The Solution:
Given that Joe has
[tex]3\frac{1}{4}\text{gallons of punch}[/tex]He added
[tex]\begin{gathered} 1\frac{1}{2}\text{quarts of juice to the punch} \\ \text{Converting this to gallon, (since 1 gallon =4 quarts), we get} \\ 1\frac{1}{2}\text{quarts}=\frac{3}{2}\times\frac{1}{4}=\frac{3}{8}\text{gallon} \end{gathered}[/tex]He drinks
[tex]\frac{1}{4}\text{quart}=\frac{1}{4}\times\frac{1}{4}=\frac{1}{16}\text{gallon}[/tex]Now, the expression that gives what is left with Joe is
[tex]\begin{gathered} \text{ Let x represent Punch and y represent juice.} \\ So,\text{ we have} \\ 3\frac{1}{4}x+\frac{3}{8}y-\frac{1}{16}x \end{gathered}[/tex]Simplifying, we get
[tex]\begin{gathered} \frac{13}{4}x-\frac{1}{16}x+\frac{3}{8}y \\ \\ \frac{52x-x}{16}+\frac{3}{8}y \\ \\ \frac{51x}{16}+\frac{3}{8}y \end{gathered}[/tex]Therefore, the required expression is
[tex]\frac{51}{16}x+\frac{3}{8}y[/tex]