Fractions

An equivalent fraction is a fraction that is equal to another fraction.

Example:

A fraction can be changed into an equivalent fraction by multiplying or dividing the numerator and denominator by the same number.

Example:

A fraction may be reduced by dividing both the numerator and the denominator of a fraction by

the same number.

Example:

Addition and Subtraction of Fractions

When two or more fractions have the same denominator, they are said to have a common denominator. The rules for adding fractions with a common denominator will first be explored. Consider the example.

First of all, the fraction means three segments, i.e. . Looking at this as the addition of pie segments:

It is obvious that three of these segments the plus one of these segments the equal four

of these segments

This graphic illustration can be done for any addition of fractions with common denominators. The sum of the fractions is obtained by adding the numerators and dividing this sum by the common denominator.

Also, this general method applies to subtraction, for example,

The general method of subtraction of fractions with common denominators is to subtract the numerators and place this difference over the common denominator.

When fractions do not have a common denominator, this method must be modified. For example,

consider the problem:

This presents a problem, the same problem one would have if he were asked to add 6 feet to 3 yards. In this case the entities (units) aren't equal, so the 6 feet are first converted to 2 yards and then they are added to 3 yards to give a total of 5 yards.

6 feet + 3 yards = 2 yards + 3 yards = 5 yards

Going back to the fraction addition example, then and must both be expressed in the same segments to be added. Without developing the general method, is ths. Multiply by or (one) to give the equivalent fraction. Similarly, equals .

Then,

The general method of adding or subtracting fractions which do not have a common denominator is to convert the individual fractions to equivalent fractions with a common denominator. These equally sized segments can then be added or subtracted.

The simplest method to calculate a common denominator is to multiply the denominators. This is obtained if each fraction is multiplied top and bottom by the denominator of the other fraction (and thus by one, giving an equivalent fraction).

For more than two fractions, each fraction is multiplied top and bottom by each of the other denominators. This method works for simple or small fractions. If the denominators are large or many fractions are to be added, this method is cumbersome.

Example:

would require the denominator to be equal to 64 x 32 x 6 = 12,288. This kind of number is very hard to use.

In the earlier example

You notice that both 30 and 18 can be divided by 6; if this is done:

By doing this we arrive at a smaller and more useful number: takes the place of .

The sum of two or more fractions reduced to its simplest form contains the smallest possible denominator common to both fractions. This denominator is called the least common denominator (LCD).

Example:

Using trial and error we can find that 24 is the LCD or smallest number that 3, 6, and 8 will all

divide into evenly. Therefore, if each fraction is converted into 24^ths, the fractions can be added.

This is the simplest form the fraction can have. To eliminate the lengthy process of trial and error used in finding the LCD, you can reduce the denominators to their prime numbers.