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Q.
Let the harmonic mean and the geometric mean of two positive numbers be in the ratio $4: 5$. The two numbers are in the ratio
Sequences and Series
Solution:
Harmonic mean of $a, b$ is $H=\frac{2 a b}{a+b}$
Geometric mean $G=\sqrt{a b}$
Given: $\frac{H}{G}=\frac{4}{5}$,
so $\frac{2 \sqrt{a b}}{a+b}=\frac{4}{5}$
or, $\frac{a+b}{2 \sqrt{a b}}=\frac{5}{4}$
By componendo and dividendo
$\frac{(\sqrt{a}+\sqrt{b})^{2}}{(\sqrt{a}-\sqrt{b})^{2}}=\frac{9}{1}$
or $\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}=\frac{3}{1}$
Again, by componendo and dividendo $\frac{2 \sqrt{a}}{2 \sqrt{b}}=\frac{3+1}{3-1}$
$\frac{\sqrt{a}}{\sqrt{b}}=2$
or $\frac{a}{b}=\frac{4}{1}$