Evaluate the decimal expression (0.2 × 0.2 + 0.01) divided by (0.1 × 0.1 + 0.02), and express the result as a simplified fraction.

Introduction / Context:
This problem checks your ability to work with decimal operations and algebraic structure. The expression contains products and sums of decimals in both the numerator and denominator, and the final answer needs to be expressed as a simplified fraction. Such questions are common in aptitude tests to see if you can handle precise decimal arithmetic and recognize patterns.

Given Data / Assumptions:

• Numerator: 0.2 × 0.2 + 0.01.
• Denominator: 0.1 × 0.1 + 0.02.
• We must compute (numerator) / (denominator).
• All values are exact decimals, and we assume standard arithmetic rules.

Concept / Approach:
First we simplify the numerator and denominator separately by performing multiplication and addition. Once both parts are simplified to single decimal numbers, we divide the numerator by the denominator. When possible, we then convert the resulting decimal into a fraction and simplify that fraction. Recognizing that 0.2 = 2/10 and 0.1 = 1/10 can also help to see the fraction relationships more clearly.

Step-by-Step Solution:
Step 1: Compute the numerator: 0.2 × 0.2 = 0.04. Step 2: Add 0.01 to the numerator: 0.04 + 0.01 = 0.05. Step 3: Compute the denominator: 0.1 × 0.1 = 0.01. Step 4: Add 0.02 to the denominator: 0.01 + 0.02 = 0.03. Step 5: The expression becomes 0.05 / 0.03. Step 6: Write 0.05 as 5/100 and 0.03 as 3/100. Step 7: Now 0.05 / 0.03 = (5/100) / (3/100) = (5/100) × (100/3) = 5/3.

Verification / Alternative check:
We can also treat 0.05 / 0.03 as a direct decimal division. 0.05 divided by 0.03 equals 1.6666 recurring, which is the decimal form of 5/3. Checking the fraction by converting 5/3 to decimal gives 1.6666 recurring, confirming alignment. Both methods support the same result, so our answer is consistent and reliable.

Why Other Options Are Wrong:
9/3 simplifies to 3, which is much larger than the computed value near 1.67.
11/3 is approximately 3.67, again far from the correct ratio.
13/3 is about 4.33, clearly not matching 0.05 / 0.03.

Common Pitfalls:
A common mistake is to add or multiply decimals incorrectly, such as misplacing decimal points. Some learners also forget that dividing by a decimal can be simplified by converting to fractions or by scaling both numerator and denominator by 100. Always calculate products first, then sums, and only finally perform the division.

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