To solve this problem, we need to follow these steps:
1. Calculate the initial value of the index:
The index comprises three stocks: Stock X, Stock Y, and Stock Z.
- The value of Stock X is calculated by multiplying the number of shares by the price per share:
[tex]\( \text{Value of Stock X} = 2000 \times 3.80 = \$7600 \)[/tex]
- The value of Stock Y is calculated similarly:
[tex]\( \text{Value of Stock Y} = 1000 \times 3.50 = \$3500 \)[/tex]
- The value of Stock Z is also calculated in a similar manner:
[tex]\( \text{Value of Stock Z} = 3000 \times 4.30 = \$12900 \)[/tex]
- The initial value of the index is the sum of the values of the three stocks:
[tex]\[
\text{Initial Index Value} = 7600 + 3500 + 12900 = \$24000
\][/tex]
2. Calculate the increase in the index value:
The index rises by 5.4%. To find the increase in the index value, we calculate 5.4% of the initial index value:
[tex]\[
\text{Index Increase} = 0.054 \times 24000 = \$1296
\][/tex]
3. Calculate the final index value:
The final value of the index is obtained by adding the index increase to the initial index value:
[tex]\[
\text{Final Index Value} = 24000 + 1296 = \$25296
\][/tex]
4. Round the final index value to the nearest hundred:
To round [tex]\( 25296 \)[/tex] to the nearest hundred, we look at the tens digit (which is 9). Since 9 is greater than 5, we round up:
[tex]\[
\text{Rounded Final Index Value} = \$25300
\][/tex]
Thus, the value of the index at the end of the day, rounded to the nearest hundred, is [tex]\( \$25300 \)[/tex].
The correct answer is [tex]\( \boxed{A.\ \$25300} \)[/tex].
