The difference between the numerator and the denominator of a fraction is 5. If 5 is added to its denominator, the fraction is decreased by `1 1/4`. Find the value of the fraction.

Options

A. `1/6`
B. `2 1/4`
C. `3 1/4`
D. 6

Answer

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

### Step 1: Define the variables
Let the numerator of the fraction be \( x \) and the denominator be \( y \). According to the problem, the difference between the numerator and the denominator is 5. Therefore, we can write the first equation as:
\[ x - y = 5 \]
or
\[ x = y + 5 \] (Equation 1)

### Step 2: Set up the second condition
The problem states that if 5 is added to the denominator, the fraction decreases by \( 1 \frac{1}{4} \) (which is \( \frac{5}{4} \)). The original fraction can be expressed as \( \frac{x}{y} \). When 5 is added to the denominator, the new fraction becomes \( \frac{x}{y + 5} \). The condition can be expressed as:
\[ \frac{x}{y} - \frac{x}{y + 5} = \frac{5}{4} \] (Equation 2)

### Step 3: Substitute Equation 1 into Equation 2
From Equation 1, we know \( x = y + 5 \). We will substitute this into Equation 2:
\[ \frac{y + 5}{y} - \frac{y + 5}{y + 5} = \frac{5}{4} \]

### Step 4: Simplify the equation
Now we simplify the left-hand side:
\[ \frac{y + 5}{y} - \frac{y + 5}{y + 5} = \frac{y + 5}{y} - 1 \]
This simplifies to:
\[ \frac{y + 5 - y}{y} = \frac{5}{y} \]
Thus, we have:
\[ \frac{5}{y} = \frac{5}{4} \]

### Step 5: Cross-multiply to solve for \( y \)
Cross-multiplying gives us:
\[ 5 \cdot 4 = 5 \cdot y \]
This simplifies to:
\[ 20 = 5y \]
Dividing both sides by 5, we find:
\[ y = 4 \]

### Step 6: Find \( x \)
Now that we have \( y \), we can find \( x \) using Equation 1:
\[ x = y + 5 = 4 + 5 = 9 \]

### Step 7: Write the fraction
Now we can write the fraction:
\[ \frac{x}{y} = \frac{9}{4} \]

### Step 8: Convert to mixed fraction
The fraction \( \frac{9}{4} \) can be converted to a mixed fraction:
\[ 9 \div 4 = 2 \text{ remainder } 1 \]
Thus, \( \frac{9}{4} = 2 \frac{1}{4} \).

### Final Answer
The value of the fraction is:
\[ 2 \frac{1}{4} \]

---