The sum of the numerator and denominator of a positive fraction is 11. If 2 is added to both numerator and denominator, the fraction is increased by `(1)/(24)` The difference of numerator and denominator of the fraction is

Options

A. 5
B. 3
C. 1
D. 9

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information provided.

### Step 1: Define the Variables
Let the numerator of the fraction be \( x \) and the denominator be \( y \).

### Step 2: Set Up the First Equation
From the problem, we know that the sum of the numerator and denominator is 11:
\[ x + y = 11 \]

### Step 3: Express the Denominator in Terms of the Numerator
We can express \( y \) in terms of \( x \):
\[ y = 11 - x \]

### Step 4: Set Up the Fraction
The fraction can be expressed as:
\[ \frac{x}{y} = \frac{x}{11 - x} \]

### Step 5: Set Up the Second Equation
According to the problem, if we add 2 to both the numerator and the denominator, the fraction increases by \( \frac{1}{24} \):
\[ \frac{x + 2}{y + 2} = \frac{x}{y} + \frac{1}{24} \]

### Step 6: Substitute for \( y \)
Substituting \( y = 11 - x \) into the equation gives:
\[ \frac{x + 2}{(11 - x) + 2} = \frac{x}{11 - x} + \frac{1}{24} \]
This simplifies to:
\[ \frac{x + 2}{13 - x} = \frac{x}{11 - x} + \frac{1}{24} \]

### Step 7: Cross-Multiply to Eliminate the Fractions
Cross-multiplying gives:
\[ (x + 2)(11 - x) = (13 - x)\left(\frac{x}{11 - x} + \frac{1}{24}\right) \]

### Step 8: Simplify the Left Side
Expanding the left side:
\[ 11x + 22 - x^2 - 2x = 9x + 22 - x^2 \]
This simplifies to:
\[ 9x + 22 - x^2 \]

### Step 9: Simplify the Right Side
Now, we need to simplify the right side:
\[ (13 - x)\left(\frac{x}{11 - x} + \frac{1}{24}\right) \]
This will require finding a common denominator and simplifying.

### Step 10: Combine and Rearrange the Equation
After simplifying both sides, we will collect all terms on one side to form a quadratic equation:
\[ x^2 + 72x - 385 = 0 \]

### Step 11: Factor the Quadratic Equation
Factoring the quadratic equation gives:
\[ (x + 77)(x - 5) = 0 \]
This results in two potential solutions for \( x \):
\[ x = 5 \quad \text{or} \quad x = -77 \]

### Step 12: Choose the Positive Solution
Since we are looking for a positive fraction, we take \( x = 5 \).

### Step 13: Find the Denominator
Now substituting \( x = 5 \) back into the equation for \( y \):
\[ y = 11 - 5 = 6 \]

### Step 14: Calculate the Difference
The difference between the numerator and denominator is:
\[ y - x = 6 - 5 = 1 \]

### Final Answer
The difference of the numerator and denominator of the fraction is \( 1 \).