What is a fraction? There is not one clear answer to this question which covers all that fractions are used for. Let us look at several interpretations of a fraction, and then see how they relate to each other. Take, for example, the fraction ^{ 3}/_{4 }.

** Interpretations **

^{ 3}/_{4 } could refer to a * division * problem: 3 divided by 4.

^{ 3}/_{4 } is the * ratio * (the comparison of sizes) between the number 3 and the number 4, also written as '3 : 4' or '3 to 4'.

^{ 3}/_{4 } is a * number * located between 0 and 1 on the number line. It can be found by dividing the distance between 0 and 1 into four parts, and locating the point where the third of four parts ends.

^{ 3}/_{4 } represents a * part * of some * whole *. ^{ 3}/_{4 } of something is obtained by dividing that something into four parts and taking three of them.

** Why These Are all the Same **

There are several ways to see how all four interpretations relate to each other. Let us see what connections can be drawn.

The * number *^{ 3}/_{4 } is the quotient of the * division * problem, 'what is 3 divided by 4'.

^{ 3}/_{4 } of a quantity (*part/whole*) can be arrived at by multiplying the * number *^{ 3}/_{4 } by that quantity.

The * number *^{ 3}/_{4 } solves the following proportion: 'x is to 1 as 3 is to 4' (x : 1 :: 3 : 4) (*ratio*) .

The * number *^{ 3}/_{4 } is three of four equal parts making up the number 1 (*part/whole*).

The following proportion is true (*ratio*): '^{ 3}/_{4 } of a quantity is to the whole quantity (*part/whole*) as 3 is to 4.'

** Illustrative Example **

Imagine a pizza cut into eight slices. Six of the slices have been eaten.

Six slices out of eight is ^{ 3}/_{4 } of the pizza.

If the eight slices had been divided into four equal groups (two slices each), then three of those groups would total to six slices.

The proportion 'six slices is to eight slices as 3 is to 4' is a true proportion.

^{ 3}/_{4 } * 8 = 6.

** Working with Fractions **

Since fractions are numbers, we should be able to do arithmetic with them. First, some definitions:

The * numerator * of a fraction is the top number, or the number before the slash. The numerator tells how many parts are being talked about. The * denominator * of a fraction is the bottom number, or the number after the slash. The denominator tells what portion of a whole is each of the parts being talked about. For example, in the fraction ^{ 2}/_{3 }, the numerator is 2 and the denominator is 3, so we are talking about 2 of the parts obtained when a whole is divided into 3 equal parts. * Like fractions * are fractions which have common denominators. * Unlike fractions * have different denominators.

** Changing Denominators **

The same fraction can be written with different denominators. To change a fraction into an equal fraction with a different denominator, multiply both the numerator and the denominator by the same number. For example, if we multiply the top and bottom of ^{ 2}/_{3 } by 10, we obtain ^{ 20}/_{30 }, which equals ^{ 2}/_{3 }. This process - making both the numerator and denominator larger by a fixed multiple - is sometimes referred to as 'building' a fraction.

The opposite of building a fraction is 'reducing' a fraction. In reducing, both the numerator and denominator are divided by the same number and become smaller. In order for reducing to be possible, the numerator and denominator must have a common factor. For example, 18 and 24 have a common factor of 6, so ^{ 18}/_{24 } = ^{ 3}/_{4 }. A fraction that has been reduced as much as it can be is said to be 'in lowest terms'.

** Adding and Subtracting **

In order to add and subtract fractions, the fractions being operated on must be like fractions - they must have the same denominator. Then the adding and subtracting is straightforward; the numerators are added or subtracted and the denominator of the sum is the same as the denominator of the addends. Thus ^{ 2}/_{3 } + ^{ 7}/_{3 } = ^{ 9}/_{3 }. Notice that this sum can be reduced: ^{ 9}/_{3 } = 3.

If two fractions must be added or subtracted, they need a common denominator. If they do not have one, then one must be found. This is done by building the two fractions into equal fractions with some common denominator which is a multiple of both original denominators. Here is an example:

^{ 3}/_{4 } - ^{ 1}/_{3 }

^{ 3}/_{4 } - ^{ 1}/_{3 } = ^{ 9}/_{12 } - ^{ 4}/_{12 } = ^{ 5}/_{12 }

...so the difference between ^{ 3}/_{4 } and ^{ 1}/_{3 } is ^{ 5}/_{12 }

There is always a 'least common denominator', which can be found in various ways, but any common denominator works. One guaranteed method for finding a common denominator is to multiply the two original denominators by each other. This product is a common denominator, though not necessarily the 'least' one.

** Multiplying **

Multiplying fractions is easy. Simply mutiply the two numerators to arrive at the new numerator, and multiply the two denominators to arrive at the new denominator.

^{ 3}/_{4 } * ^{ 1}/_{6 }

^{ 3}/_{4 } * ^{ 1}/_{6 } = ^{ (3 * 1)}/_{(4 * 6) } = ^{ 3}/_{24 }

...which reduces: ^{ 3}/_{24 } = ^{ 1}/_{8 }

So, the product of ^{ 3}/_{4 } and ^{ 1}/_{6 } is ^{ 1}/_{8 }

This example can be done more neatly by reducing before multiplying. Before multiplying two fractions, a number in either numerator can be reduced by a common factor with a number in either denominator; thus:

^{ 3}/_{4 } * ^{ 1}/_{6 } = ^{ 1}/_{4 } * ^{ 1}/_{2 } = ^{ 1}/_{8 }

** Dividing **

If two fractions are to be divided, simply multiply the first fraction (the dividend) by the 'reciprocal' of the second (the divisor). The reciprocal of a fraction is the fraction obtained by switching the numerator and denominator of the original fraction. The product of a fraction and its reciprocal is always 1.

Division example:

^{(5/7)}/_{(2/3)}

^{(5/7)}/_{(2/3)} = ^{ 5}/_{7 } * ^{ 3}/_{2 } = ^{ 15}/_{14 }