Least Common Denominator

What is the least common denominator?

When fractions have different denominators, they represent different-sized pieces of a whole. To combine or compare them accurately, you must express them using pieces of the same size.

This is why finding a common denominator is necessary. While you can multiply the denominators together to find any shared denominator, the least common denominator is the smallest possible number that works.

Working with LCD fractions is often called finding the lowest common denominator, and both terms describe the same mathematical idea.

LCD and least common multiple

The least common denominator is simply the least common multiple of the denominators in the set of fractions.

Because the denominators are the numbers you are trying to match, you apply the exact same methods used for finding a least common multiple. The only difference is that you apply the resulting number exclusively to the bottom of the fractions.

Find the LCD by listing multiples

The most direct way to find least common denominator values is to write down the multiples of each denominator until you find a match.

This method works best when the denominators are relatively small numbers.

1. Write down the first few multiples of the first denominator.
2. Write down the first few multiples of the second denominator.
3. Identify the smallest number that appears in both lists.
   

The smallest shared multiple is the least common denominator.

For example, to find the LCD for $\dfrac{1}{6}$ and $\dfrac{7}{15}$, list their multiples. The multiples of $6$ are $6, 12, 18, 24, 30$, and so on. The multiples of $15$ are $15, 30, 45$. Because $30$ is the smallest number that appears in both lists, the LCD is $30$.

Find the LCD using prime factors

When the denominators are large, listing multiples can take too long. Instead, you can find the least common denominator using prime factorization.

1. Find the prime factors of each denominator.
2. Write out the factors, aligning matching prime numbers where possible.
3. Multiply each prime factor the greatest number of times it appears in any single factorization.
   

For example, to find the LCD for fractions with denominators of $8$ and $12$, break each number down into prime factors. The prime factorization of $8$ is $2 \times 2 \times 2$. The prime factorization of $12$ is $2 \times 2 \times 3$.

The prime factor $2$ appears a maximum of three times in the first factorization. The prime factor $3$ appears a maximum of one time in the second factorization. Multiply these together to find the LCD of $24$.

Rewrite fractions using the LCD

Once you find the least common denominator, you must use it to create equivalent fractions. This ensures the overall value of the fraction stays the same even though the denominator changes.

Divide the new least common denominator by the original denominator to find the multiplier. Then, multiply both the numerator and the denominator by that number.

Repeating this step for every fraction allows you to combine or compare them accurately.

Worked examples

Review these least common denominator examples to see how the method is used in practice.

Example 1: Comparing fractions using the LCD

Question: Which fraction is larger: $\dfrac{5}{6}$ or $\dfrac{7}{9}$?

Method:

1. Find the LCD for $6$ and $9$. Multiples of $9$ are $9, 18, 27$. Multiples of $6$ include $18$. The LCD is $18$.
2. Rewrite each fraction with a denominator of $18$.
3. Multiply the numerator and denominator of $\dfrac{5}{6}$ by $3$ to get $\dfrac{15}{18}$.
   
4. Multiply the numerator and denominator of $\dfrac{7}{9}$ by $2$ to get $\dfrac{14}{18}$.
5. When comparing fractions with the same denominator, you only need to compare the numerators.

Answer: Because $15$ is larger than $14$, $\dfrac{15}{18}$ is larger. This means $\dfrac{5}{6}$ is the larger fraction.

Check: Cross-multiply: $5 \times 9 = 45$ and $7 \times 6 = 42$. Since $45 > 42$, the fraction $\dfrac{5}{6}$ is indeed larger.

Example 2: Adding fractions with different denominators

Question: Calculate $\dfrac{1}{6} + \dfrac{7}{15}$.

Method:

1. Find the LCD of $6$ and $15$. By listing multiples, the least common multiple is $30$.
2. Rewrite $\dfrac{1}{6}$ with a denominator of $30$. Multiply the numerator and denominator by $5$ to get $\dfrac{5}{30}$.
3. Rewrite $\dfrac{7}{15}$ with a denominator of $30$. Multiply the numerator and denominator by $2$ to get $\dfrac{14}{30}$.
4. Add the numerators together and keep the denominator exactly the same.

Answer: $\dfrac{5}{30} + \dfrac{14}{30} = \dfrac{19}{30}$.

Check: Ensure the fraction is in its simplest form. The number $19$ is a prime number and does not divide evenly into $30$, so the answer is fully simplified.

Example 3: Ordering three fractions

Question: Arrange $\dfrac{2}{3}$, $\dfrac{5}{8}$, and $\dfrac{7}{12}$ from least to greatest.

Method:

1. Find the LCD of $3$, $8$, and $12$. Using prime factors, $3$ is prime, $8 = 2 \times 2 \times 2$, and $12 = 2 \times 2 \times 3$. The LCD is $2 \times 2 \times 2 \times 3 = 24$.
2. Convert each fraction to an equivalent fraction with a denominator of $24$.
3. For $\dfrac{2}{3}$, multiply the numerator and denominator by $8$ to get $\dfrac{16}{24}$.
   
4. For $\dfrac{5}{8}$, multiply the numerator and denominator by $3$ to get $\dfrac{15}{24}$.
   
5. For $\dfrac{7}{12}$, multiply the numerator and denominator by $2$ to get $\dfrac{14}{24}$.
6. Compare the numerators: $14 < 15 < 16$. This technique works perfectly for ordering fractions.

Answer: The correct order is $\dfrac{7}{12}$, $\dfrac{5}{8}$, $\dfrac{2}{3}$.

Check: Convert each original fraction to a decimal. $\dfrac{7}{12} \approx 0.583$, $\dfrac{5}{8} = 0.625$, and $\dfrac{2}{3} \approx 0.667$. The decimal values confirm the correct order.

Common mistakes

It is easy to make calculation errors when working with denominators. Watch out for these common traps.

Multiplying the denominators

A frequent error is always multiplying the denominators together to find the LCD. While multiplying $4 \times 6 = 24$ provides a valid common denominator, the least common denominator of $4$ and $6$ is actually $12$. Always check for smaller shared multiples first.

Forgetting the numerator

Students sometimes change the denominator to the LCD but forget to multiply the numerator by the same amount. The entire fraction must be multiplied by the same value to remain equivalent.

Adding the denominators

When adding fractions, never add the denominators together. Once the denominators are exactly the same, only the numerators are added.

Frequently asked questions

Here are short answers to common questions about finding and using the least common denominator.

Can the LCD be one of the original denominators?

Yes. If one denominator is a multiple of the other, the larger denominator is the least common denominator. For example, the LCD of $\dfrac{1}{4}$ and $\dfrac{3}{8}$ is $8$.

Do I need the LCD to multiply fractions?

No. Common denominators are only required for addition, subtraction, and comparison. To multiply fractions, you simply multiply straight across.

What if the fractions have variables in the denominator?

The same rules apply. The least common denominator will be the simplest algebraic expression that includes all factors from each individual denominator.