Question 2 of 10

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 & \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 Stock ABC & 8000 & [tex]$\$[/tex] 4.25[tex]$ & 8000 & $[/tex]\[tex]$ 3.90$[/tex] \\
\hline Stock XYZ & 5000 & [tex]$\$[/tex] 2.90[tex]$ & 5000 & $[/tex]\[tex]$ 2.50$[/tex] \\
\hline Stock QRS & 2000 & [tex]$\$[/tex] 6.40[tex]$ & 2000 & $[/tex]\[tex]$ 6.10$[/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]$-8.4 \%$[/tex]
B. [tex]$-7.7 \%$[/tex]
C. [tex]$7.7 \%$[/tex]
D. [tex]$8.4 \%$[/tex]



Answer :

To find the rate of change of a portfolio of three stocks over one week, we need to follow a detailed step-by-step approach:

1. Calculate the total value of the portfolio on Day 1:
- For Stock ABC: [tex]\( \text{No. of shares} \times \text{Price per share} = 8000 \times 4.25 = 34000 \)[/tex]
- For Stock XYZ: [tex]\( \text{No. of shares} \times \text{Price per share} = 5000 \times 2.90 = 14500 \)[/tex]
- For Stock QRS: [tex]\( \text{No. of shares} \times \text{Price per share} = 2000 \times 6.40 = 12800 \)[/tex]

Summing these values gives us the total value for Day 1:
[tex]\[ 34000 + 14500 + 12800 = 61300 \][/tex]

2. Calculate the total value of the portfolio on Day 8:
- For Stock ABC: [tex]\( \text{No. of shares} \times \text{Price per share} = 8000 \times 3.90 = 31200 \)[/tex]
- For Stock XYZ: [tex]\( \text{No. of shares} \times \text{Price per share} = 5000 \times 2.50 = 12500 \)[/tex]
- For Stock QRS: [tex]\( \text{No. of shares} \times \text{Price per share} = 2000 \times 6.10 = 12200 \)[/tex]

Summing these values gives us the total value for Day 8:
[tex]\[ 31200 + 12500 + 12200 = 55900 \][/tex]

3. Calculate the rate of change of the portfolio’s value from Day 1 to Day 8:
- The formula for rate of change is:
[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]
- Plugging in the values:
[tex]\[ \text{Rate of Change} = \left( \frac{55900 - 61300}{61300} \right) \times 100 = \left( \frac{-5400}{61300} \right) \times 100 \approx -8.809135 \][/tex]

4. Round the rate of change to the nearest tenth:
[tex]\[ -8.809135 \approx -8.8 \][/tex]

Therefore, the rate of change of this simple index over one week, rounded to the nearest tenth, is [tex]\(-8.8\%\)[/tex].

The correct answer is:
A. [tex]\(-8.4\%\)[/tex]
B. [tex]\(-7.7\%\)[/tex]
C. [tex]\(7.7\%\)[/tex]
D. [tex]\(8.4\%\)[/tex]

The correct choice is not available, which may be a mistake. However, based on our careful calculations and rounding to the nearest tenth, the correct rate of change should be [tex]\(-8.8\%\)[/tex].