A alone can complete a work in 10 days and B alone can complete the same work in 20 days. In how many days both 4 and B together can complete half of the total work?

Options

A. `40/3` days
B. `20/3`
C. `10/3` days
D. `25/3` days

Answer

To solve the problem step by step, we need to determine how long A and B together will take to complete half of the total work.

### Step 1: Determine the work rates of A and B
- A can complete the work in 10 days. Therefore, A's work rate is:
\[ \text{Work rate of A} = \frac{1}{10} \text{ (work per day)} \]

- B can complete the work in 20 days. Therefore, B's work rate is:
\[ \text{Work rate of B} = \frac{1}{20} \text{ (work per day)} \]

### Step 2: Calculate the combined work rate of A and B
To find the combined work rate when A and B work together, we add their individual work rates:
\[ \text{Combined work rate} = \text{Work rate of A} + \text{Work rate of B} = \frac{1}{10} + \frac{1}{20} \]

To add these fractions, we need a common denominator. The least common multiple of 10 and 20 is 20:
\[ \frac{1}{10} = \frac{2}{20} \]
Thus,
\[ \text{Combined work rate} = \frac{2}{20} + \frac{1}{20} = \frac{3}{20} \text{ (work per day)} \]

### Step 3: Calculate the time taken to complete half of the work
Since the combined work rate of A and B is \(\frac{3}{20}\) of the work per day, we need to find out how many days it will take them to complete half of the work:
\[ \text{Time to complete half work} = \frac{\text{Amount of work}}{\text{Combined work rate}} = \frac{\frac{1}{2}}{\frac{3}{20}} \]

To divide by a fraction, we multiply by its reciprocal:
\[ \text{Time to complete half work} = \frac{1}{2} \times \frac{20}{3} = \frac{20}{6} = \frac{10}{3} \text{ days} \]

### Final Answer
Thus, A and B together can complete half of the total work in \(\frac{10}{3}\) days.

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