Quotition and partition

In arithmetic, quotition and partition are two ways of viewing fractions and division. In quotitive division one asks "how many parts are there?" while in partitive division one asks "what is the size of each part?"

In general, a quotient ${\displaystyle Q=N/D,}$ where Q, N, and D are integers or rational numbers, can be conceived of in either of 2 ways:

1. Quotition: "How many parts of size D must be added to get a sum of N?" $${\displaystyle N=Q\times D=\underbrace {D+D+\cdots +D} _{Q{\text{ parts}}}.}$$
2. Partition: "What is the size of each of D equal parts whose sum is N?" $${\displaystyle N=D\times Q=\underbrace {Q+Q+\cdots +Q} _{D{\text{ parts}}}.}$$

For example, the quotient ${\displaystyle 6/2=3}$ can be conceived of as representing either of the decompositions:

$${\displaystyle 6=\underbrace {2+2+2} _{\text{3 parts}}=\underbrace {3+3} _{\text{2 parts}}.}$$

In the rational number system used in elementary mathematics, the numerical answer is always the same no matter which way you put it, as a consequence of the commutativity of multiplication.