$480 is the price after the discount was added. We can then assume that $480 is 3/4 the price of the original item. To find 1/4, we simply divide by 3, which equals $160. since $160 is one quarter of the price, the original price was $640.
The conversion factor is used to convert from inches to cm. The inches unit in the denominator cancels with one of the inches unit from the original 112 square inches figure.
We use two identical copies of the same conversion factor to convert from square inches to square centimeters. This is the "multple conversion" your teacher is referring to.
Think of "square inches", aka , as "inches times inches". Or think of a 1 inch by 1 inch square that has an area of 1 square inch. The same applies for the square centimeter unit as well.