A alone can do a plece of work in 20 days and B alone in 30 days. They begin to work together. They will finish half of the work in :

Options

A. 8 days
B. 9 days
C. 12 days
D. 6 days

Answer

To solve the problem step by step, we will determine how long it takes for A and B to finish half of the work together.

### Step 1: Determine the work done by A and B in one day.
- A can complete the work in 20 days, so the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{20} \]
- B can complete the work in 30 days, so the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{30} \]

### Step 2: Calculate the total work done by A and B together in one day.
- To find the total work done by A and B together in one day, we add their individual contributions:
\[ \text{Total work done in one day} = \frac{1}{20} + \frac{1}{30} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 20 and 30 is 60.
\[ \frac{1}{20} = \frac{3}{60} \quad \text{and} \quad \frac{1}{30} = \frac{2}{60} \]
- Now we can add them:
\[ \text{Total work done in one day} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \]

### Step 3: Calculate the time taken to complete the entire work.
- Since A and B together can complete \(\frac{1}{12}\) of the work in one day, the time taken to complete the entire work is:
\[ \text{Time to complete the work} = \frac{1}{\frac{1}{12}} = 12 \text{ days} \]

### Step 4: Calculate the time taken to complete half of the work.
- To find the time taken to complete half of the work, we simply take half of the total time:
\[ \text{Time to complete half of the work} = \frac{12}{2} = 6 \text{ days} \]

### Final Answer:
A and B together will finish half of the work in **6 days**.

---