Is a Quotient the number of times one value fits inside another, or the value of one of the groups produced by the operation My question is a simple one, but one I haven't been able to figure out through research.
When a simple division is performed suppose 10/2 = 5, is that 5 classified as the frequency or the number of times the value of 2 "fits" into 10? In other words we are saying that the group with a value of 2 can be created 5 times out of ten.
Or are we saying that 10/2 = 5 means that we are splitting 10 into 2 groups, and the "value" of ONE of those groups is 5.
Some places denote that the quotient is a frequency while other places they denote that the quotient is a value.
If the quotient is a frequency, or the number of times that a group with value 2 can fit into ten, how are we allowed to perform further operations on a number that is a frequency.
For example:
We usually go along with the procedure below as being valid
(10/2) + 3 = 8
However if the answer of 10/2 = 5 is a frequency, how can we we add 3 to this quotient if 3 is considered a value and not a frequency. 3 must be defined as a frequency too. In my mind its like adding apples and oranges. Frequency + Value
I hope someone can help me 
 A: This related question still remains unanswered.
One of these operations is quotition and the other is partition and they are the same operation in the sense of having the same value in every instance.  However, one could still wonder whether, when applied to transfinite ordinals, or to surreal numbers, or to some other exotic "numbers", they might differ from each other.
\begin{align}
& \frac{\$6}{\$2} = (3\text{ with no “}\$\text{''}) \\[4pt]
= {} & \text{how many }\$2\text{ items you can buy if you have }\$6. \\
& \text{This is “quotition.''  It asks how many parts there are.} \\[15pt]
& \frac{\$6} 2 = (\$3 \text{ with a “}\$\text{''}) \\[4pt]
= {} & \text{how much money each of two partners gets if they split the whole } \$6 \text{ equally}. \\
& \text{This is “partition.'' It asks what the size of each part is.}
\end{align}
That the $\text{“}\$\text{''}$ either cancels from the numerator and denominator or fails to cancel, as the case may be, just as if it were a number, is a phenomenon used incessantly in the physical sciences and engineering, applied to unit of measurement of physical quantities rather than of money.
A: All of the above.
The example quotients you have given ask one to arrange objects into a rectangular array.  You may label either the width or the height "frequency" and the other "value".  You can also swap the labels.  This is a consequence of the commutativity of multiplication -- it doesn't matter what order you take the multiplicands.
This means that the labels "frequency" and "value" are not permanent, they are transient.  They only exist long enough to complete the computation, then they vanish.  In your example of $(10/2) + 3 = 8$ whether the $10$ is arranged as $5$ rows of $2$ or $2$ rows of $5$ is a detail that exists only long enough to get "$5$" and then that detail vanishes.  Once a division is completed, the details of how it was performed are dropped.  The result of a division is a number, just like $3$, and has no memory of being a width or a height.
