A fraction is greater than twice its reciprocal by `7/15` what is the fraction

Options

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

Answer

To solve the problem step by step, we will define the fraction and set up the equation based on the information given in the question.

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

### Step 2: Write 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 twice its reciprocal by \( \frac{7}{15} \). This can be expressed as:
\[ x = 2 \cdot \frac{1}{x} + \frac{7}{15} \]

### Step 4: Simplify the Equation
Multiply both sides by \( x \) to eliminate the fraction:
\[ x^2 = 2 + \frac{7x}{15} \]

### Step 5: Clear the Fraction
To eliminate the fraction, multiply the entire equation by 15:
\[ 15x^2 = 30 + 7x \]

### Step 6: Rearrange the Equation
Rearranging gives us a standard quadratic equation:
\[ 15x^2 - 7x - 30 = 0 \]

### Step 7: Use the Quadratic Formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 15 \), \( b = -7 \), and \( c = -30 \).

### Step 8: Calculate the Discriminant
First, calculate the discriminant:
\[ b^2 - 4ac = (-7)^2 - 4 \cdot 15 \cdot (-30) = 49 + 1800 = 1849 \]

### Step 9: Calculate the Roots
Now substitute back into the quadratic formula:
\[ x = \frac{7 \pm \sqrt{1849}}{30} \]
Since \( \sqrt{1849} = 43 \):
\[ x = \frac{7 \pm 43}{30} \]

### Step 10: Find the Two Possible Values
Calculating the two possible values:
1. \( x = \frac{7 + 43}{30} = \frac{50}{30} = \frac{5}{3} \)
2. \( x = \frac{7 - 43}{30} = \frac{-36}{30} = -\frac{6}{5} \)

### Step 11: Determine the Valid Fraction
Since we are looking for a positive fraction, we take:
\[ x = \frac{5}{3} \]

### Step 12: Verify the Solution
To verify, check if \( \frac{5}{3} \) satisfies the original condition:
- Twice the reciprocal of \( \frac{5}{3} \) is \( 2 \cdot \frac{3}{5} = \frac{6}{5} \).
- Now, check if \( \frac{5}{3} - \frac{6}{5} = \frac{7}{15} \).

Finding a common denominator (15):
\[ \frac{5}{3} = \frac{25}{15}, \quad \frac{6}{5} = \frac{18}{15} \]
Thus,
\[ \frac{25}{15} - \frac{18}{15} = \frac{7}{15} \]
This confirms our solution is correct.

### Final Answer
The fraction is \( \frac{5}{3} \).