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Finding Common Denominators
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In order to add fractions, the denominators must be the same. In the example of pizza, this means each slice has to be the same size. If we are working with pizza slices that have been cut into different sizes, we need to find a way to cut all of them into slices that are the same size. This is done using Least Common Multiples. LCM’s are how we change the denominator of a fraction. We multiply by 1, but the 1 doesn’t look like a 1. It is in the form of something like $$\dfrac{3}{3}$$ or $$\dfrac{7}{7}$$. (Remember, anything divided by itself equals 1.) The following video will give more details and work through some examples.

Finding Common Denominators
Example of Adding Fractions with Different Denominators (inch)

To find the common denominator, find the LCM of the existing denominators. To find the new numerator, multiply the existing numerator by the same number multiplied to its denominator to get the LCM.

### Practice Problems

1. What is the common denominator you would use if you wanted to add the fractions $$\dfrac{1}{4}$$ and $$\dfrac{1}{3}$$? (
Video Solution
Solution: 12

Details:

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2. What is the common denominator you would use if you wanted to add the fractions $$\dfrac{1}{6}$$ and $$\dfrac{1}{9}$$? (
Solution
Solution:
18
)
3. What do you get when you add the fractions $$\dfrac{1}{4}$$ and $$\dfrac{1}{3}$$? (
Solution
Solution:
$$\displaystyle \frac{1}{4}+\frac{1}{3}=\frac{3}{12}+\frac{4}{12}=\frac{7}{12}$$
)
4. Add: $$\displaystyle \frac{1}{6}+\frac{1}{9}$$ (
Solution
Solution: $$\displaystyle \frac{1}{6}+\frac{1}{9}=\frac{3}{18}+\frac{2}{18}=\frac{5}{18}$$

Details:
We are adding $$\dfrac{1}{6}$$ to $$\dfrac{1}{9}$$. We can represent that as $$\dfrac{1}{6}$$ of a whole, added to $$\dfrac{1}{9}$$ of a whole: Step 1: Find the least common denominator.
There are two ways that we can find the common denominator: skip counting using multiples or by factoring each denominator. For this example, we will use multiples:

Multiples of $$6$$: $$6, 12, {\color{red}18}, 24, etc.$$

Multiples of $$9$$: $$9, {\color{red}18}, 27, etc.$$

The smallest number that both $$6$$ and $$9$$ divide into evenly is $$\color{red}18$$

Step 2: Write both fractions as an equivalent fraction with a denominator of $$18$$.
Let’s start with $$\dfrac{1}{6}$$.
$$6 \times {\color{red}3} = 18$$

Then multiply the numerator and denominator of $$\dfrac{1}{6}$$ by $$3$$:
$$\displaystyle \dfrac{1\times\color{RED}3}{6\times\color{RED}3}=\dfrac{3}{18}$$

We do the same for $$\dfrac{1}{9}$$.
$$9 \times {\color{red}2} = 18$$

Then multiply the numerator and the denominator by $$2$$:
$$\dfrac{1\times\color{RED}2}{9\times\color{RED}2}=\dfrac{2}{18}$$ Which gives us $$\displaystyle \dfrac{3}{18}+\dfrac{2}{18}$$: Note that both circles now have 18 parts. $$\dfrac{1}{6}$$ is the same amount as $$\dfrac{3}{18}$$ and $$\dfrac{1}{9}$$ is the same amount as $$\dfrac{2}{18}$$.

$$\displaystyle \dfrac{3}{18}+\dfrac{2}{18}=\frac{5}{18}$$ )
5. Subtract: $$\displaystyle \frac{1}{6}-\frac{1}{9}$$ (
Solution
Solution: $$\displaystyle \frac{1}{6}-\frac{1}{9}=\frac{3}{18}-\frac{2}{18}=\frac{1}{18}$$

Details:
We are subtracting $$\dfrac{1}{9}$$ from $$\dfrac{1}{6}$$. We can represent that as $$\dfrac{1}{6}$$ of a whole, minus $$\dfrac{1}{9}$$ of a whole: Step 1: Find the least common denominator.

There are two ways that we can find the common denominator: skip counting using multiples or by factoring each denominator. For this example, we will use multiples.

Multiples of $$6$$: $$6, 12, {\color{red}18}, 24, etc.$$
Multiples of $$9$$: $$9, {\color{red}18}, 27, etc.$$

The smallest number that both $$6$$ and $$9$$ divide into evenly is $$\color{red}18$$

Step 2: Write both fractions as an equivalent fraction with a denominator of $$18$$.

Let’s start with $$\dfrac{1}{6}$$, $$6 \times 3 = 18$$ so we multiply the numerator and denominator of $$\dfrac{1}{6}$$ by $$3$$:
$$\displaystyle \frac{1\times\color{RED}3}{6\times\color{RED}3}=\frac{3}{18}$$

We do the same for $$\dfrac{1}{9}$$, $$9 \times {\color{red} 2} = 18$$, so we will multiply the numerator and the denominator by $$2$$:
$$\displaystyle \frac{1\times\color{RED}2}{9\times\color{RED}2}=\frac{2}{18}$$ Which gives us the following: Note that both circles now have 18 parts. $$\dfrac{1}{6}$$ is the same amount as $$\dfrac{3}{18}$$ and $$\dfrac{1}{9}$$ is the same amount as $$\dfrac{2}{18}$$.

Step 3: Subtract

We are subtracting $$\dfrac{2}{18}$$ from the $$\dfrac{3}{18}$$: Which gives us $$\displaystyle \frac{3}{18}-\frac{2}{18}=\frac{1}{18}$$: )
6. Subtract: $$\displaystyle \frac{1}{4}-\frac{5}{8}$$ (
Video Solution
Solution: $$\displaystyle \frac{1}{4}-\frac{5}{8}=\frac{2}{8}-\frac{5}{8}=-\frac{3}{8}$$

Details:

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