of days for which they worked together.

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the work done by A and B in one day.
- A can complete the work in 20 days, so A's work rate is \( \frac{1}{20} \) of the work per day.
- B can complete the work in 30 days, so B's work rate is \( \frac{1}{30} \) of the work per day.

### Step 2: Calculate the combined work rate of A and B.
- The combined work rate of A and B when they work together is:
\[ \text{Combined work rate} = \frac{1}{20} + \frac{1}{30} \]
- To add these fractions, we need a common denominator. The least common multiple of 20 and 30 is 60.
- Converting the fractions:
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
- Now adding them:
\[ \text{Combined work rate} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \]
- This means A and B together can complete \( \frac{1}{12} \) of the work in one day.

### Step 3: Determine the amount of work A completes alone in 10 days.
- After B leaves, A completes the remaining work in 10 days.
- The amount of work A can complete in 10 days is:
\[ \text{Work done by A in 10 days} = 10 \times \frac{1}{20} = \frac{10}{20} = \frac{1}{2} \]
- This means A completes half of the total work alone.

### Step 4: Calculate the total work done before B leaves.
- Since A completed \( \frac{1}{2} \) of the work alone, the remaining work done by A and B together before B left is also \( \frac{1}{2} \).
- Let \( x \) be the number of days they worked together. The work done by A and B together in \( x \) days is:
\[ \text{Work done together} = x \times \frac{1}{12} \]
- Setting this equal to the remaining work:
\[ x \times \frac{1}{12} = \frac{1}{2} \]

### Step 5: Solve for \( x \).
- To find \( x \), we can multiply both sides by 12:
\[ x = \frac{1}{2} \times 12 = 6 \]

### Conclusion:
- Therefore, A and B worked together for **6 days**.