$[(125)^2 \div 50 \times 20] \div 25 = x$

Concept & Strategy This problem tests simplification using the properties of fractions and the order of operations (BODMAS). Converting consecutive multiplication and division into a single fraction allows for rapid cancellation of terms without calculating large squares. Step-by-Step Solution * Given expression: $$ [(125)^2 \div 50 \times 20] \div 25 = x $$ * Step 1: Simplify the expression inside the square brackets first. Rewrite the division and multiplication as a fraction. $$ [(125 \times 125) \times \frac{20}{50}] $$ * Step 2: Simplify the fraction $\frac{20}{50}$ to $\frac{2}{5}$. $$ [125 \times 125 \times \frac{2}{5}] $$ * Step 3: Cancel out the $5$ in the denominator with one of the $125$s ($125 \div 5 = 25$). $$ [125 \times 25 \times 2] $$ * Step 4: Multiply the remaining terms. It is easier to multiply $25 \times 2$ first. $$ 125 \times 50 = 6250 $$ So, the inner bracket evaluates to $6250$. * Step 5: Perform the final division outside the bracket. $$ 6250 \div 25 = 250 $$ Exam Strategy & Shortcut **Factoring out 25:** You can solve this almost entirely mentally by keeping factors intact. $$ \frac{125 \times 125 \times 20}{50 \times 25} $$ Notice that $50 \times 25 = 1250$. So the expression is: $$ \frac{125 \times 125 \times 20}{1250} $$ Divide $1250$ by $125$ to get $10$ in the denominator. $$ \frac{125 \times 20}{10} $$ Cancel the zeros: $$ 125 \times 2 = 250 $$ Common Pitfall The biggest time-waster here is brute-force calculation. Students often calculate $125^2 = 15625$, then divide by $50$ to get $312.5$, then multiply by $20$ to get $6250$, and finally divide by $25$. This creates massive opportunities for arithmetic errors. Final Answer **Therefore, the correct answer is 250.**