Let's solve the problem step-by-step:
1. Calculate the initial value of the index:
The initial value of the index is determined by the sum of the products of the number of shares and the price per share for each stock.
- For Stock X:
[tex]\[
\text{Value of Stock X} = 2000 \text{ shares} \times \$ 3.80 = \$ 7600
\][/tex]
- For Stock Y:
[tex]\[
\text{Value of Stock Y} = 1000 \text{ shares} \times \$ 3.50 = \$ 3500
\][/tex]
- For Stock Z:
[tex]\[
\text{Value of Stock Z} = 3000 \text{ shares} \times \$ 4.30 = \$ 12900
\][/tex]
- Adding these values together, the initial value of the index is:
[tex]\[
\text{Initial value of the index} = \$ 7600 + \$ 3500 + \$ 12900 = \$ 24000
\][/tex]
2. Calculate the rise in the index:
The index rises by 5.4% over the course of the day. To find the increase in the index value, we multiply the initial value by 5.4%.
[tex]\[
\text{Rise in the index} = \$ 24000 \times \frac{5.4}{100} = \$ 24000 \times 0.054 = \$ 1296
\][/tex]
3. Calculate the final value of the index:
The final value of the index is the initial value plus the rise in the index.
[tex]\[
\text{Final value of the index} = \$ 24000 + \$ 1296 = \$ 25296
\][/tex]
4. Round the final value to the nearest hundred:
To round \[tex]$ 25296 to the nearest hundred, we look at the tens digit, which is 9. Since it is 5 or greater, we round up.
\[
\text{Final value rounded to the nearest hundred} = \$[/tex] 25300
\]
Therefore, the value of the index at the end of the day, rounded to the nearest hundred, is [tex]\(\$ 25300\)[/tex].
So, the correct answer is:
C. [tex]\(\$ 25,300\)[/tex]
