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 $Q=N/D,$ 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?" $$N=Q\times D=\underbrace {D+D+\cdots +D} _{\text{Q parts}}.$$
2. Partition: "What is the size of each of D equal parts whose sum is N?" $$N=D\times Q=\underbrace {Q+Q+\cdots +Q} _{\text{D parts}}.$$

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

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

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.

See also

• List of partition topics

References

• Klapper, Paul (1916). The teaching of arithmetic: A manual for teachers. p. 202.
• Solomon, Pearl Gold (2006). The math we need to know and do in grades preK–5 : concepts, skills, standards, and assessments (2nd ed.). Thousand Oaks, Calif.: Corwin Press. pp. 105–106. ISBN 9781412917209.

External links

• A University of Melbourne web page shows what to do when the fraction is a ratio of integers or rational.