Vijay and Siddhartha go to a pub and order a pithcer full of beer. In an hour together they finish half the pitcher of beer. Vijay alone can finish a pithcer of beer in x hours and Siddhartha alone can finish a pitcher of beer in y hours. If x and y are the integral numbers, then the maximum value of x+y is

Options

A. a. 4
B. b. 8
C. c. 9
D. d. 12

Answer

To solve the problem, we need to determine the maximum value of \( x + y \) where \( x \) is the time taken by Vijay to finish a pitcher of beer alone, and \( y \) is the time taken by Siddhartha to finish a pitcher of beer alone.

### Step-by-step Solution:

1. **Understanding the Problem**:
- Vijay and Siddhartha together finish half a pitcher of beer in 1 hour. Therefore, in 1 hour, their combined rate of drinking beer is \( \frac{1}{2} \) pitcher/hour.

2. **Setting Up the Equations**:
- Vijay can finish a pitcher in \( x \) hours, so his rate is \( \frac{1}{x} \) pitcher/hour.
- Siddhartha can finish a pitcher in \( y \) hours, so his rate is \( \frac{1}{y} \) pitcher/hour.
- Together, their combined rate is:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{2} \]

3. **Finding a Common Denominator**:
- To solve the equation, we can express it with a common denominator:
\[ \frac{y + x}{xy} = \frac{1}{2} \]

4. **Cross Multiplying**:
- Cross multiplying gives us:
\[ 2(y + x) = xy \]
- Rearranging this, we get:
\[ xy - 2x - 2y = 0 \]

5. **Factoring the Equation**:
- We can factor this quadratic equation:
\[ (x - 2)(y - 2) = 4 \]
- This means that the product of \( (x - 2) \) and \( (y - 2) \) must equal 4.

6. **Finding Integer Solutions**:
- The pairs of integers that multiply to 4 are:
- \( (1, 4) \)
- \( (2, 2) \)
- \( (4, 1) \)
- This gives us the following values for \( (x, y) \):
- From \( (1, 4) \): \( (3, 6) \) → \( x + y = 9 \)
- From \( (2, 2) \): \( (4, 4) \) → \( x + y = 8 \)
- From \( (4, 1) \): \( (6, 3) \) → \( x + y = 9 \)

7. **Finding the Maximum Value**:
- The maximum value of \( x + y \) from these pairs is \( 9 \).

### Conclusion:
The maximum value of \( x + y \) is \( 9 \).