The numerator of a fraction is 4 less than denominator. If 1 is added to both its numerator and denominator, it becomes 1/2. Find the fraction.

Answer

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

### Step 1: Define the Variables
Let:
- \( a \) = numerator of the fraction
- \( b \) = denominator of the fraction

### Step 2: Set Up the First Equation
According to the question, the numerator is 4 less than the denominator. This can be expressed as:
\[ a = b - 4 \]
This is our first equation.

### Step 3: Set Up the Second Equation
The question states that if 1 is added to both the numerator and the denominator, the fraction becomes \( \frac{1}{2} \). This can be expressed mathematically as:
\[ \frac{a + 1}{b + 1} = \frac{1}{2} \]
Cross-multiplying gives us:
\[ 2(a + 1) = 1(b + 1) \]
Expanding this, we get:
\[ 2a + 2 = b + 1 \]
Rearranging it leads to:
\[ 2a - b = -1 \]
This is our second equation.

### Step 4: Solve the Equations
Now we have a system of two equations:
1. \( a = b - 4 \)
2. \( 2a - b = -1 \)

We can substitute the expression for \( a \) from the first equation into the second equation:
Substituting \( a \) in the second equation:
\[ 2(b - 4) - b = -1 \]
Expanding this:
\[ 2b - 8 - b = -1 \]
Simplifying gives:
\[ b - 8 = -1 \]
Adding 8 to both sides:
\[ b = 7 \]

### Step 5: Find the Value of \( a \)
Now that we have \( b \), we can find \( a \) using the first equation:
\[ a = b - 4 \]
Substituting \( b = 7 \):
\[ a = 7 - 4 = 3 \]

### Step 6: Write the Fraction
Now we have both the numerator and the denominator:
- \( a = 3 \)
- \( b = 7 \)

Thus, the fraction is:
\[ \frac{a}{b} = \frac{3}{7} \]

### Final Answer
The fraction is \( \frac{3}{7} \).

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