What is Fraction?

A fraction
(sometimes, a common fraction) is a way of expressing a number
that is a ratio of two integers:

The top (or the
first) number is called the numerator, the bottom (or the
second) number is called the denominator. The numerator answers
the question, How many? The denominator specifies the answer to
that question, How many of what? Accordingly, "p/q" is
pronounced "p q-th," meaning p parts each equal to the q-th part
of a whole. (This
we may meaningfully refer to as the u-property or u-aspect
of a fraction, that is interpreted as the number of fractional
units.) When the numeric aspect of the fraction needs to be
emphasized, "p/q" is pronounced simply "p over q" or "p divided
by q."

**What is the
whole?** It depends. What is 1, or what is 5? These are
abstractions that are used to indicate quantity: 1 apple, 1
acre, 1 dollar, 5 apples, 5 acres, 5 dollars. In the same manner
2/3 is an abstraction that indicates a quantity of apples,
areas, or the balance in a bank account. 2/3 of an apple is two
parts of an apple that was divided into three parts. To
determine 2/3 of 5 apples one needs to divide 5 apples into
three equal parts and take any two of them. How does one divided
5 apples into three equal parts? Divide each of the apples into
three parts and combine five of the pieces at a time. (This of
course assumes that the apples are equal and one is sufficiently
skillful to divide each into three equal parts.) Incidentally,
it shows that 2/3 of 5 apples equals 10/3 of an apple. When Mr.
Smith's backyard is said to be 2/3 of Mr. Johnes' the intention
is to their areas: the latter is bigger than the former and,
more accurately, if Mr. Johnes' backyard measures, say, 3 acres,
then that of Mr. Smith's is 2 acres. 2/3 of a bank account
balance is $6 if the whole is $9, and is $60 if the whole is
$90.
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