Finding the Least Common Multiple (LCM) of fractions is useful when adding or subtracting fractions, comparing fractional values, and solving many math problems. Although fractions look more complicated than whole numbers, finding their LCM becomes simple once you understand the formula.
In this guide, you’ll learn how to find the LCM of fractions step by step, with easy examples and practical tips.
What Is the LCM of Fractions?
The Least Common Multiple (LCM) is the smallest positive number that is a multiple of two or more numbers.
For fractions, the LCM is commonly found using this formula:
LCM of fractions = LCM of numerators ÷ GCD of denominators
Where:
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LCM = Least Common Multiple
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GCD = Greatest Common Divisor (also called HCF)
Formula
For two fractions:
LCM(ab,cd)=LCM(a,c)GCD(b,d)\text{LCM}\left(\frac{a}{b},\frac{c}{d}\right) = \frac{\text{LCM}(a,c)}{\text{GCD}(b,d)}
This formula works for positive fractions in their simplest form.
How to Find the LCM of Fractions
Follow these four simple steps.
Step 1: Write the Fractions
Identify the numerators and denominators of the fractions.
For example:
23,45\frac{2}{3},\quad \frac{4}{5}
The numerators are 2 and 4, while the denominators are 3 and 5.
Step 2: Find the LCM of the Numerators
Find the LCM of the numerators.
For 2 and 4:
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Multiples of 2: 2, 4, 6, 8, ...
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Multiples of 4: 4, 8, 12, ...
The smallest common multiple is 4.
Therefore:
LCM(2,4)=4\text{LCM}(2,4)=4
Step 3: Find the GCD of the Denominators
Now find the GCD of the denominators.
For 3 and 5, the only common factor is 1.
GCD(3,5)=1\text{GCD}(3,5)=1
Step 4: Apply the Formula
Substitute the values into the formula:
LCM(23,45)=LCM(2,4)GCD(3,5)\text{LCM}\left(\frac{2}{3},\frac{4}{5}\right) = \frac{\text{LCM}(2,4)}{\text{GCD}(3,5)} =41=4=\frac{4}{1}=4
Answer:
4\boxed{4}
So, the LCM of 2/3 and 4/5 is 4.
More Examples of LCM of Fractions
Example 1: LCM of 1/4 and 3/8
Step 1: Find the LCM of the numerators.
LCM(1,3)=3\text{LCM}(1,3)=3
Step 2: Find the GCD of the denominators.
GCD(4,8)=4\text{GCD}(4,8)=4
Step 3: Apply the formula.
LCM(14,38)=34\text{LCM}\left(\frac{1}{4},\frac{3}{8}\right) = \frac{3}{4}
Answer:
34\boxed{\frac{3}{4}}
Example 2: LCM of 2/3 and 5/6
Step 1: Find the LCM of the numerators.
LCM(2,5)=10\text{LCM}(2,5)=10
Step 2: Find the GCD of the denominators.
GCD(3,6)=3\text{GCD}(3,6)=3
Step 3: Apply the formula.
LCM(23,56)=103\text{LCM}\left(\frac{2}{3},\frac{5}{6}\right) = \frac{10}{3}
Answer:
103\boxed{\frac{10}{3}}
Example 3: LCM of 3/4 and 5/6
Step 1: Find the LCM of the numerators.
LCM(3,5)=15\text{LCM}(3,5)=15
Step 2: Find the GCD of the denominators.
GCD(4,6)=2\text{GCD}(4,6)=2
Step 3: Apply the formula.
LCM(34,56)=152\text{LCM}\left(\frac{3}{4},\frac{5}{6}\right) = \frac{15}{2}
Answer:
152\boxed{\frac{15}{2}}
How to Find the LCM of Three or More Fractions
The same formula can be extended to three or more fractions:
LCM of fractions=LCM of all numeratorsGCD of all denominators\text{LCM of fractions} = \frac{\text{LCM of all numerators}}{\text{GCD of all denominators}}
Example: LCM of 1/2, 2/3, and 3/4
Step 1: Find the LCM of the numerators.
