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

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.

Quotition

Thought of quotitively, a division problem can be solved by repeatedly subtracting groups of the size of the divisor. For instance, suppose each egg carton fits 12 eggs, and the problem is to find how many cartons are needed to fit 36 eggs in total. Groups of 12 eggs at a time can be separated from the main pile until none are left, 3 groups:

$$36{\text{ eggs}}{}-{}\!\!\!\ \ \underbrace {12{\text{ eggs}}-12{\text{ eggs}}-12{\text{ eggs}}} _{3{\text{ groups}}}=0\implies {\frac {36}{12}}=3.$$

Partition

Thought of partitively, a division problem might be solved by sorting the initial quantity into a specific number of groups by adding items to each group in turn. For instance, a deck of 52 playing cards could be divided among 4 players by dealing the cards to into 4 piles one at a time, eventually yielding piles of 13 cards each.

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.