There are many examples where adding quantities with unlike dimensions is not useful. However, there are examples where multiplying quantities with unlike dimensions is also not useful. For example if a weight of 3 newtons is hanging from a thread of length 2 meters, the quantity 6 newton-meters doesn’t seem very informative.

The special status granted the multiplication of units seems (to me) to result from the empirical fact that there are many situations where the information lost in summarizing a situation by a product is unimportant. For example, a torque of 6 ft-lbs allows us to predict certain effects without knowing whether it came from a force of 6 lbs on a 1 ft lever or a force of 3 lbs on a 2 ft lever.

Inventing a situation where adding unlike units is useful is difficult. For example the quantity 6 apples+oranges might describe a situation where there are 3 apples and 2 oranges – or zero apples and 6 oranges – or 19 apples and -13 oranges. Perhaps, for integer values, we could imagine a fruit counting machine and pretend we are only interested in effects resulting from the number of fruits, not from their species.

It’s interesting that the “+” in “apples+oranges” suggests the logical connective “or”. Are we willing to make the grand generality that no useful physical predictions can be made from a quantity that summarizes a total that is “one type of thing or another”? If we assert that generality, are we to prove it from other assumptions? Or do we take it as an axiom supported by empirical observations?

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Stoichiometry is indeed a method where we can find the various amounts of ingredients in a chemical reaction, using the conservation of matter.

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I messed up with building up with this last step.

What is needed is

But

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What is the mistake here and how can I correct it?

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