Evaluate the expression (0.2 × 0.2 + 0.01) ÷ (0.1 × 0.1 + 0.02) and express the result as a simplified fraction.

Introduction / Context:
This question involves evaluation of an expression containing decimals in both the numerator and denominator. It tests your ability to perform decimal multiplication and addition correctly and then convert the resulting ratio into a simplified fraction. Such manipulation is common in decimal fraction and algebra sections of aptitude tests.

Given Data / Assumptions:

• The expression is (0.2 × 0.2 + 0.01) ÷ (0.1 × 0.1 + 0.02).
• All operations inside the numerator and denominator are standard multiplication and addition.
• We must evaluate the expression exactly and then express the result as a fraction in simplest form.
• No rounding is needed because all decimals are finite with a small number of digits.

Concept / Approach:
The plan is to calculate the numerator and denominator separately as decimals, then form their quotient. Once we have the decimal result, we convert it into a fraction and simplify. Because the numbers are small and neat, each step involves straightforward calculations and leads to a simple rational number.

Step-by-Step Solution:
First evaluate the numerator: 0.2 × 0.2 + 0.01.Compute 0.2 × 0.2 = 0.04.Add 0.01: numerator = 0.04 + 0.01 = 0.05.Next evaluate the denominator: 0.1 × 0.1 + 0.02.Compute 0.1 × 0.1 = 0.01.Add 0.02: denominator = 0.01 + 0.02 = 0.03.Now form the ratio: 0.05 ÷ 0.03.Write 0.05 as 5/100 and 0.03 as 3/100.Then 0.05 ÷ 0.03 = (5/100) ÷ (3/100) = (5/100) × (100/3) = 5/3.Therefore, the value of the expression is 5/3.

Verification / Alternative check:
We can also treat 0.05 ÷ 0.03 directly as a decimal division. Calculate 0.05 ÷ 0.03 by multiplying numerator and denominator by 100: 5 ÷ 3. This gives 1.6666..., whose exact fractional representation is 5/3. Both the fraction manipulation method and the direct decimal division lead to the same result, confirming that 5/3 is correct.

Why Other Options Are Wrong:
Option 9/5 equals 1.8, which would require the numerator and denominator to be in a different ratio than 5 to 3. Option 41/4 is 10.25, far too large for a quotient of two small decimals less than 1. Option 41/12 is about 3.4167, also too large. Option 3/5 equals 0.6, which would correspond to a smaller ratio than 5/3. None of these match the computed decimal 1.6666..., so they are incorrect.

Common Pitfalls:
Errors often arise from miscalculating the initial decimal products, for example confusing 0.2 × 0.2 with 0.2 or 0.04 with 0.004. Another mistake is incorrectly handling the decimal division and not clearing the decimals in numerator and denominator properly. Converting each decimal to a fraction and simplifying step by step is a reliable way to avoid such mistakes.

Final Answer:
The simplified value of the expression (0.2 × 0.2 + 0.01) ÷ (0.1 × 0.1 + 0.02) is 5/3.