A fraction is a number that represents a part of a whole. Fractions are commonly written as where a and b are any number and b is not equal to 0. In a fraction, the top number is referred to as the numerator, and the bottom number is referred to as the denominator:
While the term "fraction" commonly refers to "common fraction," (the form described here) fractional numbers can also be expressed as decimal fractions and percents.
Slices of pizza can be a good way to picture fractions. In a pizza that has 8 slices, 1 slice is a fraction of the whole pizza, written as . Two slices would be , 3 slices , and so on. In this example, the top number, known as the numerator is the number of slices of pizza we have, while the bottom number, the denominator, is the total number of slices of pizza in the whole pizza.
In the above example with pizza, we may talk about having 2, 4, 6, or even 8 slices of pizza. In these cases the fraction would be written as
respectively; the fractions can further be simplified or reduced as follows:
These are known as equivalent fractions. Having 2 slices out of 8 of a pizza is equivalent to one-fourth of the pizza, and so on. Fractions can be reduced when the numerator or denominator share a factor. The fraction is equal to 1, and is an equivalent fraction to any number divided by itself (except for 0).
Adding fractions with the same denominator is much like adding non-fractions. You add the numerator as you would normally, and keep the denominator the same. For example:
Adding fractions when they have a different denominator is slightly more involved. To add fractions, the denominator of the fractions must be the same. If the denominator is not the same, the fractions being added need to be turned into equivalent fractions with the same denominator, then added. To do this we need to find a common denominator. Because of the nature of fractions, it is possible to find numerous common denominators, but generally, we want to find the least common denominator to make the arithmetic simpler.
1. Add the fractions :
Notice that in (1) we cannot just add the numerators because the denominators are not the same. So we must first find a common denominator. To do this, we need to form equivalent fractions for , that share the same denominator. We can do this by multiplying each of the fractions by some multiple of the fraction such that these fractions will have the same denominator:
Now that all the fractions share the same denominator, we can add them as we would normally, as shown in (2). Note that in this particular problem, we did not need to change all the fractions, since one of the fractions has a denominator of 12, a multiple that all the fractions share.
2. Add the fractions :
In this case, the first common multiple that 4 and 7 share is 4 × 7 = 28. Multiplying all of the denominators is one way to find a common denominator, but it often won't be the least common denominator. In Example 1, we could have multiplied 3 × 6 × 12 = 216, and formed the equivalent fractions:
and are equivalent fractions, but using the least common denominator of 12 makes the problem easier to work with.
Subtracting fractions follows the same process as addition. As long as the denominator of the fractions being subtracted are the same, the fractions can be subtracted as they would be normally.