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I feel there is some clarification needed.
 Say you have a pie and cut it into 1/4 (fourths). You now have 4/4 pieces of the same pie, but it is still one pie. If you keep cutting it up, you still have only one pie. Just because you cut it into pieces does not mean that you now have that many pies.
This is assuming no one eats a piece.
 Another example is from an object perspective, to see fractions as groups of objects. If I cut an object into fourths, then you now have 4/4 pieces of that object, but the material that makes up the object still has equal quantity in all four pieces. if you take or add to one of those four pieces, that 4/4 becomes 3/3 because you can only have similar groups of objects in a fraction. The augmented object has become unique and it's fractional value is 1/1.
 Finally, going back to ratios and rates, is again looking at why 1/2 and 2/4 are ONLY equivalent ratios or rates and not equivalent quantities. One, you are not changing an object by cutting it (a physical reaction). If you have half of an object or two halves of a half of an object, the quantity of atoms does not change, but the number of that object does, and so the quantity changes. Two, if you change an object via chemical reaction, you have a product and a byproduct. The original object is lost, 1/0 of that object, and you now have created two new objects out of that: the byproduct and product. So in this case, if a chemical reaction occurs, the fraction of that object becomes 1/0, and creates two new objects, 1/1 and another 1/1. Logically we are saying that 1/0 means this object does not exist. It's like the opposite of NTFS: instead of deleting the reference, the data is deleted, but the reference is there; after a chemical reaction occurs, you can recover some data about the original object, like the size and shape, but you can't recover it.
 Notes: This could be a better perspective of objects, by using fractions and uniqueness, to group objects, which naturally is what a group is: a collection of similar objects.