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The geometric mean of the observations $\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}, \ldots \ldots \mathrm{x}_{\mathrm{n}}$ is
$\mathrm{G}_{1}$, The geometric mean of the observations $\mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{y}_{3}, \ldots . \mathrm{y}_{\mathrm{n}}$
is $\mathrm{G}_{2}$. The geometric mean of observations $\frac{\mathrm{x}_{1}}{\mathrm{y}_{1}}, \frac{\mathrm{x}_{2}}{\mathrm{y}_{2}}, \frac{\mathrm{x}_{3}}{\mathrm{y}_{3}}, \ldots \cdot \frac{\mathrm{x}_{\mathrm{n}}}{\mathrm{y}_{\mathrm{n}}}$ is
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$\mathrm{G}_{1}$, The geometric mean of the observations $\mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{y}_{3}, \ldots . \mathrm{y}_{\mathrm{n}}$
is $\mathrm{G}_{2}$. The geometric mean of observations $\frac{\mathrm{x}_{1}}{\mathrm{y}_{1}}, \frac{\mathrm{x}_{2}}{\mathrm{y}_{2}}, \frac{\mathrm{x}_{3}}{\mathrm{y}_{3}}, \ldots \cdot \frac{\mathrm{x}_{\mathrm{n}}}{\mathrm{y}_{\mathrm{n}}}$ is
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Verified Answer
The correct answer is:
$\frac{\mathrm{G}_{1}}{\mathrm{G}_{2}}$
$\quad \mathrm{G}_{1}=\left[\mathrm{x}_{1} \times \mathrm{x}_{2} \times \mathrm{x}_{3} \times \ldots \ldots \ldots \times \mathrm{x}_{\mathrm{n}}\right]^{y_{1}}$
$\mathrm{G}_{2}=\left[\mathrm{y}_{1} \times \mathrm{y}_{2} \times \mathrm{y}_{3} \times \ldots \ldots . \times \mathrm{y}_{\mathrm{s}}\right]^{1 / \mathrm{m}}$
$\Rightarrow \frac{\mathrm{G}_{1}}{\mathrm{G}_{2}}=\left[\frac{\mathrm{x}_{1}}{\mathrm{y}_{1}} \times \frac{\mathrm{x}_{2}}{\mathrm{y}_{2}} \times \frac{\mathrm{x}_{3}}{\mathrm{y}_{3}} \times \ldots \ldots \ldots \times \frac{\mathrm{x}_{\mathrm{n}}}{\mathrm{y}_{\mathrm{n}}}\right]^{1 / \mathrm{n}}$
$\frac{\mathrm{G}_{1}}{\mathrm{G}_{2}}$ is the G.M of $\frac{\mathrm{x}_{1}}{\mathrm{y}_{1}}, \frac{\mathrm{x}_{2}}{\mathrm{y}_{2}}, \frac{\mathrm{x}_{3}}{\mathrm{y}_{3}}, \ldots \ldots \ldots \ldots, \frac{\mathrm{x}_{\mathrm{a}}}{\mathrm{y}_{\mathrm{a}}}$
$\mathrm{G}_{2}=\left[\mathrm{y}_{1} \times \mathrm{y}_{2} \times \mathrm{y}_{3} \times \ldots \ldots . \times \mathrm{y}_{\mathrm{s}}\right]^{1 / \mathrm{m}}$
$\Rightarrow \frac{\mathrm{G}_{1}}{\mathrm{G}_{2}}=\left[\frac{\mathrm{x}_{1}}{\mathrm{y}_{1}} \times \frac{\mathrm{x}_{2}}{\mathrm{y}_{2}} \times \frac{\mathrm{x}_{3}}{\mathrm{y}_{3}} \times \ldots \ldots \ldots \times \frac{\mathrm{x}_{\mathrm{n}}}{\mathrm{y}_{\mathrm{n}}}\right]^{1 / \mathrm{n}}$
$\frac{\mathrm{G}_{1}}{\mathrm{G}_{2}}$ is the G.M of $\frac{\mathrm{x}_{1}}{\mathrm{y}_{1}}, \frac{\mathrm{x}_{2}}{\mathrm{y}_{2}}, \frac{\mathrm{x}_{3}}{\mathrm{y}_{3}}, \ldots \ldots \ldots \ldots, \frac{\mathrm{x}_{\mathrm{a}}}{\mathrm{y}_{\mathrm{a}}}$
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