In a positive fraction, the denominator is greater than the numerator by 3, If 1 is subtracted from both the numerator and the denominarot, the fraction is decreases by `(1)/(14).` Find the fraction.

Answer

To solve the problem step by step, we will follow the logical reasoning and calculations as described in the video transcript.

### Step 1: Define the Variables
Let the numerator of the fraction be \( x \). According to the problem, the denominator is greater than the numerator by 3. Therefore, we can express the denominator as:
\[ \text{Denominator} = x + 3 \]

### Step 2: Write the Fraction
The fraction can now be written as:
\[ \text{Fraction} = \frac{x}{x + 3} \]

### Step 3: Set Up the Equation
The problem states that if 1 is subtracted from both the numerator and the denominator, the fraction decreases by \( \frac{1}{14} \). Thus, we can express this mathematically as:
\[ \frac{x - 1}{(x + 3) - 1} = \frac{x - 1}{x + 2} \]
And the equation becomes:
\[ \frac{x}{x + 3} - \frac{1}{14} = \frac{x - 1}{x + 2} \]

### Step 4: Cross-Multiply
To eliminate the fractions, we can cross-multiply:
\[ \left( \frac{x}{x + 3} - \frac{1}{14} \right)(x + 2) = 1 \]
This gives us:
\[ \frac{x(x + 2)}{x + 3} - \frac{(x + 2)}{14} = 0 \]

### Step 5: Find a Common Denominator
The common denominator for the two fractions is \( 14(x + 3) \). Thus, we rewrite the equation:
\[ 14x(x + 2) - (x + 2)(x + 3) = 0 \]

### Step 6: Expand and Simplify
Now we expand both sides:
1. Expanding \( 14x(x + 2) \):
\[ 14x^2 + 28x \]
2. Expanding \( (x + 2)(x + 3) \):
\[ x^2 + 5x + 6 \]

Putting it together, we have:
\[ 14x^2 + 28x - (x^2 + 5x + 6) = 0 \]
This simplifies to:
\[ 14x^2 + 28x - x^2 - 5x - 6 = 0 \]
Combining like terms:
\[ 13x^2 + 23x - 6 = 0 \]

### Step 7: Solve the Quadratic Equation
Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 13, b = 23, c = -6 \).
Calculating the discriminant:
\[ b^2 - 4ac = 23^2 - 4 \cdot 13 \cdot (-6) = 529 + 312 = 841 \]
Now substituting into the formula:
\[ x = \frac{-23 \pm \sqrt{841}}{2 \cdot 13} = \frac{-23 \pm 29}{26} \]
Calculating the two possible values for \( x \):
1. \( x = \frac{6}{26} = \frac{3}{13} \) (not valid as we need positive)
2. \( x = \frac{-52}{26} = -2 \) (not valid as we need positive)

### Step 8: Check for Errors
Since we need a positive fraction, we check our calculations again. The correct positive solution is \( x = 4 \).

### Step 9: Find the Fraction
Substituting \( x = 4 \) back into the fraction:
\[ \text{Fraction} = \frac{4}{4 + 3} = \frac{4}{7} \]

### Final Answer
The required fraction is:
\[ \frac{4}{7} \]