Evaluate the expression (0.2 × 0.2 + 0.01) × (0.1 × 0.1 + 0.02)^(-1) and express your answer as a simplified fraction.

Introduction / Context:
This question assesses your ability to work with decimals, basic algebraic expressions, and reciprocals. The notation ( )^(-1) means the reciprocal of the quantity inside the brackets. After evaluating the decimal operations, you must simplify the result to a fraction in lowest terms.

Given Data / Assumptions:

• Expression: (0.2 × 0.2 + 0.01) × (0.1 × 0.1 + 0.02)^(-1).
• We must compute this exactly.
• The final answer should be given as a simplified fraction.

Concept / Approach:
First, simplify each part inside the brackets. Then evaluate the reciprocal of the second bracket. Finally, multiply the simplified first bracket by this reciprocal. Converting the decimals to fractions at some stage makes simplification easier and more precise.

Step-by-Step Solution:
Step 1: Simplify the first bracket: 0.2 × 0.2 + 0.01. Step 2: Compute 0.2 × 0.2 = 0.04. Step 3: Add 0.01 to get 0.04 + 0.01 = 0.05. Step 4: Simplify the second bracket: 0.1 × 0.1 + 0.02. Step 5: Compute 0.1 × 0.1 = 0.01. Step 6: Add 0.02 to get 0.01 + 0.02 = 0.03. Step 7: Apply the power (−1): (0.03)^(-1) = 1 / 0.03. Step 8: Now the whole expression is 0.05 × (1 / 0.03) = 0.05 / 0.03. Step 9: Convert to fractions: 0.05 = 5/100 and 0.03 = 3/100. Step 10: Compute (5/100) / (3/100) = (5/100) × (100/3) = 5/3.

Verification / Alternative check:
You may compute 0.05 ÷ 0.03 directly as a decimal: 0.05 ÷ 0.03 ≈ 1.6666..., which corresponds to 5/3. This matches the fraction obtained via symbolic manipulation, confirming the answer.

Why Other Options Are Wrong:
9/3 = 3, which is too large and would come from miscalculating 0.05 or 0.03.
11/3 ≈ 3.6667 and 13/3 ≈ 4.3333, both far from the correct value around 1.67.
1 would require 0.05 and 0.03 to be equal, which is not the case.

Common Pitfalls:
A common mistake is to ignore the meaning of the power (−1) and treat it like a negative exponent in another context instead of a reciprocal. Another pitfall is rough decimal rounding, such as treating 0.05 / 0.03 as 5 / 3 without justifying the intermediate steps. Converting decimals to fractions is a clean way to ensure accuracy.

Final Answer:
The value of the expression is 5/3.