The numerator of a fraction is one more than its denominator. If its reciprocal is subtracted from if the difference is `11/30`. Find the fraction.

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript while ensuring clarity in each step.

### Step-by-Step Solution:

1. **Define the Variables**:
Let the denominator of the fraction be \( x \). According to the problem, the numerator is one more than the denominator, so the numerator will be \( x + 1 \).

**Fraction**:
\[ \text{Fraction} = \frac{x + 1}{x} \]

2. **Find the Reciprocal**:
The reciprocal of the fraction is:
\[ \text{Reciprocal} = \frac{x}{x + 1} \]

3. **Set Up the Equation**:
According to the problem, if the reciprocal is subtracted from the fraction, the difference is \( \frac{11}{30} \). Therefore, we can set up the equation:
\[ \frac{x + 1}{x} - \frac{x}{x + 1} = \frac{11}{30} \]

4. **Find a Common Denominator**:
The common denominator for the left side of the equation is \( x(x + 1) \). Thus, we rewrite the left side:
\[ \frac{(x + 1)^2 - x^2}{x(x + 1)} = \frac{11}{30} \]

5. **Expand the Numerator**:
Now, expand \( (x + 1)^2 \):
\[ (x + 1)^2 = x^2 + 2x + 1 \]
Therefore, the numerator becomes:
\[ x^2 + 2x + 1 - x^2 = 2x + 1 \]

6. **Set Up the Equation**:
Now, we have:
\[ \frac{2x + 1}{x(x + 1)} = \frac{11}{30} \]

7. **Cross Multiply**:
Cross multiplying gives us:
\[ 30(2x + 1) = 11x(x + 1) \]

8. **Expand Both Sides**:
Expanding both sides results in:
\[ 60x + 30 = 11x^2 + 11x \]

9. **Rearrange the Equation**:
Rearranging gives:
\[ 11x^2 + 11x - 60x - 30 = 0 \]
Simplifying this results in:
\[ 11x^2 - 49x - 30 = 0 \]

10. **Factor the Quadratic Equation**:
We need to factor \( 11x^2 - 49x - 30 \). We look for two numbers that multiply to \( 11 \times -30 = -330 \) and add to \( -49 \). The numbers are \( -55 \) and \( 6 \).

11. **Split the Middle Term**:
We rewrite the equation:
\[ 11x^2 - 55x + 6x - 30 = 0 \]
Grouping gives:
\[ 11x(x - 5) + 6(x - 5) = 0 \]
Factoring out \( (x - 5) \):
\[ (x - 5)(11x + 6) = 0 \]

12. **Solve for x**:
Setting each factor to zero gives:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \]
\[ 11x + 6 = 0 \quad \Rightarrow \quad x = -\frac{6}{11} \]

13. **Find the Fractions**:
- For \( x = 5 \):
\[ \text{Fraction} = \frac{5 + 1}{5} = \frac{6}{5} \]
- For \( x = -\frac{6}{11} \):
\[ \text{Fraction} = \frac{-\frac{6}{11} + 1}{-\frac{6}{11}} = \frac{\frac{5}{11}}{-\frac{6}{11}} = -\frac{5}{6} \]

### Final Answer:
The two possible fractions are \( \frac{6}{5} \) and \( -\frac{5}{6} \).