LCM(1,2,3)=6\text{LCM}(1,2,3)=6
Step 2: Find the GCD of the denominators.
GCD(2,3,4)=1\text{GCD}(2,3,4)=1
Step 3: Apply the formula.
LCM(12,23,34)=61=6\text{LCM}\left(\frac{1}{2},\frac{2}{3},\frac{3}{4}\right) = \frac{6}{1}=6
Answer:
6\boxed{6}
Alternative Method: Convert Fractions to Equivalent Fractions
Another way to understand the LCM of fractions is to look for the smallest positive number that each fraction divides evenly into.
For example, consider:
12,13\frac{1}{2},\quad \frac{1}{3}
A common multiple must be divisible by both fractions:
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1÷12=21 \div \frac{1}{2}=2
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1÷13=31 \div \frac{1}{3}=3
Since 1 is not a multiple of both fractions in the required whole-number sense, try 2:
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2÷12=42 \div \frac{1}{2}=4
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2÷13=62 \div \frac{1}{3}=6
Both results are whole numbers, so:
LCM(12,13)=1\text{LCM}\left(\frac{1}{2},\frac{1}{3}\right)=1
This illustrates an important point: the LCM of fractions is not always the smallest common denominator used when adding fractions.
LCM of Fractions vs. Least Common Denominator
These two concepts are related but different.
| Concept | Meaning |
|---|---|
| LCM of fractions | The smallest positive number that is a multiple of each fraction. |
| Least Common Denominator (LCD) | The smallest positive common multiple of the denominators. |
For example, for:
14,16\frac{1}{4},\quad \frac{1}{6}
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LCM of the fractions = LCM(1,1)GCD(4,6)=12\frac{\text{LCM}(1,1)}{\text{GCD}(4,6)}=\frac{1}{2}
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LCD = LCM(4,6)=12\text{LCM}(4,6)=12
So, LCM of fractions and LCD are not the same thing.
When adding or subtracting fractions, you usually need the LCD, not the LCM of the fractions.
Common Mistakes to Avoid
1. Finding the LCM of the Denominators Only
Finding the LCM of denominators gives the least common denominator, not necessarily the LCM of the fractions.
2. Using the GCD of Numerators
The formula requires the LCM of numerators and the GCD of denominators.
3. Forgetting to Simplify Fractions
Always simplify fractions before applying the formula. For example:
24=12\frac{2}{4}=\frac{1}{2}
Using simplified fractions makes the calculation easier and avoids confusion.
4. Confusing LCM With GCD
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LCM finds the smallest common multiple.
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GCD finds the greatest common factor.
They are different operations, but both are used in the fraction LCM formula.
Use an LCM Calculator
If you want to calculate the LCM of whole numbers quickly, use our LCM Calculator.
It can help you find the Least Common Multiple of two or more numbers without doing the calculation manually.
For fraction problems, remember to distinguish between the LCM of fractions and the least common denominator.
Frequently Asked Questions
What is the formula for the LCM of fractions?
The formula is:
LCM of fractions=LCM of numeratorsGCD of denominators\text{LCM of fractions} = \frac{\text{LCM of numerators}}{\text{GCD of denominators}}
Can I find the LCM of three or more fractions?
Yes. Find the LCM of all numerators and divide it by the GCD of all denominators.
Is the LCM of fractions the same as the LCD?
No. The LCM of fractions and the least common denominator are different concepts.
Do fractions need to be simplified first?
Yes. Simplifying fractions first helps ensure the formula is applied correctly.
Can the LCM of fractions be a fraction?
Yes. For example:
LCM(14,16)=12\text{LCM}\left(\frac{1}{4},\frac{1}{6}\right)=\frac{1}{2}
The result can be a whole number or a fraction.
Final Thoughts
Finding the LCM of fractions becomes easy when you follow the correct formula:
LCM of fractions=LCM of numeratorsGCD of denominators\boxed{\text{LCM of fractions}=\frac{\text{LCM of numerators}}{\text{GCD of denominators}}}
Remember to simplify the fractions, calculate the LCM of the numerators, find the GCD of the denominators, and divide. With a little practice, you can solve fraction LCM problems quickly and accurately.