The sum of the numerator and the denominator of a fraction is equal to 7. Four times the numerator is 8 less than 5 times the denominator. Then, the fraction is

Options

A. `(2)/(5)`
B. `(1)/(6)`
C. `(5)/(2)`
D. `(3)/(4)`

Answer

To solve the problem step by step, we will define the variables and set up the 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 Equations
From the problem, we have two pieces of information:
1. The sum of the numerator and the denominator is equal to 7:
\[ x + y = 7 \quad \text{(Equation 1)} \]
2. Four times the numerator is 8 less than five times the denominator:
\[ 4x = 5y - 8 \quad \text{(Equation 2)} \]

### Step 3: Solve Equation 1 for \( y \)
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 7 - x \]

### Step 4: Substitute \( y \) in Equation 2
Now, substitute \( y \) in Equation 2:
\[ 4x = 5(7 - x) - 8 \]
Expanding this, we get:
\[ 4x = 35 - 5x - 8 \]
Simplifying further:
\[ 4x = 27 - 5x \]

### Step 5: Combine Like Terms
Now, add \( 5x \) to both sides:
\[ 4x + 5x = 27 \]
This simplifies to:
\[ 9x = 27 \]

### Step 6: Solve for \( x \)
Now, divide both sides by 9:
\[ x = 3 \]

### Step 7: Find \( y \)
Now that we have \( x \), we can find \( y \) using Equation 1:
\[ y = 7 - x = 7 - 3 = 4 \]

### Step 8: Write the Fraction
Now that we have both the numerator and the denominator, the fraction is:
\[ \frac{x}{y} = \frac{3}{4} \]

### Final Answer
The fraction is \( \frac{3}{4} \).

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