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 step by step, we will calculate the amount of work done by A and B, and then determine how long they worked together before B left.

### Step 1: Calculate the work rates of A and B
- A can complete the work in 20 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1}{20} \text{ (work per day)} \]
- B can complete the work in 30 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{30} \text{ (work per day)} \]

### Step 2: Calculate the combined work rate of A and B
- When A and B work together, their combined work rate is:
\[ \text{Combined work rate} = \frac{1}{20} + \frac{1}{30} \]
- To add these fractions, find a common denominator. The least common multiple of 20 and 30 is 60:
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{30} = \frac{2}{60} \]
- Therefore, the combined work rate is:
\[ \text{Combined work rate} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \text{ (work per day)} \]

### Step 3: Determine the total work done by A in 10 days
- A completes the remaining work in 10 days. Since A's work rate is \(\frac{1}{20}\), the amount of work A does in 10 days is:
\[ \text{Work done by A in 10 days} = 10 \times \frac{1}{20} = \frac{10}{20} = \frac{1}{2} \text{ of the total work} \]

### Step 4: Calculate the total work
- If A completed \(\frac{1}{2}\) of the total work in 10 days, then the total work can be represented as:
\[ \text{Total work} = 1 \text{ (whole work)} \]

### Step 5: Determine the work done before B leaves
- Since A completed \(\frac{1}{2}\) of the work alone, the other half must have been completed by A and B together. Thus, the work done by A and B together is:
\[ \text{Work done by A and B together} = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 6: Calculate the time they worked together
- Let \(T\) be the number of days they worked together. The work done by A and B together in \(T\) days is:
\[ T \times \frac{1}{12} = \frac{1}{2} \]
- Solving for \(T\):
\[ T = \frac{1/2}{1/12} = \frac{1}{2} \times \frac{12}{1} = 6 \text{ days} \]

### Final Answer
- A and B worked together for **6 days**.

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