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If the ratio of H.M. and G.M. of two quantities is $12: 13$, then the ratio of the numbers is
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Given that $\frac{H \cdot M .}{G \cdot M .}=\frac{12}{13} \Rightarrow \frac{\frac{2 a b}{a+b}}{\sqrt{a b}}=\frac{12}{13}$
or $\frac{a+b}{2 \sqrt{a b}}=\frac{13}{12}$
$\Rightarrow \quad \frac{(a+b)+2 \sqrt{a b}}{(a+b)-2 \sqrt{a b}}$ $=\frac{13+12}{13-12}=\frac{25}{1}$
$\Rightarrow \quad \frac{(\sqrt{a}+\sqrt{b})^2}{(\sqrt{a}-\sqrt{b})^2}=\frac{5^2}{1}$ $\Rightarrow \quad \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}=\frac{5}{1}$
$\Rightarrow \quad \frac{(\sqrt{a}+\sqrt{b})+(\sqrt{a}-\sqrt{b})}{(\sqrt{a}+\sqrt{b})-(\sqrt{a}-\sqrt{b})}$ $=\frac{5+1}{5-1}$
$\Rightarrow \frac{2 \sqrt{a}}{2 \sqrt{b}}=\frac{6}{4} \Rightarrow\left(\frac{a}{b}\right)^{1 / 2}$$-\frac{6}{4} \Rightarrow a: b=9: 4$
or $\frac{a+b}{2 \sqrt{a b}}=\frac{13}{12}$
$\Rightarrow \quad \frac{(a+b)+2 \sqrt{a b}}{(a+b)-2 \sqrt{a b}}$ $=\frac{13+12}{13-12}=\frac{25}{1}$
$\Rightarrow \quad \frac{(\sqrt{a}+\sqrt{b})^2}{(\sqrt{a}-\sqrt{b})^2}=\frac{5^2}{1}$ $\Rightarrow \quad \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}=\frac{5}{1}$
$\Rightarrow \quad \frac{(\sqrt{a}+\sqrt{b})+(\sqrt{a}-\sqrt{b})}{(\sqrt{a}+\sqrt{b})-(\sqrt{a}-\sqrt{b})}$ $=\frac{5+1}{5-1}$
$\Rightarrow \frac{2 \sqrt{a}}{2 \sqrt{b}}=\frac{6}{4} \Rightarrow\left(\frac{a}{b}\right)^{1 / 2}$$-\frac{6}{4} \Rightarrow a: b=9: 4$
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