We are well aware that “of” usually means “multiplied by.” Examples include “half of 4,” “10% of 80,” or even “two of those.” There is even an invisible use of the word in this way: “10 increased by 50%” really means “10 increased by 50% of itself,” i.e. “10 increased by 50% of 10.” This unstated “of” phrase is also at the heart of another article I wrote called “Decreasing at a WHAT Rate?!!” where an entire controversy could be solved by simply clarifying, “Rate of What?”
However, there are other meanings of the word “of” that often appear in our language.
THE FUNCTIONAL “OF”
There is an endless number of functional phrases like “sum of,” “difference of,” “square root of,” “third root of,” “common logarithm of,” “base 27 logarithm of,” etc. If 10 is the square root of 100, it is the square root that belongs to 100. One might also call this a “genitive of” or “possesive of.”
THREE OTHER “OF” USES
If “4 of 5 dentists” agree on something, then it is understood that “4 out of 5 dentists” agree. This is 4/5 = 0.8 = 80%, but it is not “4 times 5” dentists. This “of” basically means “divided by.”
In a more rare and pedantic instance, the time can be “a quarter of 4.” But this amounts to 4 minus a quarter of an hour = 4:00 - 0:15 = 3:45. This is sort of a “reverse subtraction,” followed by another invisible “of an hour.”
Finally, “of” is often used to mean “equals.” If “16 has a square root of 4,” this does not mean “16 has a 2,” but “16 has a square root equal to 4.” This may not be proper, but it happens a lot.
ANALYSIS
The last three uses of the word “of” - division, reverse subtraction, and equation - should be avoided, since “4 of 5” or “quarter of 4” or “square root of 4” are just asking to be taken out of context by those unfamiliar with their technical and contextual meanings.
Functional “of” statements, however, are perfectly legitimate and cannot be avoided. Therefore “of” does not always mean “times,” and one must be careful to read the problems carefully, and seek clarity where doubt exists.