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

Options

A. `7/11`
B. `5/9`
C. `3/7`
D. `4/7`

Answer

To solve the problem step by step, we will define the variables for the numerator and denominator of the fraction, set up equations based on the information given, and solve for the unknowns.

### Step 1: Define the Variables
Let the numerator of the fraction be \( x \) and the denominator be \( y \).

**Hint:** Start by defining your variables clearly to represent the unknowns in the problem.

### Step 2: Set Up the First Equation
According to the problem, the numerator is 4 less than the denominator. This can be expressed as:
\[ x = y - 4 \]

**Hint:** Translate the relationships described in the problem into mathematical equations.

### Step 3: Set Up the Second Equation
The problem states that if 1 is added to both the numerator and the denominator, the fraction becomes \( \frac{1}{2} \). This gives us the equation:
\[ \frac{x + 1}{y + 1} = \frac{1}{2} \]

**Hint:** Use the information about how the fraction changes when 1 is added to both parts to form another equation.

### Step 4: Cross-Multiply to Eliminate the Fraction
To eliminate the fraction, we can cross-multiply:
\[ 2(x + 1) = 1(y + 1) \]
This simplifies to:
\[ 2x + 2 = y + 1 \]

**Hint:** Cross-multiplication helps in simplifying equations involving fractions.

### Step 5: Rearrange the Second Equation
Rearranging the equation from Step 4 gives:
\[ 2x + 2 - 1 = y \]
So, we have:
\[ y = 2x + 1 \]

**Hint:** Rearranging equations can help isolate one variable in terms of another.

### Step 6: Substitute the First Equation into the Second Equation
Now, we can substitute the expression for \( y \) from the first equation \( y = x + 4 \) into the second equation \( y = 2x + 1 \):
\[ x + 4 = 2x + 1 \]

**Hint:** Substituting one equation into another allows you to solve for one variable.

### Step 7: Solve for \( x \)
Rearranging the equation gives:
\[ 4 - 1 = 2x - x \]
\[ 3 = x \]

**Hint:** Isolate the variable to find its value.

### Step 8: Find \( y \)
Now that we have \( x \), we can find \( y \) using the first equation:
\[ y = x + 4 = 3 + 4 = 7 \]

**Hint:** Use the value of one variable to find the other.

### Step 9: Write the Fraction
The fraction can now be expressed as:
\[ \frac{x}{y} = \frac{3}{7} \]

**Hint:** Ensure you clearly state the final answer in the context of the problem.

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

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