A can do a piece of work in certain number of days, while B takes three days more than A to complete the same work. Working together, A and B can complete the work in two days. How many days does B take to complete the work?

Options

A. 8
B. 3
C. 4
D. 6

Answer

To solve the problem step by step, let's define the variables and equations based on the information given in the question.

### Step 1: Define the variables
Let A take \( x \) days to complete the work. Therefore, B takes \( x + 3 \) days to complete the same work.

### Step 2: Determine the work done per day
The work done by A in one day is \( \frac{1}{x} \) (since A completes the work in \( x \) days), and the work done by B in one day is \( \frac{1}{x + 3} \).

### Step 3: Set up the equation for combined work
When A and B work together, they complete the work in 2 days. Therefore, their combined work done in one day is \( \frac{1}{2} \).

So, we can write the equation:
\[ \frac{1}{x} + \frac{1}{x + 3} = \frac{1}{2} \]

### Step 4: Find a common denominator
To solve the equation, we need to find a common denominator for the left side. The common denominator is \( x(x + 3) \):
\[ \frac{x + 3 + x}{x(x + 3)} = \frac{1}{2} \]
This simplifies to:
\[ \frac{2x + 3}{x(x + 3)} = \frac{1}{2} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 2(2x + 3) = x(x + 3) \]
Expanding both sides results in:
\[ 4x + 6 = x^2 + 3x \]

### Step 6: Rearrange the equation
Rearranging the equation to one side gives:
\[ x^2 + 3x - 4x - 6 = 0 \]
This simplifies to:
\[ x^2 - x - 6 = 0 \]

### Step 7: Factor the quadratic equation
Now, we can factor the quadratic equation:
\[ (x - 3)(x + 2) = 0 \]
Setting each factor to zero gives us:
\[ x - 3 = 0 \quad \text{or} \quad x + 2 = 0 \]
Thus, \( x = 3 \) or \( x = -2 \).

### Step 8: Determine the valid solution
Since \( x \) represents the number of days A takes to complete the work, it must be a positive number. Therefore, we take \( x = 3 \).

### Step 9: Calculate the days B takes
Now, we can find the number of days B takes:
\[ x + 3 = 3 + 3 = 6 \]

### Final Answer
B takes **6 days** to complete the work.

---