The geometric mean of the observations `x_(1),x_(2)x_(3)…x_(n)` is G. The geometric mean of the observations `y_(1), y_(2), y_(3).... Y_(n)` is `G_(2)` . The geometric mean of the observations `(x_(1))/(y_(1)),(x_(1))/(y_(2)),(x_(3))/(y_(3))…(x_(n))/(y_(n))` is

Options

A. `G_(1)G_(2)`
B. `ln(G_(1)G_(2))`
C. `(G_(1))/(G_(2))`
D. `ln(G_(1))/(G_(2))`

Answer

To find the geometric mean of the observations \(\frac{x_1}{y_1}, \frac{x_2}{y_2}, \frac{x_3}{y_3}, \ldots, \frac{x_n}{y_n}\), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Geometric Mean**:
The geometric mean \(G_1\) of the observations \(x_1, x_2, \ldots, x_n\) is defined as:
\[ G_1 = (x_1 \cdot x_2 \cdot x_3 \cdots x_n)^{\frac{1}{n}} \]
Similarly, the geometric mean \(G_2\) of the observations \(y_1, y_2, \ldots, y_n\) is:
\[ G_2 = (y_1 \cdot y_2 \cdot y_3 \cdots y_n)^{\frac{1}{n}} \]

2. **Setting Up the Problem**:
We need to find the geometric mean of the ratios \(\frac{x_1}{y_1}, \frac{x_2}{y_2}, \frac{x_3}{y_3}, \ldots, \frac{x_n}{y_n}\). Let's denote this geometric mean as \(G\).

3. **Expressing the Geometric Mean**:
The geometric mean \(G\) of the ratios can be expressed as:
\[ G = \left(\frac{x_1}{y_1} \cdot \frac{x_2}{y_2} \cdots \frac{x_n}{y_n}\right)^{\frac{1}{n}} \]

4. **Rearranging the Expression**:
We can rearrange the expression for \(G\):
\[ G = \left(\frac{x_1 \cdot x_2 \cdots x_n}{y_1 \cdot y_2 \cdots y_n}\right)^{\frac{1}{n}} \]

5. **Using the Definitions of \(G_1\) and \(G_2\)**:
From the definitions of \(G_1\) and \(G_2\), we can substitute:
\[ G_1 = (x_1 \cdot x_2 \cdots x_n)^{\frac{1}{n}} \quad \text{and} \quad G_2 = (y_1 \cdot y_2 \cdots y_n)^{\frac{1}{n}} \]
Therefore, we can express the product \(x_1 \cdot x_2 \cdots x_n\) and \(y_1 \cdot y_2 \cdots y_n\) in terms of \(G_1\) and \(G_2\):
\[ x_1 \cdot x_2 \cdots x_n = G_1^n \quad \text{and} \quad y_1 \cdot y_2 \cdots y_n = G_2^n \]

6. **Substituting Back**:
Substitute these back into the expression for \(G\):
\[ G = \left(\frac{G_1^n}{G_2^n}\right)^{\frac{1}{n}} = \frac{G_1}{G_2} \]

7. **Final Result**:
Thus, the geometric mean of the observations \(\frac{x_1}{y_1}, \frac{x_2}{y_2}, \ldots, \frac{x_n}{y_n}\) is:
\[ G = \frac{G_1}{G_2} \]