In number system aptitude, the difference between a positive proper fraction and its reciprocal is 9/20. What is the value of that fraction?

Introduction:
This question tests your understanding of fractions and their reciprocals in basic number system aptitude. You are given the difference between a positive proper fraction and its reciprocal and asked to determine the original fraction. This involves setting up and solving a quadratic equation in fractional form.

Given Data / Assumptions:

• The fraction is a positive proper fraction, so its value is between 0 and 1.
• The difference between the reciprocal of the fraction and the fraction itself is 9/20.
• Let the fraction be x, where 0 < x < 1.
• We need to find the exact value of x.

Concept / Approach:
A reciprocal of a non-zero number x is 1/x. Since the fraction is proper, 1/x will be greater than x. The statement “difference is 9/20” becomes an equation: 1/x − x = 9/20. We then clear denominators and solve the resulting quadratic equation to find the valid fractional solution between 0 and 1.

Step-by-Step Solution:
Step 1: Let the fraction be x.Step 2: Translate the statement.1/x − x = 9/20Step 3: Multiply both sides by 20x to clear denominators.20 − 20x^2 = 9xStep 4: Rearrange into standard quadratic form.20x^2 + 9x − 20 = 0Step 5: Solve using the quadratic formula.x = [−9 ± √(9^2 + 42020)] / (40)x = [−9 ± √1681] / 40 = [−9 ± 41] / 40Step 6: Compute both roots.x₁ = (32) / 40 = 4/5, x₂ = (−50) / 40 = −5/4

Verification / Alternative check:
Since the fraction must be a positive proper fraction, we discard −5/4. Now check x = 4/5: reciprocal is 5/4. Difference = 5/4 − 4/5 = (25 − 16)/20 = 9/20, which matches the given condition.

Why Other Options Are Wrong:
3/10 and 3/5 do not give a difference of 9/20 when compared with their reciprocals. 4/3 is not a proper fraction (it is greater than 1). 5/8 is less than 1 but its reciprocal difference does not equal 9/20.

Common Pitfalls:
Common errors include writing x − 1/x instead of 1/x − x, or forgetting that a proper fraction must lie between 0 and 1. Others try trial and error with options without forming the equation, which is slower and more error prone.

Final Answer:
The required proper fraction is 4/5.