The simplification of $$ \frac{0.2 \times 0.2 + 0.02 \times 0.02 - 0.4 \times 0.02}{0.36} $$ gives

### Concept & Formula This problem tests your ability to spot algebraic identities disguised as decimal arithmetic. The numerator is an expanded form of a perfect square. $$(a - b)^2 = a^2 - 2ab + b^2$$ ### Step-by-Step Solution * Identify the structure of the numerator: * First term: $0.2 \times 0.2 = (0.2)^2$. This is $a^2$ where $a = 0.2$. * Second term: $0.02 \times 0.02 = (0.02)^2$. This is $b^2$ where $b = 0.02$. * Third term: $-0.4 \times 0.02$. Let's rewrite $-0.4$ as $-2 \times 0.2$. The term becomes $-2 \times 0.2 \times 0.02$, which perfectly matches $-2ab$. * Compress the numerator using the identity $(a - b)^2$: $$ (0.2 - 0.02)^2 $$ * Calculate the difference inside the bracket: $$ 0.20 - 0.02 = 0.18 $$ * Square the result to finalize the numerator: $$ (0.18)^2 = 0.0324 $$ * Now, divide by the denominator ($0.36$): $$ \frac{0.0324}{0.36} $$ * To simplify, multiply the top and bottom by $10000$ to remove decimals: $$ \frac{324}{3600} $$ * Reduce the fraction. Both are divisible by $36$: $$ \frac{9}{100} = 0.09 $$ ### Exam Strategy & Shortcut Once you recognize the numerator evaluates to $(0.18)^2$, look at the denominator, $0.36$. You can rewrite the fraction as $\frac{0.18 \times 0.18}{0.36}$. Since $0.18$ is exactly half of $0.36$, the fraction simplifies immediately to $\frac{0.18}{2} = 0.09$. This avoids calculating $0.0324$ entirely. ### Common Pitfall Failing to recognize the $-2ab$ structure because $0.4$ is written instead of $2 \times 0.2$. Always check if the coefficients in decimal sequences can be factored into a "2" to reveal a square identity. ### Final Answer Therefore, the correct answer is 0.09.