The denominator of a fraction is 3 more than the numerator. If 2 is added to the numerator and 5 is added to the denominator, the fraction becomes `(1)/(2)`. Find the fraction.

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Define the variables
Let the numerator of the fraction be \( x \). According to the problem, the denominator is 3 more than the numerator. Therefore, we can express the denominator as:
\[ y = x + 3 \]

### Step 2: Set up the equation based on the problem statement
The problem states that if 2 is added to the numerator and 5 is added to the denominator, the fraction becomes \( \frac{1}{2} \). We can write this as:
\[ \frac{x + 2}{y + 5} = \frac{1}{2} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 2(x + 2) = 1(y + 5) \]

### Step 4: Expand both sides
Expanding both sides results in:
\[ 2x + 4 = y + 5 \]

### Step 5: Substitute the expression for \( y \)
Now, substitute \( y \) from Step 1 into the equation:
\[ 2x + 4 = (x + 3) + 5 \]

### Step 6: Simplify the equation
This simplifies to:
\[ 2x + 4 = x + 8 \]

### Step 7: Rearrange the equation to isolate \( x \)
Subtract \( x \) from both sides:
\[ 2x - x + 4 = 8 \]
This simplifies to:
\[ x + 4 = 8 \]

### Step 8: Solve for \( x \)
Subtract 4 from both sides:
\[ x = 4 \]

### Step 9: Find the value of \( y \)
Now that we have \( x \), we can find \( y \):
\[ y = x + 3 = 4 + 3 = 7 \]

### Step 10: Write the fraction
The original fraction is:
\[ \frac{x}{y} = \frac{4}{7} \]

### Final Answer
The required fraction is \( \frac{4}{7} \).

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