A can do of a work in 10 days. B can do of the work in 20 days. In how many days can both A and B together do the work?

Options

A. 30 days
B. 32 days
C. 24 days
D. 25 days

Answer

To solve the problem of how many days A and B can complete the work together, we can follow these steps:

### Step 1: Determine the work done by A and B individually.
- A can complete the work in 10 days. Therefore, A's work rate is \( \frac{1}{10} \) of the work per day.
- B can complete the work in 20 days. Therefore, B's work rate is \( \frac{1}{20} \) of the work per day.

### Step 2: Calculate the combined work rate of A and B.
- The combined work rate of A and B is the sum of their individual work rates:
\[ \text{Combined work rate} = \frac{1}{10} + \frac{1}{20} \]

### Step 3: Find a common denominator and add the fractions.
- The least common multiple of 10 and 20 is 20. Therefore, we convert \( \frac{1}{10} \) to have a denominator of 20:
\[ \frac{1}{10} = \frac{2}{20} \]
- Now, add the two fractions:
\[ \frac{2}{20} + \frac{1}{20} = \frac{3}{20} \]

### Step 4: Calculate the time taken for A and B to complete the work together.
- If their combined work rate is \( \frac{3}{20} \) of the work per day, then the time taken to complete the entire work (1 unit of work) is the reciprocal of their combined work rate:
\[ \text{Time} = \frac{1}{\frac{3}{20}} = \frac{20}{3} \text{ days} \]

### Final Answer:
- Therefore, A and B together can complete the work in \( \frac{20}{3} \) days, which is approximately 6 days and 2/3 of a day.

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