The sum of the numerator and denominator of a fraction is 8. If 3 is added to both of the numerator and the denominator, the fraction becomes ` (3)/(4)`. Find the fraction.

Answer

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

### 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
According to the problem, the sum of the numerator and denominator is 8. Therefore, we can write the first equation as:
\[ x + y = 8 \]

### Step 3: Set Up the Second Equation
The problem states that if 3 is added to both the numerator and the denominator, the fraction becomes \( \frac{3}{4} \). This gives us the second equation:
\[ \frac{x + 3}{y + 3} = \frac{3}{4} \]

### Step 4: Cross-Multiply to Eliminate the Fraction
To eliminate the fraction, we can cross-multiply:
\[ 4(x + 3) = 3(y + 3) \]

### Step 5: Expand Both Sides
Now we will expand both sides of the equation:
\[ 4x + 12 = 3y + 9 \]

### Step 6: Rearrange the Equation
Rearranging the equation gives us:
\[ 4x - 3y = 9 - 12 \]
This simplifies to:
\[ 4x - 3y = -3 \]

### Step 7: Solve the System of Equations
Now we have a system of equations:
1. \( x + y = 8 \) (Equation 1)
2. \( 4x - 3y = -3 \) (Equation 2)

From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 8 - x \]

### Step 8: Substitute \( y \) in Equation 2
Substituting \( y \) in Equation 2:
\[ 4x - 3(8 - x) = -3 \]
Expanding this gives:
\[ 4x - 24 + 3x = -3 \]
Combining like terms:
\[ 7x - 24 = -3 \]

### Step 9: Solve for \( x \)
Adding 24 to both sides:
\[ 7x = 21 \]
Dividing by 7:
\[ x = 3 \]

### Step 10: Find \( y \)
Now substitute \( x \) back into Equation 1 to find \( y \):
\[ 3 + y = 8 \]
So,
\[ y = 8 - 3 = 5 \]

### Step 11: Write the Fraction
The fraction is:
\[ \frac{x}{y} = \frac{3}{5} \]

### Final Answer
Thus, the fraction is \( \frac{3}{5} \).

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