Let P = (0.08)^2, Q = 1/(0.08)^2, and R = (1 − 0.08)^2 − 1. Identify the correct ordering/relationship among P, Q, and R.

Introduction / Context: This question examines decimal computation and comparison, including a small square, its reciprocal square, and a shifted square minus 1.

Given Data / Assumptions:

• P = (0.08)^2
• Q = 1/(0.08)^2
• R = (1 − 0.08)^2 − 1

Concept / Approach: Compute each value exactly and then compare. Beware that squaring a small decimal gives a very small positive number, while the reciprocal square becomes large. A near-one square minus 1 will be negative.

Step-by-Step Solution:

Verification / Alternative check: Estimate check: 0.08^2 is clearly tiny positive; reciprocal square is very large; a number close to 1 squared then minus 1 is a small negative. The ordering matches intuition.

Why Other Options Are Wrong: Any option claiming P = R is false since P ≈ 0.0064 and R ≈ −0.1536. Options reversing the order ignore signs or magnitude.

Common Pitfalls: Misplacing decimal points; forgetting that reciprocals of numbers less than 1 are greater than 1; sign errors when subtracting 1.

Final Answer: R < P < Q