Re:Can 1/3 and 1/3 = 2/6? It seemed so
by tsimionescu on Tuesday May 26, 2020 @12:29PM
(#23313157)
You can't perform operations with apples, oranges and fruits unless you use them completely interchangeably ,i.e. 1 apple = 1 orange. Otherwise, you will break algebraic equalities.
You can convert apples to fruit and oranges to fruit and then do operations on fruit, but you can't go back from the result to apples and/or oranges. For example, 1/7 of (3 apples + 4 oranges) is 1 fruit, but we can't tell if it's one orange, or 1 apple, or 1/3 apple and 2/3 orange.
The grandparent's note about units and your mention of ratios get at the same problem for me:
The lesson mixes two of the physical world concepts that are often modeled by fractions.
One is an incomplete whole (the six pack) and the other is the proportions of a mixture (table demographics).
Fractions always have the same rules, but the rules map differently to the different territory being modeled.
Note that I'm using ratio notation because the answer for the above is not the same as adding 5:10 apples and 1:2 oranges; in other words, the exact numerator and denominator both matter, so it's not really a simple fraction; a simple fraction can be reduced to its lowest terms (e.g. 5/15 becomes 1/3), but you can't do that here and still support the mediant operation. https://en.wikipedia.org/wiki/Mediant_(mathematics)