When you need to add or subtract fractions, you will need to first make sure that the fractions have the same denominator. The denominator tells you how many pieces the whole has been broken into, and the numerator tells you how many of those pieces you are using.
You need a common denominator, technically called the least common multiple. Remember that if a number is a multiple of another, you can divide them and have no remainder.
One way to find the least common multiple of two or more numbers is to first multiply each by [latex]1[/latex], [latex]2[/latex], [latex]3[/latex], [latex]4[/latex], etc. For example, find the least common multiple of [latex]2[/latex] and [latex]5[/latex].
First, list all the multiples of [latex]2[/latex]:
Then list all the multiples of [latex]5[/latex]:
[latex]2\cdot 1 = 2[/latex]
[latex]5\cdot 1 = 5[/latex]
[latex]2\cdot 2 = 4[/latex]
[latex]5\cdot 2 = 10[/latex]
[latex]2\cdot 3 = 6[/latex]
[latex]5\cdot 3 = 15[/latex]
[latex]2\cdot 4 = 8[/latex]
[latex]5\cdot 4 = 20[/latex]
[latex]2\cdot 5 = 10[/latex]
[latex]5\cdot 5 = 25[/latex]
The smallest multiple they have in common will be the common denominator for the two!
How To: Adding Fractions with Unlike Denominators
Find a common denominator.
Rewrite each fraction using the common denominator.
Now that the fractions have a common denominator, you can add the numerators.
Simplify by canceling out all common factors in the numerator and denominator.
Add [latex]\frac{2}{3}+\frac{1}{5}[/latex]. Simplify the answer.
Since the denominators are not alike, find a common denominator by multiplying the denominators.
[latex]3\cdot5=15[/latex]
Rewrite each fraction with a denominator of [latex]15[/latex].
You can find a common denominator by finding the common multiples of the denominators. The least common multiple is the easiest to use.Add [latex]\frac{3}{7}+\frac{2}{21}[/latex]. Simplify the answer.
Since the denominators are not alike, find the least common denominator by finding the least common multiple (LCM) of [latex]7[/latex] and [latex]21[/latex].
Multiples of [latex]7: 7, 14, 21[/latex]
Multiples of [latex]21: 21[/latex]
Rewrite each fraction with a denominator of [latex]21[/latex].
Subtracting fractions follows a similar process to adding fractions. When you subtract fractions, you must think about whether they have a common denominator, just like with adding fractions.
How To: Subtracting Fractions with Unlike Denominators
Find a common denominator.
Rewrite each fraction using the common denominator.
Now that the fractions have a common denominator, you can subtract the numerators.
Simplify by canceling out all common factors in the numerator and denominator.
Below are some examples of subtracting fractions whose denominators are not alike.
Subtract [latex]\frac{1}{5}-\frac{1}{6}[/latex]. Simplify the answer.
The fractions have unlike denominators, so you need to find a common denominator. Recall that a common denominator can be found by multiplying the two denominators together.
[latex]5\cdot6=30[/latex]
Rewrite each fraction as an equivalent fraction with a denominator of [latex]30[/latex].
The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of those integers without leaving a remainder.