To find out how many cupcakes Eloise baked, we need to work backwards.
We know that her family ate the remainder of the cupcakes after she gave 1/9 of them to her friend and sold 3/4 of the remainder. So, let's start with the number of cupcakes her family ate, which is 28.
Her family ate all the remaining cupcakes, so let's say the number of cupcakes after she sold 3/4 of them is x.
Since her family ate 28 cupcakes, we have the equation x = 28.
Now, let's find the number of cupcakes before she sold 3/4 of them. We can represent this number as y.
To find y, we need to undo the operation of selling 3/4 of the cupcakes. To do that, we need to divide x by 1 - 3/4, which is 1/4.
So, y = x / (1 - 3/4) = x / (1/4) = 4x.
Substituting x = 28, we find that y = 4 * 28 = 112.
Therefore, Eloise baked 112 cupcakes.