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A simple index of three stocks opens the day with these values:

\begin{tabular}{|l|c|c|}
\cline {2-3}
\multicolumn{1}{c|}{} & No. of shares & Price per share \\
\hline
Stock [tex]$X$[/tex] & 5000 & [tex]$\$[/tex] 4.30[tex]$ \\
\hline
Stock $[/tex]Y[tex]$ & 2000 & $[/tex]\[tex]$ 3.20$[/tex] \\
\hline
Stock [tex]$Z$[/tex] & 8000 & [tex]$\$[/tex] 4.90[tex]$ \\
\hline
\end{tabular}

The index rises $[/tex]4.9\%[tex]$ over the course of the day. What is the value of the index at the end of the day? Round your answer to the nearest hundred.

A. $[/tex]\[tex]$ 70,400$[/tex]
B. [tex]$\$[/tex] 67,100[tex]$
C. $[/tex]\[tex]$ 68,700$[/tex]
D. [tex]$\$[/tex] 68,300$



Answer :

GinnyAnswer
To determine the value of the index at the end of the day, follow these steps:

1. Calculate the initial value of the index:
- For Stock [tex]\(X\)[/tex]: The value is [tex]\(5000 \text{ shares} \times \$4.30 \text{ per share} = \$21500\)[/tex].
- For Stock [tex]\(Y\)[/tex]: The value is [tex]\(2000 \text{ shares} \times \$3.20 \text{ per share} = \$6400\)[/tex].
- For Stock [tex]\(Z\)[/tex]: The value is [tex]\(8000 \text{ shares} \times \$4.90 \text{ per share} = \$39200\)[/tex].
- Summing these values gives the total initial value of the index:
[tex]\[ \$21500 + \$6400 + \$39200 = \$67100 \][/tex]

2. Determine the percentage increase over the day:
- The index rises by [tex]\(4.9\%\)[/tex].
- The new value of the index at the end of the day is calculated as follows:
[tex]\[ \text{New total value} = \text{Initial total value} \times (1 + \text{Percentage increase}) \][/tex]
- Substituting the given values:
[tex]\[ \text{New total value} = \$67100 \times (1 + 0.049) = \$67100 \times 1.049 = \$70387.9 \][/tex]

3. Round the final value to the nearest hundred:
- The value \[tex]$70387.9 rounds to \$[/tex]70400.

Thus, the value of the index at the end of the day, rounded to the nearest hundred, is:
[tex]\[ \boxed{\$70400} \][/tex]

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