Simplify the compound expression: (1/5 ÷ (1/5 of 1/5)) / ((1/5 of 1/5) ÷ 1/5). Use the convention that “of” means multiplication.

Introduction / Context:Expressions mixing “÷” and “of” (multiplication) are common in aptitude tests. Correctly applying operator precedence and translating “of” to multiplication simplifies the problem. Here we compute a fraction whose numerator and denominator are themselves small fractional divisions.

Given Data / Assumptions:

• Use “of” as multiplication.
• Perform multiplication/division before addition/subtraction (standard precedence).
• Expression: (1/5 ÷ (1/5 * 1/5)) / ((1/5 * 1/5) ÷ 1/5).

Concept / Approach:Evaluate each inner part separately, then divide the results. Division by a fraction equals multiplication by its reciprocal, which greatly simplifies the arithmetic with small numbers like fifths and twenty-fifths.

Step-by-Step Solution:Compute the inner product: (1/5 of 1/5) = 1/25.Numerator: (1/5) ÷ (1/25) = (1/5) * (25/1) = 5.Denominator: (1/25) ÷ (1/5) = (1/25) * (5/1) = 1/5.Overall: 5 ÷ (1/5) = 5 * 5 = 25.

Verification / Alternative check:Rewrite the whole fraction at once: [(1/5)/(1/25)] / [(1/25)/(1/5)] = (5) / (1/5) = 25 — consistent.

Why Other Options Are Wrong:

• 1 / 5 / 1/5 / 125: These stem from inverting the wrong fraction or misinterpreting “of”. Only 25 follows correct precedence and reciprocal rules.

Common Pitfalls:Forgetting “of” means multiply; doing the outer division before simplifying the inner divisions; mishandling reciprocals.

Final Answer:25