A simple index of three stocks has opening values on Day 1 and Day 8 as shown in the table below.

\begin{tabular}{|l|c|c|c|c|}
\hline
\multirow{2}{*}{} & \multicolumn{2}{|c|}{Day 1} & \multicolumn{2}{c|}{Day 8} \\
\cline{2-5}
& No. shares & \begin{tabular}{c}Price per\\ share\end{tabular} & No. shares & \begin{tabular}{c}Price per\\ share\end{tabular} \\
\hline
StockABC & 4000 & [tex]$\$[/tex] 3.15[tex]$ & 4000 & $[/tex]\[tex]$ 3.50$[/tex] \\
\hline
StockXYZ & 5000 & [tex]$\$[/tex] 4.30[tex]$ & 5000 & $[/tex]\[tex]$ 3.90$[/tex] \\
\hline
StockQRS & 6000 & [tex]$\$[/tex] 4.60[tex]$ & 6000 & $[/tex]\[tex]$ 4.50$[/tex] \\
\hline
\end{tabular}

What is the rate of change of this simple index over one week? Round your answer to the nearest tenth.

A. [tex]$1.2 \%$[/tex]

B. [tex]$-1.2 \%$[/tex]

C. [tex]$2.0 \%$[/tex]

D. [tex]$-2.0 \%$[/tex]



Answer :

To determine the rate of change of the simple index over one week, we'll follow these steps:

1. Calculate the total value of each stock on Day 1 and Day 8:

- Stock ABC:
- Day 1: [tex]\( 4000 \times 3.15 = 12600 \)[/tex] dollars
- Day 8: [tex]\( 4000 \times 3.50 = 14000 \)[/tex] dollars

- Stock XYZ:
- Day 1: [tex]\( 5000 \times 4.30 = 21500 \)[/tex] dollars
- Day 8: [tex]\( 5000 \times 3.90 = 19500 \)[/tex] dollars

- Stock QRS:
- Day 1: [tex]\( 6000 \times 4.60 = 27600 \)[/tex] dollars
- Day 8: [tex]\( 6000 \times 4.50 = 27000 \)[/tex] dollars

2. Sum up the total values for each day:

- Total value on Day 1:
[tex]\[ 12600 + 21500 + 27600 = 61700 \text{ dollars} \][/tex]

- Total value on Day 8:
[tex]\[ 14000 + 19500 + 27000 = 60500 \text{ dollars} \][/tex]

3. Calculate the rate of change of the index: The rate of change is calculated as follows:
[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]

Substitute the values:
[tex]\[ \text{Rate of change} = \left( \frac{60500 - 61700}{61700} \right) \times 100 \][/tex]
[tex]\[ \text{Rate of change} = \left( \frac{-1200}{61700} \right) \times 100 \approx -1.9\% \][/tex]

4. Round the rate of change to the nearest tenth:
The rate of change to the nearest tenth is [tex]\(-1.9\%\)[/tex].

Thus, the rate of change of the simple index over one week is [tex]\(\boxed{-1.9\%}\)[/tex]. Therefore, the correct answer is not listed in the options provided. Please confirm the choices or the problem details.