Let's solve the problem step by step to find out the total amount of special drink needed for the grand movie premiere.
1. Convert mixed numbers to improper fractions if necessary:
- 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
2. First, convert the mixed numbers into improper fractions:
- [tex]\( 4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2} \)[/tex]
- [tex]\( 2 \frac{3}{4} = 2 + \frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{11}{4} \)[/tex]
3. Now, add up all the quantities:
- Sparkling soda: [tex]\( \frac{9}{2} \)[/tex] gallons
- Berry extract: 1 gallon
- Frosted lemon fizz: [tex]\( \frac{11}{4} \)[/tex] gallons
- Crushed starfruit: [tex]\( \frac{1}{8} \)[/tex] gallon
We need to get a common denominator to add these fractions:
- For [tex]\( \frac{9}{2} \)[/tex]: Multiply by 4: [tex]\( \frac{9}{2} \times \frac{4}{4} = \frac{36}{8} \)[/tex]
- For 1 gallon: Multiply by 8: [tex]\( 1 \times \frac{8}{8} = \frac{8}{8} \)[/tex]
- For [tex]\( \frac{11}{4} \)[/tex]: Multiply by 2: [tex]\( \frac{11}{4} \times \frac{2}{2} = \frac{22}{8} \)[/tex]
- For [tex]\( \frac{1}{8} \)[/tex]: Already has the denominator 8
4. Add these fractions:
- [tex]\( \frac{36}{8} + \frac{8}{8} + \frac{22}{8} + \frac{1}{8} \)[/tex]
5. Combine the numerators over the common denominator:
- [tex]\( \frac{36 + 8 + 22 + 1}{8} = \frac{67}{8} \)[/tex]
6. 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.
Therefore, the correct choice is:
[tex]\( 8 \frac{3}{8} \)[/tex] gallons
