# fraction

[A] Before fractions can be added they must all be expressed in terms of the same denominator. To add 1/2, 1/3, and 1/4 they must all be stated in twelfths (in this example, 12 is the lowest common denominator) as 6/12, 4/12, and 3/12. They can then be added to give 13/12, an improper fraction that simplifies to 1 1/12. This sum explains why it is impossible to divide anything into shares of 1/2, 1/3, and 1/4 – their sum is larger than 1.

[B] To multiply fractions merely multiply the numerators and then multiply the denominators. A third of a half is 1/3 × 1/2 = 1/6 (the same as a half of a third – order doesn't matter).

[C] To divide fractions invert the divisor and multiply. For example, 5/6 divided by 5 is 5/6 × 1/5 = 1/6, which is exactly the same as the quantity described as a third of a half, as shown in illustration B.

A fraction is a rational number that represents
a part, or several equal parts, of a whole; examples include one-half, two-thirds,
and three-fifths. The word comes from the Latin *frangere*, meaning
"to break." A **simple, common, or vulgar fraction** is of
the form *a*/*b*, where *a* (the numerator) may be
any integer and *b* (the denominator)
may be any integer greater than 0. If *a* < *b*, the fraction
is said to be **proper** ("bottom heavy"); otherwise it is **improper** ("top heavy"). A decimal
fraction has a denominator of 10, 100, 1000, etc.

In an **algebraic fraction**, the denominator, or the numerator and denominator, are algebraic expressions, for example *x*/(*x*^{2} + 2). In a **composite fraction**, both the numerator and denominator are themselves fractions.

Two fractions *a/b* and *c/d* can be added, subtracted, multiplied, and divided according to the rules:

*a/b* + *c/d* = (*ad* + *bc*)/*b/d*

*a/b* - *c/d* = (*ad* - *bc*)/*bd*

*a/b* × *c/d* = *ac/bd*

*a/b* ÷ *c/d* = *ad/bc*