Least Common Denominator – Definition, Examples, Facts, FAQs

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What Is the Least Common Denominator?

The least common denominator (LCD) is the smallest number divisible by all denominators of the given set of fractions. It is the smallest number among the common multiples of the denominators. 

In simple words, LCD is the LCM of the denominators of the given fractions.

The concept of LCD in math is really useful when it comes to comparing, adding or subtracting unlike fractions.

Example: Add the fractions 19 and 35

To add any two fractions, firstly we check if the denominators are the same.

Here, the denominators are 9 and 5.  

Find the least common denominator.

Multiples of 9=9,18,27,36,45, …

Multiples of 5=5,10,15,20,25,30,35,40,45, …

Common multiples of 9 and 5=45,50,95, …

LCM (9, 5) = LCD (19 and 35)=45

Definition of Least Common Denominator

The least common denominator of a set of fractions is the smallest number of all the common multiples of denominators. It is also known as the Lowest Common Denominator (abbreviated as LCD). 

How to Find the Least Common Denominator

To find the least common denominator, we can use either of the ways as given below: 

Listing Method

One way is to list the multiples of both the denominators. This method is convenient to use when the denominators are small numbers.

Example: Find the least common denominator of 58 and 1112

Multiples of 8=8,16,24,32,40,48,… 

Multiples of 12=12,24,36,48,… 

Common Multiples of 8 and 12=24,48,… 

LCD (58,1112)= LCM (8,12) =24

We can make the denominators of 58 and 1112 same by finding the LCD. Multiply both numerator and denominator of 58 with 3. Multiply both numerator and denominator of 1112 with 2.

58×33=1524

1112×22=2224

Prime Factorization Method

Find the prime factorization of the denominators. Identify the common (matching) factors. Note down the remaining factors. Multiply them together. 

Example:  521,330

Prime factorization of 21=3×7

Prime factorization of 30=3×2×5

Common factors =3

Uncommon factors =2,7,5

LCD =2×7×5×3=210

NOTE: If the two or more denominators have HCF =1, simply multiply the denominators to find the LCD. 

For example, 19 and 47

Since the HCF of 9 and 7 is 1, the Least Common Denominator is the product of two denominators. On multiplying the denominators, we get 9×7=63

Applications of Least Common Denominator

The concept of LCD in math is really helpful when working with fractions. Let’s see how to simplify operations on fractions using the least common denominator.

We will discuss two points.

  • Comparing & ordering fractions using the least common denominator
  • Adding and subtracting fractions using the least common denominator

Comparing and Ordering Fractions Using LCD

We can easily compare and order unlike fractions by finding LCD.

Example: Find the LCD of the fractions: 35,46,920

5 5 10 15 20 25 30 35 40 45 50 55 60
6 6 12 18 24 30 36 42 48 54 60 66 72
20 20 40 60 80 100 120 140 160 180 200 220 240

Using the table of multiples above, we can observe that

LCM of 5, 20 and 6=60.

Thus, LCD of the given fractions is 60 

The fractions can be rewritten as: 3660,4060,2760

Ascending order: 2760<3660<4060920<35<46

Descending order: 4060>3660>276046>35>920

Adding and Subtracting Fractions Using LCD

Using the least common denominator, fractions can be added and subtracted.

Example 1: Find: 56920.

6=2×3

20=2×2×5

LCM (6, 20) =2×2×3×5=60

LCD (56,920)=60

5×106×10=5060

9×320×3=2760

We get

56920=50602760=1360

Example 2: Find 34+15

Since GCD(4,5)=1, LCM (4,5)=4×5=20

LCD(34,15)=20

The fractions can be rewritten as 1520 and 420.

Sum =1520+420=1920

Conclusion

In this article, we learned about Least Common Denominator, its definition, applications along with examples on how to find LCD. Let’s solve a few more examples and practice problems for better understanding.

Solved Examples on Least Common Denominator

1. Find the LCD for 25,17 and 49

Solution:

The denominators 5, 7, and 9 have no common factors other than 1.

HCF (5, 7 and 9) =1

Thus, LCM (5, 7 and 9) =5×7×9=315

LCD(25,17,49)=315.

2. Simplify: 21473

Solution:

We will first find the LCD of the denominators.

LCM (3, 4) =12

LCD (214,73)=12

21×34×3=6312 and 7×43×4=2812

21473=63122812=3512

3. Find the LCD of 78 and 16 by listing multiples.

Solution:

Multiples of 8=8,16,24,32,40,48, …

Multiples of 6=6,12,18,24,30,36, …

LCM(8, 6) =24

Thus, LCD(78,16)=24

4. Compare the fractions 29,34.

Solution:

9 and 4 have no common factor other  than 1.

Thus, LCM(4, 9) =9×4=36

Thus, LCD(78 and 16)=36

Let’s rewrite the fractions using the common denominator.

29=836 and 34=2736

Here, 836<2736

Thus, 29<34

Practice Problems on Least Common Denominator

Least Common Denominator

Attend this Quiz & Test your knowledge.

1

Which of the following holds true?

16<58
27<111
23<813
110>78
CorrectIncorrect
Correct answer is: 16<58
LCD = LCM (6,8)=24
1×46×4=424 and 5×38×3=1524
424<1524
2

The Least Common Denominator of fractions is simply the ____ of all denominators.

GCF
HCF
GCD
LCM
CorrectIncorrect
Correct answer is: LCM
The LCD of fractions is calculated by finding the LCM of the denominators.
3

The LCD of 13 and 14 is ____.

112
712
512
1112
CorrectIncorrect
Correct answer is: 512
3 and 4 are coprimes. So, HCF(3,4)=1
LCM (3,4)=12
Thus, LCD of 13 and 14 is 3×4=12.
4

The LCD is the smallest number that is _____ all denominators.

divisible by
a factor of
a divisor of
None of the above
CorrectIncorrect
Correct answer is: divisible by
Since the LCD is a LCM of denominators. Thus, it is basically a multiple of denominators. Thus, it is divisible by all denominators.

Frequently Asked Questions on Least Common Denominator

LCD of fractions is the LCM of the denominators of the fractions. LCM of two or more numbers is the smallest number of common multiples of given numbers.

Least Common Denominator is the smallest common multiple of the common multiples of the denominators of a set of fractions. On the other hand, the common denominator is the common multiple of the denominators. For example: For the fractions 35 and 27, the least common denominator is 35. The common denominator can be 35, 70, 105, etc.

LCD stands for Least Common Denominator and GCF stands for Greatest Common Factor. They are just about opposites. LCD is the least multiple that is the same for two or more denominators whereas, the GCF of two or more numbers is the greatest factor that these numbers share.

Multiplying all of the denominators results in a common denominator between the fractions, it does not always give you the LCD. If the GCF of denominators is 1, then the LCD of fractions can be calculated by simply multiplying the denominators.