Let's define the next variables:
x: quarts of sparkling water
y: quarts of grape juice
We want to make 100 quarts of punch, then
x + y = 100
In order to follow the recipe we need to keep the next ratio:
[tex]\begin{gathered} \frac{x}{y}=\frac{1\frac{1}{2}}{\frac{3}{4}} \\ \frac{x}{y}=\frac{1.5}{0.75} \\ \frac{x}{y}=2 \\ \text{Isolating x} \\ x=2y \end{gathered}[/tex]Replacing this result into the first equation:
[tex]\begin{gathered} 2y+y=100 \\ 3y=100 \\ y=\frac{100}{3} \\ y=33\frac{1}{3} \end{gathered}[/tex]Replacing into the previous formula:
[tex]\begin{gathered} x=2y \\ x=2\cdot33\frac{1}{3} \\ x=66\frac{2}{3} \end{gathered}[/tex]You would need 66 2/3 quarts of sparkling water and 33 1/3 quarts of grape juice to make 100 quarts of punch