[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/(x2 + 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