Common Denominator: Definition, Method and Examples

What is a common denominator?

A fraction consists of a numerator (the top number) and a denominator (the bottom number). The denominator represents the total number of equal parts that make up a whole.

When two or more fractions share the exact same denominator, they have a common denominator. Having a shared denominator or common bottom number means the fractions represent pieces of the exact same size. Fractions that share the same denominator fall into the category of like and unlike fractions.

Why make denominators the same?

If you want to add, subtract, or evaluate fractions, their parts must be the same size. For instance, evaluating $\dfrac{1}{3}$ and $\dfrac{1}{4}$ together is difficult because a third is a larger slice than a fourth. You cannot combine pieces of different sizes into a single accurate number.

Finding a common denominator splits both fractions into identical, smaller pieces. Once the parts represent the same size, you can count the total number of parts effortlessly. This makes comparing fractions directly or ordering fractions much simpler.

Find a common denominator

To perform calculations with fractions that have different denominators, you must first find a common denominator. There are two primary methods to do this.

Method 1: Product of the denominators

The fastest way to find a valid common denominator is to multiply the two bottom numbers together. For example, to find a common denominator for $\dfrac{1}{4}$ and $\dfrac{2}{5}$, multiply $4 \times 5 = 20$. Both original denominators divide evenly into $20$.

Method 2: Least Common Multiple

To keep the final numbers as small as possible, list the multiples of each denominator and find the smallest number that appears in both lists. This shared value is called the least common multiple. For $\dfrac{1}{6}$ and $\dfrac{3}{8}$, list their multiples:

• Multiples of $6$: $6, 12, 18, 24, 30$
• Multiples of $8$: $8, 16, 24, 32$

The lowest number in both lists is $24$. The common denominator is $24$.

Rewrite equivalent fractions

Once you identify the common denominator, the next step is to rewrite the original fractions. This creates equivalent fractions that retain their original value but are expressed using the identical part size.

To do this, determine what factor was used to multiply the original denominator. Then, multiply the numerator by that exact same factor.

Multiply the numerator and the denominator by the exact same value.

For example, to convert $\dfrac{2}{3}$ to a denominator of $12$, multiply the denominator by $4$. You must also multiply the numerator by $4$, giving $\dfrac{8}{12}$.

Common versus least common denominator

Any shared multiple of the original denominators can serve as a common denominator. However, the lowest possible shared multiple is known as the least common denominator.

While multiplying denominators together is fast, using the least common denominator keeps the calculations simpler and avoids the need to simplify large fractions at the end.

Worked examples

Example 1: Using the product method

Question: Find a common denominator for $\dfrac{2}{5}$ and $\dfrac{3}{8}$, then rewrite the fractions.

Method:

1. Multiply the two denominators together to find a quick common denominator.
2. Multiply $5 \times 8 = 40$.
3. Rewrite $\dfrac{2}{5}$ by multiplying both parts by $8$, resulting in $\dfrac{16}{40}$.
   
4. Rewrite $\dfrac{3}{8}$ by multiplying both parts by $5$, resulting in $\dfrac{15}{40}$.

Answer: The common denominator is $40$, and the equivalent fractions are $\dfrac{16}{40}$ and $\dfrac{15}{40}$.

Example 2: Adding using a least common denominator

Question: Add $\dfrac{5}{6}$ and $\dfrac{3}{4}$.

Method:

1. List the multiples of $6$ ($6, 12, 18$) and $4$ ($4, 8, 12$).
2. Identify the lowest shared multiple, which is $12$.
3. Rewrite $\dfrac{5}{6}$ as $\dfrac{10}{12}$.
   
4. Rewrite $\dfrac{3}{4}$ as $\dfrac{9}{12}$.
5. Add the numerators while keeping the common denominator the same: $10 + 9 = 19$.

Answer: The sum of $\dfrac{5}{6}$ and $\dfrac{3}{4}$ is $\dfrac{19}{12}$.

Check: The product method gives a common denominator of $24$. The equivalent fractions are $\dfrac{20}{24} + \dfrac{18}{24} = \dfrac{38}{24}$. Simplifying $\dfrac{38}{24}$ by dividing by $2$ results in $\dfrac{19}{12}$, confirming the answer.

Example 3: Subtraction with a related denominator

Question: A recipe requires $\dfrac{7}{8}$ of a liter of water. You have already poured in $\dfrac{1}{4}$ of a liter. How much more water is needed?

Method:

1. Identify that the denominator $8$ is a multiple of $4$.
2. Use $8$ as the common denominator.
3. Rewrite $\dfrac{1}{4}$ by multiplying both the numerator and the denominator by $2$, making it $\dfrac{2}{8}$.
4. Subtract the equivalent fractions: $\dfrac{7}{8} - \dfrac{2}{8} = \dfrac{5}{8}$.

Answer: You need $\dfrac{5}{8}$ of a liter more water.

Common mistakes

Adding the denominators

Some learners mistakenly believe they can find a common denominator by adding the bottom numbers together. Denominators represent the fixed size of the parts, so you must find a common multiple. Adding denominators changes the mathematical meaning of the fractions.

Forgetting to update the numerator

If you change the denominator to match another fraction but leave the numerator the same, you have completely changed the value of the fraction. You must always multiply both parts of the fraction by the identical number to keep it equivalent.

Frequently asked questions

Can a common denominator be zero?

No. A denominator shows how many equal parts make up a whole. A denominator of zero means dividing a whole into zero parts, which is mathematically undefined.

Can a common denominator be $1$?

Yes. When working with whole numbers, they can all be written as fractions with a shared denominator of $1$.

What is the common denominator of $3$ and $4$?

Because $3$ and $4$ do not share any common factors other than $1$, their least common denominator is their product, which is $12$.