$\sqrt{110\frac{1}{4}} = x$

### Concept & Strategy Convert the mixed fraction into an improper fraction, take the square root of the numerator and denominator independently, and then convert back into decimal format. Fraction conversion identity: $$ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $$ ### Step-by-Step Solution - **Step 1: Convert the mixed fraction to an improper fraction.** - Multiply the integer by the denominator and add the numerator: $110 \times 4 + 1 = 440 + 1 = 441$. - The expression becomes: $\sqrt{\frac{441}{4}}$ - **Step 2: Extract the square roots.** - Split the radical: $\frac{\sqrt{441}}{\sqrt{4}}$ - We know $\sqrt{441} = 21$ and $\sqrt{4} = 2$. - Result: $\frac{21}{2}$ - **Step 3: Convert the fraction to a decimal value.** - $\frac{21}{2} = 10.5$ ### Exam Strategy & Shortcut Alternatively, express the value directly as a decimal from the start: $\sqrt{110\frac{1}{4}} = \sqrt{110.25}$. We know that $10^2 = 100$ and $11^2 = 121$, so the answer must be between $10$ and $11$. Since the number ends in $.25$, its square root must end in $.5$. This points straight to $10.5$. ### Common Pitfall Mistakenly picking option (a) $10.25$ because the original question had a $\frac{1}{4}$ fractional component, assuming that the root splits directly onto the component parts. Never split a radical across individual terms of an addition or mixed fraction format. ### Final Answer Therefore, the correct answer is 10.5.