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Fractions



This article takes a look at fraction basics: what a fraction is, different ways to represent fractions, and how to work with fractions, turning them into decimals and percentages, for example, and converting fractions by finding the Least Common Denominator (LCD).

Fractions are ubiquitous in our lives, even in situations in which we may not consciously think of them. When we eat one piece of the leftover pizza and leave the rest for others, when we slice a sandwich before eating it, when we check the gas gauge in our vehicles, when we deal out cards for a game of “War” - in all these cases, we’re working with fractions. For more about fractions, read the rest of this article.

What Are Fractions?

Fractions tell us about parts in relationship to a whole. That is, they give information about proportions or ratios: they do not tell absolute amounts. For example, saying that you have 1/8 of a pizza tells us about the portion in relation to the whole, but gives no information about the size of the piece. It could be from a personal size pizza with a 6-inch diameter or an extra-large pizza with a 24-inch diameter. The fraction doesn’t convey that type of information.

Fractions have three parts: two numbers and a slash. Depending on the style, the slash may be horizontal or at an angle. Above the slash (and somewhat to the left, if the slash is at an angle) is the numerator. Below the slash (and somewhat to the right, if the slash is at an angle) is the denominator.

The denominator tells how many equal parts make a whole. The numerator tells how many of those parts are under consideration. The relationship of the numerator to the denominator is called a ratio. A ratio is a mathematical expression used to compare quantities relative to each other without reference to outside information.

So far, we have been speaking about fractions in relation to whole and parts of objects in the world. But it is important to realize that fractions aren’t bound by the laws that govern such objects. For example, it is possible to have negative fractions, such as -2/3, and fractions that are more than a whole, such as 13/9. Fractions can also have a denominator that is much larger than any division we are likely to make, like 1/24,579. Fractions can also contain one or more variables, like this:

x/4                  y + z / 12              5/b

Different Representations of Fractions

Numbers that can be represented as fractions can also be represented as decimals and as percentages. This is easier to understand if we turn to a different consideration of fractions, one that considers it as a quotient, or the numerator divided by the denominator. For example, let’s take the fraction ½. If we rephrase it as a quotient, we would have:

1 ¸ 2  =

and solving for that, we would get 0.5, which is the decimal expression of the fraction ½. If we recall that a decimal can be expressed as a percentage by choosing to

  • move the decimal point two places to the right
  • add zeroes in any added places that have no number
  • and remove any trailing zeroes on the left

we can convert 0.5 to 50%, which again is the equivalent of the fraction ½.

Any fraction in which the numerator and the denominator are the same is equal to 1.

Working With Fractions

Sometimes when working with fractions, it is necessary to have identical denominators. This is true for adding and subtracting fractions. As with equations, fractions remain equivalent if the same operation - adding, subtracting, multiplying or dividing - is carried out on both the numerator and denominator. This allows one to create equivalent fractions from fractions that have different denominators, which is useful when, for example, adding or subtracting them.

Suppose you wanted to add 1/3 + 1/2. First, you would find the Least Common Denominator (LCD), that is, the lowest integer of which both 3 and 2 are factors. That would be 6. In order to make the first fraction have a denominator of 6, you could multiply it by 2/2 . When you consider that 2/2 = 1, it is clear that this operation does not change the ratio of the fraction. Similarly, one could follow the same procedure, and multiple the second fraction by 3/3. One would then have:

2/6 + 3/6 = 5/6