$\sqrt{110.25} \times \sqrt{0.01} \div \sqrt{0.0025} - \sqrt{420.25}$ equals

### Concept & Logic This problem combines square roots of decimals with BODMAS/PEMDAS order of operations. Evaluate all square roots first by treating them as perfect squares divided by powers of 10, then apply standard arithmetic precedence (Division/Multiplication, then Subtraction). ### Step-by-Step Solution * Resolve the square roots of the decimal numbers: * $\sqrt{110.25} = \sqrt{\frac{11025}{100}} = \frac{105}{10} = 10.5$ * $\sqrt{0.01} = 0.1$ * $\sqrt{0.0025} = 0.05$ * $\sqrt{420.25} = \sqrt{\frac{42025}{100}} = \frac{205}{10} = 20.5$ * Substitute these values back into the equation: * $10.5 \times 0.1 \div 0.05 - 20.5$ * Apply BODMAS - Perform Division first: * $0.1 \div 0.05 = \frac{0.10}{0.05} = 2$ * Perform Multiplication next: * $10.5 \times 2 = 21$ * Finally, perform Subtraction: * $21 - 20.5 = 0.5$ ### Exam Strategy & Shortcut To quickly find the square roots of numbers ending in 25 (like 11025 and 42025), use the "number ending in 5" squaring trick in reverse. A number ending in 5 squares to something ending in 25. The preceding digits are formed by $n(n+1)$. For 110, $10 \times 11 = 110$, so the root is 105. For 420, $20 \times 21 = 420$, so the root is 205. This instantly gives you $10.5$ and $20.5$ without tedious manual square root division. ### Common Pitfall Misplacing decimal points during the square root phase. For example, incorrectly thinking $\sqrt{0.0025}$ is $0.005$ instead of $0.05$. Always check your root by squaring it mentally ($5 \times 5 = 25$, need 4 decimal places, so $0.05 \times 0.05 = 0.0025$). ### Final Answer **Therefore, the correct answer is 0.50.**