To determine the total amount of special drink needed for the grand movie premiere, we'll add up the quantities of each ingredient specified in the recipe:
- Sparkling soda: [tex]\(4 \frac{1}{2}\)[/tex] gallons
- Berry extract: 1 gallon
- Frosted lemon fizz: [tex]\(2 \frac{3}{4}\)[/tex] gallons
- Crushed starfruit: [tex]\(\frac{1}{8}\)[/tex] gallon
First, we'll convert the mixed fractions to improper fractions to make addition easier:
1. [tex]\(4 \frac{1}{2}\)[/tex] gallons of sparkling soda can be converted as follows:
[tex]\[
4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2}
\][/tex]
2. 1 gallon of berry extract remains as:
[tex]\[
1 = \frac{2}{2}
\][/tex]
3. [tex]\(2 \frac{3}{4}\)[/tex] gallons of frosted lemon fizz can be converted as follows:
[tex]\[
2 \frac{3}{4} = 2 + \frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{11}{4}
\][/tex]
4. [tex]\(\frac{1}{8}\)[/tex] gallon of crushed starfruit is already in its simplest form.
Now, we need to find a common denominator to add these fractions. The least common multiple of 2, 4, and 8 is 8.
Convert each fraction to have a denominator of 8:
- [tex]\(\frac{9}{2}\)[/tex] becomes [tex]\(\frac{36}{8}\)[/tex]
- 1 (which is [tex]\(\frac{2}{2}\)[/tex]) becomes [tex]\(\frac{8}{8}\)[/tex]
- [tex]\(\frac{11}{4}\)[/tex] becomes [tex]\(\frac{22}{8}\)[/tex]
- [tex]\(\frac{1}{8}\)[/tex] remains [tex]\(\frac{1}{8}\)[/tex]
Now, add all these fractions together:
[tex]\[
\frac{36}{8} + \frac{8}{8} + \frac{22}{8} + \frac{1}{8} = \frac{36 + 8 + 22 + 1}{8} = \frac{67}{8}
\][/tex]
Convert the improper fraction back to a mixed number:
[tex]\[
\frac{67}{8} = 8 \frac{3}{8}
\][/tex]
Thus, the total amount of special drink needed is [tex]\(8 \frac{3}{8}\)[/tex] gallons.
The correct answer is:
[tex]\[
\boxed{8 \frac{3}{8} \text{ gallons}}
\][/tex]
