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Fractions

A fraction is a mathematical way of expressing a value that is less than a whole unit or one. It can also be use to express a ratio of two values. A fraction consist of two parts, a numerator and a denominator. The denominator is the number that the numerator is being divided by. There are three types of fractions: proper, improper and mixed fractions. A proper fraction consists of a numerator that is smaller than the denominator while an improper fraction contains a larger numerator than the denominator.

Mixed fractions contain an integer and a proper fraction. Mixed fractions and improper fractions are interchangeable.

Comparing

In order to find out which of two fractions is larger, a technique called cross multiplying is used. The first cross product is the multiplication of the first numerator with the second denominator. The second cross product is the multiplication of the second numerator with the first denominator. If the first cross product is larger than the second, then the first fraction is larger than the second. Below, two fractions are compared using this method and it can be seen that the second fraction is larger.

Adding

In order to add fractions, there are several rules about what format they can be in. First off, mixed fractions must be converted to improper fractions before they can be added. Secondly, fractions can only be added if they have a common denominator. In order to add fractions with different denominators, it is necessary to convert them so they have the same denominator. There are two methods of adding fractions. The first method is the method of the lowest common denominator (LCD). This method simply involves making a list of the multiples of each denominator and finding the lowest number common to both lists. This is the LCD. Once the LCD is found, the multiple needed to convert each denominator to the LCD is determined and both the numerator and the denominator of each fraction is multiplied by this number. Once the fractions are converted, the numerators can be added.
 

The second method involves simply multiplying the denominator of the second fraction by both the numerator and the denominator of the first fraction and multiplying the denominator of the first fraction by both the numerator and the denominator of the second fraction. Then the numerators are added as for the first method. However, in many cases the resulting fraction is not in its lowest terms and needs to be reduced. This means that it is possible to factor out a common number from both the numerator and denominator. Once the fraction is reduced, the answer should match the answer obtained using the first method. This method makes the first step easier but adds a step at the end.

If the fractions to be added are mixed fractions, the fractions need to be converted to improper fractions as stated above. When adding a fraction and a whole number, the whole number is treated as a mixed fraction and also needs to be converted into an improper fraction before it can be added.

Subtracting 

Subtracting fractions is just like adding fractions. The denominators of all of the fractions must be the same. The only difference is that the numerators are subtracted rather than added.

Multiplying

Multiplying fractions does not require that the fractions have the same denominator. Instead, the numerators are multiplied with each other and the denominators are multiplied with each other. However, the resulting fraction may not be in lowest terms so it may need to be reduced.

It is possible to reduce before multiplying. This is useful if there are many numbers to multiply or if the numbers are large. To do this, factor each of the numerators and each of the denominators. Once the numerators and denominators are factored, numbers that appear in both the numerators and the denominators can be eliminated. It does not matter which numerator or which denominator the number originally appeared in.

Just as in addition, the fractions need to be proper or improper fractions in order to multiply them. Therefore, mixed fractions need to be converted to improper fractions. Whole numbers however do not need to be converted to improper fractions. The whole number can simply be multiplied to the numerator.

Dividing Fractions

In order to divide fractions it is necessary to only consider two fractions at a time. To divide fractions the second fractions is inverted. The numerator becomes the denominator and vice versa. Once the fraction is inverted the fractions are then multiplied in the usual fashion. In order to consider more than two fractions, first the first two fractions are divided. Then the resulting new fraction and the next fraction are divided.
 
 

As before with adding and multiplying, mixed numbers need to be converted to improper fractions before dividing. When dividing a whole number by a fraction, use the same rule. Invert the fraction and multiply. However, if the fraction is being divided by the whole number, the whole number must be inverted and then multiplied. Inverting a whole number is done by creating a fraction with one in the numerator and the whole number in the denominator.