To determine the rate of change of the simple index over one week, we'll follow these steps so that we know the variations in the total value of the stocks from Day 1 to Day 8. Here is a detailed, step-by-step solution:
1. Calculate the total value of stocks on Day 1:
- StockABC: 4000 shares at \[tex]$3.15 per share
- StockXYZ: 5000 shares at \$[/tex]4.30 per share
- StockQRS: 6000 shares at \[tex]$4.60 per share
The total value is computed as follows:
\[
\text{Total value on Day 1} = (4000 \times 3.15) + (5000 \times 4.30) + (6000 \times 4.60)
\]
Simplifying this:
\[
\text{Total value on Day 1} = 12600 + 21500 + 27600 = 61700.0
\]
2. Calculate the total value of stocks on Day 8:
- StockABC: 4000 shares at \$[/tex]3.50 per share
- StockXYZ: 5000 shares at \[tex]$3.90 per share
- StockQRS: 6000 shares at \$[/tex]4.50 per share
The total value is:
[tex]\[
\text{Total value on Day 8} = (4000 \times 3.50) + (5000 \times 3.90) + (6000 \times 4.50)
\][/tex]
Simplifying this:
[tex]\[
\text{Total value on Day 8} = 14000 + 19500 + 27000 = 60500.0
\][/tex]
3. Compute the rate of change of the index:
We utilize the formula for the percentage rate of change:
[tex]\[
\text{Rate of Change} = \left( \frac{\text{Total value on Day 8} - \text{Total value on Day 1}}{\text{Total value on Day 1}} \right) \times 100
\][/tex]
Substituting the calculated totals:
[tex]\[
\text{Rate of Change} = \left( \frac{60500.0 - 61700.0}{61700.0} \right) \times 100
\][/tex]
Simplifying the fraction:
[tex]\[
\text{Rate of Change} = \left( \frac{-1200.0}{61700.0} \right) \times 100 \approx -1.9449 \%
\][/tex]
4. Round the rate of change to the nearest tenth:
[tex]\[
\text{Rounded Rate of Change} = -1.9 \%
\][/tex]
Therefore, the rate of change of the simple index over one week is [tex]\( \boxed{-1.9 \%} \)[/tex], corresponding to option D.
