Certainly! Let's find the Simple Average (Geometric Mean, GM) of the Relative Quantity indices step-by-step:
In the given table, we have quantities for two different years (Base Year and Current Year) for certain items:
[tex]\[
\begin{array}{|c|c|c|}
\hline
n & \text{Base Year Quantity} & \text{Current Year Quantity} \\
\hline
1 & 23 & 111.45 \\
\hline
2 & 8 & 12 \\
\hline
3 & 10 & 11 \\
\hline
4 & 15 & 10 \\
\hline
\end{array}
\][/tex]
### Step 1: Calculate the Relative Quantity Indices
First, we need to find the relative quantity for each item by dividing the Current Year Quantity by the Base Year Quantity:
1. For [tex]\( n_1 \)[/tex]:
[tex]\[
\text{Relative Quantity} = \frac{\text{Current Year Quantity}}{\text{Base Year Quantity}} = \frac{111.45}{23} \approx 4.845652173913043
\][/tex]
2. For [tex]\( n_2 \)[/tex]:
[tex]\[
\text{Relative Quantity} = \frac{\text{Current Year Quantity}}{\text{Base Year Quantity}} = \frac{12}{8} = 1.5
\][/tex]
3. For [tex]\( n_3 \)[/tex]:
[tex]\[
\text{Relative Quantity} = \frac{\text{Current Year Quantity}}{\text{Base Year Quantity}} = \frac{11}{10} = 1.1
\][/tex]
4. For [tex]\( n_4 \)[/tex]:
[tex]\[
\text{Relative Quantity} = \frac{\text{Current Year Quantity}}{\text{Base Year Quantity}} = \frac{10}{15} \approx 0.6666666666666666
\][/tex]
Now we have the relative quantities:
[tex]\[
[4.845652173913043, 1.5, 1.1, 0.6666666666666666]
\][/tex]
### Step 2: Calculate the Geometric Mean (GM) of the Relative Quantities
The formula for the Geometric Mean of [tex]\( n \)[/tex] numbers [tex]\( x_1, x_2, \ldots, x_n \)[/tex] is:
[tex]\[
\text{GM} = \sqrt[n]{x_1 \times x_2 \times \cdots \times x_n}
\][/tex]
In our case, we have four relative quantities: [tex]\( 4.845652173913043, 1.5, 1.1, 0.6666666666666666 \)[/tex].
[tex]\[
\text{GM} = \sqrt[4]{4.845652173913043 \times 1.5 \times 1.1 \times 0.6666666666666666}
\][/tex]
By multiplying these relative quantities:
[tex]\[
4.845652173913043 \times 1.5 \times 1.1 \times 0.6666666666666666 \approx 5.329375000000001
\][/tex]
Now, we take the fourth root (since there are four numbers):
[tex]\[
\text{GM} = \sqrt[4]{5.329375000000001} \approx 1.5194493597867031
\][/tex]
Therefore, the Simple Average (Geometric Mean) of the Relative Quantity indices is approximately:
[tex]\[
1.5194493597867031
\][/tex]
