$$ \left( \frac{0.05}{0.25} + \frac{0.25}{0.05} \right)^3 = x $$

### Concept & Strategy When faced with fractions composed of decimals, shift the decimal point equally in both the numerator and denominator to convert them into integers before simplifying or applying exponents. ### Step-by-Step Solution * **Given:** $$ \left( \frac{0.05}{0.25} + \frac{0.25}{0.05} \right)^3 $$ * **Calculation:** Simplify the internal fractions by multiplying numerator and denominator by $100$ (shifting the decimal two places). $$ \frac{0.05}{0.25} = \frac{5}{25} = \frac{1}{5} = 0.2 $$ $$ \frac{0.25}{0.05} = \frac{25}{5} = 5 $$ * Add the simplified terms together: $$ 0.2 + 5 = 5.2 $$ * Cube the result: $$ (5.2)^3 = 5.2 \times 5.2 \times 5.2 $$ $$ (5.2)^2 = 27.04 $$ $$ 27.04 \times 5.2 = 140.608 $$ * Rounding to one decimal place to match the options gives $140.6$. ### Exam Strategy & Shortcut Instead of calculating $(5.2)^3$ exactly, use algebraic approximation. We know $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$. $$ (5 + 0.2)^3 \approx 5^3 + 3(5^2)(0.2) $$ $$ 125 + 3(25)(0.2) $$ $$ 125 + 75(0.2) = 125 + 15 = 140 $$ Since we ignored the smaller trailing terms ($3ab^2 + b^3$), the actual answer must be slightly greater than $140$. Looking at the options, $140.6$ is the only logical choice. ### Common Pitfall Students often struggle with calculating $(5.2)^3$ by hand, wasting precious time multiplying decimals. Failing to approximate or use binomial expansion leads to time drains and misalignment errors. ### Final Answer **Therefore, the correct answer is 140.6.**