To determine the total number of cups of punch Peter makes, we'll add the three given amounts together: [tex]\(4 \frac{1}{2}\)[/tex] cups of orange juice, [tex]\(1 \frac{1}{3}\)[/tex] cups of ginger ale, and [tex]\(6 \frac{1}{3}\)[/tex] cups of strawberry lemonade.
Step 1: Convert each mixed number to an improper fraction
For [tex]\(4 \frac{1}{2}\)[/tex] cups of orange juice:
[tex]\[
4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{9}{2}
\][/tex]
For [tex]\(1 \frac{1}{3}\)[/tex] cups of ginger ale:
[tex]\[
1 \frac{1}{3} = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3}
\][/tex]
For [tex]\(6 \frac{1}{3}\)[/tex] cups of strawberry lemonade:
[tex]\[
6 \frac{1}{3} = 6 + \frac{1}{3} = \frac{18}{3} + \frac{1}{3} = \frac{19}{3}
\][/tex]
Step 2: Convert these improper fractions to decimal equivalents, which are easier to add
[tex]\[
\frac{9}{2} = 4.5
\][/tex]
[tex]\[
\frac{4}{3} \approx 1.3333333333333333
\][/tex]
[tex]\[
\frac{19}{3} \approx 6.333333333333333
\][/tex]
Step 3: Add these decimal equivalents to find the total number of cups
[tex]\[
4.5 + 1.3333333333333333 + 6.333333333333333 = 12.166666666666666
\][/tex]
Step 4: Convert the decimal back to a mixed number if needed
The decimal [tex]\(12.166666666666666\)[/tex] is equivalent to:
[tex]\[
12 \frac{1}{6}
\][/tex]
Therefore, the total number of cups of punch Peter makes is:
[tex]\[
\boxed{12 \frac{1}{6}}
\][/tex]
