4. If the difference between the reciprocal of a positive proper fraction and the fraction itself be \( \frac{9}{20} \), the fraction is :

Let's call the positive proper fraction \( x \). Its reciprocal is \( \frac{1}{x} \). According to the problem, the difference between the reciprocal and the fraction itself can be expressed as: \[ \frac{1}{x} - x = \frac{9}{20} \] To eliminate the fraction, we can multiply through by \( 20x \): \[ 20 - 20x^2 = 9x \] Rearranging gives us: \[ 20x^2 + 9x - 20 = 0 \] Now, we can solve this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 20 \), \( b = 9 \), and \( c = -20 \). Calculating the discriminant: \[ b^2 - 4ac = 9^2 - 4 \times 20 \times (-20) = 81 + 1600 = 1681 \] Now plug into the quadratic formula: \[ x = \frac{-9 \pm \sqrt{1681}}{2 \times 20} = \frac{-9 \pm 41}{40} \] Calculating the two possible values: 1. \( x = \frac{32}{40} = \frac{4}{5} \) 2. \( x = \frac{-50}{40} \), which is negative and hence not valid for a positive fraction. Thus, the fraction is \( \frac{4}{5} \).