The numerator of a fraction is three less than the denominator. If 4 is added to both the numerator and the denominator, the value of the fraction increases by 1/8. Find the fraction.

Options

A. `7//8`
B. `3//8`
C. `9//8`
D. `5//8`

Answer

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

### Step 1: Define the Variables
Let the denominator of the fraction be \( x \). According to the problem, the numerator is three less than the denominator. Therefore, we can express the numerator as:
\[ \text{Numerator} = x - 3 \]

### Step 2: Write the Original Fraction
The original fraction can be written as:
\[ \text{Fraction} = \frac{x - 3}{x} \]

### Step 3: Set Up the New Fraction
When 4 is added to both the numerator and the denominator, the new fraction becomes:
\[ \text{New Fraction} = \frac{(x - 3) + 4}{x + 4} = \frac{x + 1}{x + 4} \]

### Step 4: Set Up the Equation
According to the problem, adding 4 to both the numerator and the denominator increases the value of the fraction by \( \frac{1}{8} \). Therefore, we can set up the equation:
\[ \frac{x + 1}{x + 4} = \frac{x - 3}{x} + \frac{1}{8} \]

### Step 5: Clear the Fractions
To eliminate the fractions, we can find a common denominator. The common denominator for the right-hand side is \( 8x \):
\[ \frac{x + 1}{x + 4} = \frac{8(x - 3) + x}{8x} \]
This simplifies to:
\[ \frac{x + 1}{x + 4} = \frac{8x - 24 + x}{8x} = \frac{9x - 24}{8x} \]

### Step 6: Cross Multiply
Cross multiplying gives us:
\[ 8x(x + 1) = (9x - 24)(x + 4) \]

### Step 7: Expand Both Sides
Expanding both sides:
\[ 8x^2 + 8x = 9x^2 + 36x - 24x - 96 \]
This simplifies to:
\[ 8x^2 + 8x = 9x^2 + 12x - 96 \]

### Step 8: Rearrange the Equation
Rearranging the equation gives:
\[ 8x^2 + 8x - 9x^2 - 12x + 96 = 0 \]
This simplifies to:
\[ -x^2 - 4x + 96 = 0 \]
Multiplying through by -1 gives:
\[ x^2 + 4x - 96 = 0 \]

### Step 9: Factor the Quadratic Equation
Now we will factor the quadratic equation:
\[ (x + 12)(x - 8) = 0 \]

### Step 10: Solve for \( x \)
Setting each factor to zero gives us:
\[ x + 12 = 0 \quad \Rightarrow \quad x = -12 \quad \text{(not valid since denominator cannot be negative)} \]
\[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \]

### Step 11: Find the Numerator
Now substituting \( x = 8 \) back into the expression for the numerator:
\[ \text{Numerator} = x - 3 = 8 - 3 = 5 \]

### Step 12: Write the Final Fraction
Thus, the fraction is:
\[ \frac{5}{8} \]

### Summary
The fraction is \( \frac{5}{8} \).

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