Lynne bought a bag of grapefruit, 1frac{5}{8} pounds of apples, and 2frac{3}{16} pounds of bananas. The total weight of her purchases was 7frac{1}{2} pounds. How much did the bag of grapefruit weigh?

Let us assume that the weight of the bag of grapefruit be x pounds 

Given:

The weight of the bag of apples is: $1{\displaystyle \frac{5}{8}}\text{ pounds or }{\displaystyle \frac{13}{8}}\text{ pounds}$

The weight of the bag of bananas is:

$2{\displaystyle \frac{3}{16}}\text{ pounds or }{\displaystyle \frac{35}{16}}\text{ pounds}$

The total weight of all of the bags that was purchased by Lynne is:

$7{\displaystyle \frac{1}{2}}\text{ pounds or }{\displaystyle \frac{15}{2}}\text{ pounds }$

So, we can write:

$$\begin{gathered} x+{\displaystyle \frac{13}{8}}+{\displaystyle \frac{35}{16}}={\displaystyle \frac{15}{2}} \\ \Rightarrow x+{\displaystyle \frac{26+35}{16}}={\displaystyle \frac{15}{2}} \\ \Rightarrow x+{\displaystyle \frac{61}{16}}={\displaystyle \frac{15}{2}} \\ \Rightarrow x={\displaystyle \frac{15}{2}}-{\displaystyle \frac{61}{16}} \\ \Rightarrow x={\displaystyle \frac{120-61}{16}} \\ \Rightarrow x={\displaystyle \frac{59}{16}} \\ \Rightarrow x=3{\displaystyle \frac{11}{16}} \end{gathered}$$

Hence, the weight of the bag of grapefruit is $\boxed{{\displaystyle 3{\displaystyle \frac{11}{16}}}}\text{ pounds }$