A is twice as fast as B and together they can complete a work in 20 days. In how many days can A alone complete the work?

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define the relationship between A and B
Let the time taken by B to complete the work alone be \( x \) days. Since A is twice as fast as B, the time taken by A to complete the work alone will be \( \frac{x}{2} \) days.

**Hint:** Remember that if one person is faster than another, their time to complete the task will be shorter.

### Step 2: Determine the work done by A and B together
If A and B work together, they can complete the work in 20 days. This means that in one day, they complete \( \frac{1}{20} \) of the work.

**Hint:** When two workers collaborate, their combined work rate is the sum of their individual work rates.

### Step 3: Express the work rates of A and B
The work rate of A is \( \frac{1}{\frac{x}{2}} = \frac{2}{x} \) (since A completes the work in \( \frac{x}{2} \) days).
The work rate of B is \( \frac{1}{x} \).

**Hint:** The work rate is the reciprocal of the time taken to complete the work.

### Step 4: Set up the equation for combined work rates
The combined work rate of A and B can be expressed as:
\[ \frac{2}{x} + \frac{1}{x} = \frac{1}{20} \]
This simplifies to:
\[ \frac{3}{x} = \frac{1}{20} \]

**Hint:** Combine the fractions on the left side to find a common work rate.

### Step 5: Solve for \( x \)
Cross-multiply to solve for \( x \):
\[ 3 \cdot 20 = x \implies x = 60 \]

**Hint:** Make sure to multiply both sides correctly to isolate \( x \).

### Step 6: Find the time taken by A to complete the work alone
Since \( x = 60 \) days (the time taken by B), the time taken by A to complete the work alone is:
\[ \frac{x}{2} = \frac{60}{2} = 30 \text{ days} \]

**Hint:** Use the relationship between A and B to find A's time based on B's time.

### Final Answer:
A alone can complete the work in **30 days**.