The difference between a positive proper fraction and its reciprocal is 9/20. Find the fraction.

Given data

• Let the proper fraction be x, 0 < x < 1.
• Then its reciprocal is 1/x (which > 1).
• Difference: (1/x − x) = 9/20.

Concept / Approach

• Solve the quadratic arising from the equation.

Step-by-step calculation

1/x − x = 9/20Multiply by 20x: 20 − 20x^2 = 9xRearrange: 20x^2 + 9x − 20 = 0Discriminant Δ = 9^2 + 4·20·20 = 81 + 1600 = 1681 = 41^2x = (−9 ± 41) / 40 ⇒ positive root x = 32/40 = 4/5

Verification

1/x − x = 5/4 − 4/5 = (25 − 16)/20 = 9/20 ✔

Common pitfalls

• Using x − 1/x instead of 1/x − x (the latter is positive for a proper fraction).

Final Answer

The fraction is 4/5.