Let the proper fraction be \(x\) \(\therefore\) According to the question, \(\frac{1}{x} - x = \frac{9}{20}\) i.e., \(\frac{1 - x^2}{x} = \frac{9}{20}\) i.e., \(20 - 20x^2 = 9x\) i.e., \(20 - 9x - 20x^2 = 0\) i.e., \(20x^2 + 9x - 20 = 0\) i.e., \(20x^2 + 25x - 16x - 20 = 0\) i.e., \(5x(4x + 5) - 4(4x + 5) = 0\) i.e., \((4x + 5)(5x - 4) = 0\) \(\therefore\) \(x = -\frac{5}{4}\) or \(x = \frac{4}{5}\) Since the fraction is a proper fraction, the value must be \(\frac{4}{5}\)