A fraction is greater than its reciprocal by `(9)/(20)`.What is the fraction?

Options

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

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Fraction
Let the fraction be \( x \).

### Step 2: Define the Reciprocal
The reciprocal of the fraction \( x \) is \( \frac{1}{x} \).

### Step 3: Set Up the Equation
According to the problem, the fraction \( x \) is greater than its reciprocal \( \frac{1}{x} \) by \( \frac{9}{20} \). This can be expressed as:
\[ x - \frac{1}{x} = \frac{9}{20} \]

### Step 4: Clear the Fraction
To eliminate the fraction, multiply both sides of the equation by \( 20x \) (the least common multiple of the denominators):
\[ 20x \left( x - \frac{1}{x} \right) = 20x \cdot \frac{9}{20} \]
This simplifies to:
\[ 20x^2 - 20 = 9x \]

### Step 5: Rearrange the Equation
Rearranging the equation gives:
\[ 20x^2 - 9x - 20 = 0 \]

### Step 6: Factor the Quadratic Equation
To factor the quadratic equation \( 20x^2 - 9x - 20 = 0 \), we look for two numbers that multiply to \( 20 \times (-20) = -400 \) and add to \( -9 \). The numbers are \( -25 \) and \( 16 \):
\[ 20x^2 - 25x + 16x - 20 = 0 \]
Grouping the terms:
\[ (20x^2 - 25x) + (16x - 20) = 0 \]
Factoring by grouping:
\[ 5x(4x - 5) + 4(4x - 5) = 0 \]
Factoring out \( (4x - 5) \):
\[ (4x - 5)(5x + 4) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives:
1. \( 4x - 5 = 0 \) → \( 4x = 5 \) → \( x = \frac{5}{4} \)
2. \( 5x + 4 = 0 \) → \( 5x = -4 \) → \( x = -\frac{4}{5} \)

### Step 8: Determine Valid Fraction
Since we are looking for a positive fraction, we take:
\[ x = \frac{5}{4} \]

### Final Answer
The fraction is \( \frac{5}{4} \).

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