In most cases when you are adding fractions, the denominators will not be the same. When the denominators of two fractions are not the same they cannot be added as is. There is however a way to convert any two fractions so that they do have the same denominator. Lets take the following example:
In this problem 3 is not equal to 7 so we cannot add them. We need to convert these two fractions into fractions that mean the same thing as the originals, but also share a common denominator. To find the equivalent fractions we must first decide on a denominator. We do this by multiplying the denominators together, in this case 3 x 7 = 21. Now that we have our new denominator we must convert the fractions.
In order to convert the fractions, we must perform an operation on them that will change them without changing their value. According to the
Multiplicative Identity we can multiply any number by 1 and get that number. Since any fraction in which the numerator is the same as the denominator is equal to 1, we must multiply both fractions by some fraction equal to 1. In this case we can multiply 1/3 x 7/7 and 4/7 x 3/3 since 7/7 and 3/3 are the same as 1.
In order to do the multiplication, you simply just have to multiply the numerators and the denominators. (
See Multiplying Fractions). For the first part of our problem we multiply the numerators 7 x 1 = 7 and the denominators 7 x 3 = 21 to get a result of 7/21. The second part we multiply the numerators 3 x 4 = 12 and the denominators 3 x 7 = 21 to get 12/21.
We now have a problem that is easy to add. Since we now have like denominators we simply add 7 + 12 to get 19. We use the common denominator in our answer to get our result of 19/21.
*Note: Instead of multiplying the denominators together to get a common denominator, you can also find and use the
Least Common Multiple of the denominators. In this case the Least Common Multiple of the denominators is called the
Least Common Denominator.