Question
Compare the following : $\frac{3 \sqrt{5}}{5 \sqrt{7}}, \frac{\sqrt{63}}{\sqrt{125}}$
✓
Answer
$\begin{aligned}
\frac{3 \sqrt{5}}{5 \sqrt{7}}, \frac{\sqrt{63}}{\sqrt{125}} & \\
3 \sqrt{5} \times \sqrt{125} & =3 \times \sqrt{5 \times 125} \\
& =3 \times \sqrt{5 \times 25 \times 5} \\
& =3 \times 5 \times 5 \\
& =3 \times 25=75 \\
5 \sqrt{7} \times \sqrt{63} & =5 \times \sqrt{7 \times 63} \\
& =5 \times \sqrt{7 \times 9 \times 7} \\
& =5 \times 3 \times 7=105
\end{aligned}$
Here, $75<105$
$\therefore \quad \frac{3 \sqrt{5}}{5 \sqrt{7}}<\frac{\sqrt{63}}{\sqrt{125}}$
\frac{3 \sqrt{5}}{5 \sqrt{7}}, \frac{\sqrt{63}}{\sqrt{125}} & \\
3 \sqrt{5} \times \sqrt{125} & =3 \times \sqrt{5 \times 125} \\
& =3 \times \sqrt{5 \times 25 \times 5} \\
& =3 \times 5 \times 5 \\
& =3 \times 25=75 \\
5 \sqrt{7} \times \sqrt{63} & =5 \times \sqrt{7 \times 63} \\
& =5 \times \sqrt{7 \times 9 \times 7} \\
& =5 \times 3 \times 7=105
\end{aligned}$
Here, $75<105$
$\therefore \quad \frac{3 \sqrt{5}}{5 \sqrt{7}}<\frac{\sqrt{63}}{\sqrt{125}}$
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