$\sqrt{\frac{48.4}{0.289}}$ is equal to

Concept & Math Similar to evaluating other decimal fractions under roots, equalize the number of decimal places before removing the decimal point. Then extract the whole-number roots and convert the resulting improper fraction to a mixed number. Step-by-Step Solution * Given expression: $\sqrt{\frac{48.4}{0.289}}$ * The numerator has 1 decimal place, and the denominator has 3. Equalize them by adding two trailing zeros to the numerator: $\sqrt{\frac{48.400}{0.289}}$ * Once equalized, drop the decimal points entirely: $\sqrt{\frac{48400}{289}}$ * Recognize the core perfect squares: $484 = 22^2$ and $289 = 17^2$. * Evaluate the root: $\frac{\sqrt{484 \times 100}}{\sqrt{289}} = \frac{22 \times 10}{17} = \frac{220}{17}$ * Convert the improper fraction $\frac{220}{17}$ into a mixed fraction. Divide $220$ by $17$: $17 \times 10 = 170$, and $17 \times 2 = 34$, so $17 \times 12 = 204$. * The remainder is $220 - 204 = 16$. * This translates to the mixed fraction $12 \frac{16}{17}$. Exam Strategy & Shortcut You can quickly estimate the final division: $220 \div 17$. Since $17 \times 10 = 170$, the quotient is clearly greater than 10 (eliminating Option a). Because $17 \times 13 = 221$, the quotient must be just under 13. This logic points directly to $12 \frac{16}{17}$ without having to perform long division mechanically. Common Pitfall A frequent error is dropping the decimal point prematurely without padding, writing $\sqrt{\frac{484}{289}} = \frac{22}{17}$. This directly leads to the incorrect mixed fraction $1 \frac{5}{17}$ (Option a). Always equalize the decimal places first. Final Answer Therefore, the correct answer is $12 \frac{16}{17}$.