What Is the Least Common Denominator?
The least common denominator, also called the lowest common denominator or LCD, is the smallest denominator that two or more fractions can share.
In order to add or subtract fractions, the denominators must be the same. When the denominators are different, we find the LCD so we can rewrite the fractions with a common denominator.
For example, you can add \large \frac{1}{5}+\frac{2}{5} because the denominators are already the same. Since both fractions are in fifths, you simply add the numerators:
\large \frac{1}{5}+\frac{2}{5}=\frac{3}{5}But in a problem like \large \frac{3}{20}+\frac{1}{24} , the denominators are different, so we first need to find the least common denominator.
Note: If you have mixed numbers, rewrite them as improper fractions before finding the LCD. For example:
\large 3\frac{1}{2}=\frac{7}{2}Why Do You Need the LCD?
You need the LCD when adding or subtracting fractions with different denominators. Using the least common denominator keeps the numbers as small as possible, which makes the problem easier to solve.
For example, if you want to add \large \frac{1}{4}+\frac{1}{6} , you need a common denominator before you can combine the fractions.
The least common denominator of 4 and 6 is 12, so we rewrite each fraction using 12 as the denominator:
\large \frac{1}{4}=\frac{3}{12} \quad \text{and} \quad \frac{1}{6}=\frac{2}{12}Now the fractions can be added:
\large \frac{3}{12}+\frac{2}{12}=\frac{5}{12}Steps to Find the Least Common Denominator
Here is a simple step-by-step method to find the LCD.
-
- Prime factor each denominator.
- Write repeating prime factors as powers, if needed.
- Use each unique prime factor with the highest power that appears.
- Multiply those factors together. That product is the LCD.
Example: How to Find the LCD
Find the LCD for \large \frac{3}{20}+\frac{1}{24}
Step 1: Prime factor each denominator
Prime factors of 20 = 2 × 2 × 5
Prime factors of 24 = 2 × 2 × 2 × 3
Step 2: Write repeating factors as powers
20 = 2^2 \times 5
24 = 2^3 \times 3
The highest power of 2 is 2^3, so that is the power of 2 we use.
Step 3: Multiply the unique prime factors using the highest power of each
LCD = 2^3 \times 3 \times 5
LCD = 8 \times 3 \times 5 = 120
So, the least common denominator is 120.
Rewrite the Fractions Using the LCD
Once you know the LCD, the next step is to rewrite each fraction using that denominator.
For \large \frac{3}{20}+\frac{1}{24} , the LCD is 120.
To change \large \frac{3}{20} into an equivalent fraction with denominator 120, multiply the denominator by 6. So multiply the numerator by 6 as well:
\large \frac{3}{20}=\frac{18}{120}To change \large \frac{1}{24} into an equivalent fraction with denominator 120, multiply the denominator by 5. So multiply the numerator by 5 as well:
\large \frac{1}{24}=\frac{5}{120}Now add the fractions:
\large \frac{18}{120}+\frac{5}{120}=\frac{23}{120}Another Example of Finding the LCD
Find the LCD for \large \frac{5}{12}+\frac{7}{18}
Step 1: Prime factor each denominator
12 = 2 × 2 × 3 = 2^2 \times 3
18 = 2 × 3 × 3 = 2 \times 3^2
Step 2: Use the highest power of each prime factor
Highest power of 2 is 2^2
Highest power of 3 is 3^2
Step 3: Multiply
LCD = 2^2 \times 3^2 = 4 \times 9 = 36
So, the LCD of 12 and 18 is 36.
Least Common Denominator vs. Least Common Multiple
A lot of students ask whether the least common denominator and least common multiple are the same thing. The answer is yes and no.
The LCD is the least common multiple (LCM) of the denominators.
So when you are finding the least common denominator of fractions, you are really finding the least common multiple of the denominator numbers.
For example, if the denominators are 6 and 8, the LCD is 24 because 24 is the least common multiple of 6 and 8.
Common Mistakes When Finding the LCD
- Using a common denominator that is not the least. A common denominator works, but the LCD is usually the best choice because it keeps numbers smaller.
- Forgetting to use the highest power of each prime factor.
- Not rewriting both fractions correctly after finding the LCD.
- Trying to add fractions before the denominators match.
What Is the Fastest Way to Find the LCD?
The fastest and most reliable method is usually prime factorization, especially when the denominators are larger numbers.
For smaller numbers, you can also list multiples until you find the first multiple both denominators share.
For example, to find the LCD of 4 and 6:
Multiples of 4: 4, 8, 12, 16, 20, …
Multiples of 6: 6, 12, 18, 24, …
The first common multiple is 12, so the LCD is 12.
Watch How to Find the LCD
Check out this video where John explains the exact steps on how to find the lowest common denominator or LCD:
Need More Help with Fractions?
If you want more help with fractions, check out John’s full courses. They include clear instruction, practice problems, and step-by-step solution videos.
👉 Explore the Pre-Algebra Course
👉 Explore the Foundations Math Course
Related Fraction Topics
Lowest Common Denominator FAQs
What is the LCD in math?
Do you always need the LCD to add fractions?
You need a common denominator to add or subtract fractions. The LCD is usually the best choice because it keeps the numbers smaller and easier to work with.