Question

Compare the following pairs of ratios.

(i) $\frac{\sqrt{5}}{3},\frac{3}{\sqrt{7}}$ (ii) $\frac{3\sqrt{5}}{5\sqrt{7}},\frac{\sqrt{63}}{\sqrt{125}}$ (iii) $\frac{5}{18},\frac{17}{121}$ (iv) $\frac{\sqrt{80}}{\sqrt{48}},\frac{\sqrt{45}}{\sqrt{27}}$ (v) $\frac{9.2}{5.1},\frac{3.4}{7.1}$

Answer 1

(i)
$$\begin{gathered} \sqrt{5}\times \sqrt{7}=\sqrt{5\times 7}=\sqrt{35} \\ 3\times 3=9=\sqrt{81} \end{gathered}$$
Now, $\sqrt{35}<\sqrt{81}$

$\therefore \frac{\sqrt{5}}{3}<\frac{3}{\sqrt{7}}$

(ii)
$$\begin{gathered} 3\sqrt{5}\times \sqrt{125}=3\sqrt{5}\times \sqrt{25\times 5}=3\sqrt{5}\times 5\sqrt{5}=75 \\ 5\sqrt{7}\times \sqrt{63}=5\sqrt{7}\times \sqrt{9\times 7}=5\sqrt{7}\times 3\sqrt{7}=105 \end{gathered}$$
Now, 75 < 105

$\therefore \frac{3\sqrt{5}}{5\sqrt{7}}<\frac{\sqrt{63}}{\sqrt{125}}$

(iii)
$$\begin{gathered} 5\times 121=605 \\ 18\times 17=306 \end{gathered}$$
Now, 605 > 306

$\therefore \frac{5}{18}>\frac{17}{121}$

(iv)
$$\begin{gathered} \sqrt{80}\times \sqrt{27}=\sqrt{16\times 5}\times \sqrt{9\times 3}=4\sqrt{5}\times 3\sqrt{3}=12\sqrt{15} \\ \sqrt{45}\times \sqrt{48}=\sqrt{9\times 5}\times \sqrt{16\times 3}=3\sqrt{5}\times 4\sqrt{3}=12\sqrt{15} \end{gathered}$$
Now, $12\sqrt{15}=12\sqrt{15}$

$\therefore \frac{\sqrt{80}}{\sqrt{48}}=\frac{\sqrt{45}}{\sqrt{27}}$

(v)
$$\begin{gathered} 9.2\times 7.1=65.32 \\ 5.1\times 3.4=17.34 \end{gathered}$$
Now, 65.32 > 17.34

$\therefore \frac{9.2}{5.1}>\frac{3.4}{7.1}$