Jabberwock: If you rip a dollar bill in half, is it not true that each half, independent of each other, are worth 50 cents? (as long as the serial number is fully intact) Nope. Whichever half is more than 50% of the bill is worth $1; the other half is worthless.
For most torn bills, only 1-15% of the bill is missing, so there's no problem with cashing the larger part. (And of course, no merchant would take the small piece.) But if it was real close to 50%, a merchant probably has the right to refuse it, even if it's at 55 or 60% there.
My sisters pulled a cute trick in their cards to my wife and me after our wedding: they tore a $50 bill into two near-50% pieces and put a piece each in their envelopes. Taping the two halves together was enough to insure that we got full value for it. (My wife thought the first sister was trying to tell us something; I knew my family and immediately told her to open the 2nd sister's card.)
This 50% rule is around in case a stash of cash was in a fire, or was forgotten and mildewed away. If the notes are taken to a proper facility (some banks might do this, also the US Treasury office), they'll carefully examine each note and determine if it's at least 50% there. If so, they'll give you full credit. Afterward, they'll destroy all the bills that you had assessed.