The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, then the denominator becomes eight times the numerator. Find the fraction.

Options

A. `(3)/(8)`
B. `(3)/(7)`
C. `(4)/(8)`
D. `(2)/(7)`

Answer

To solve the problem step by step, let's denote the numerator of the fraction as \( x \) and the denominator as \( y \).

### Step 1: Set up the equations based on the problem statement.
According to the problem, the numerator is 4 less than the denominator. This can be expressed as:
\[ x = y - 4 \]

### Step 2: Modify the numerator and denominator.
Next, we are told that if the numerator is decreased by 2 and the denominator is increased by 1, the new denominator becomes eight times the new numerator. This gives us the second equation:
\[ y + 1 = 8(x - 2) \]

### Step 3: Substitute the first equation into the second equation.
We can substitute \( x \) from the first equation into the second equation:
\[ y + 1 = 8((y - 4) - 2) \]
This simplifies to:
\[ y + 1 = 8(y - 6) \]

### Step 4: Expand and simplify the equation.
Expanding the right side:
\[ y + 1 = 8y - 48 \]
Now, rearranging the equation to isolate \( y \):
\[ y + 1 + 48 = 8y \]
\[ 49 = 8y - y \]
\[ 49 = 7y \]

### Step 5: Solve for \( y \).
Dividing both sides by 7 gives:
\[ y = \frac{49}{7} = 7 \]

### Step 6: Substitute back to find \( x \).
Now that we have \( y \), we can find \( x \) using the first equation:
\[ x = y - 4 = 7 - 4 = 3 \]

### Step 7: Write the fraction.
The fraction is given by:
\[ \text{Fraction} = \frac{x}{y} = \frac{3}{7} \]

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

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