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 $\frac{1}{9}$ and $\frac{3}{5}$.
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 $(\frac{1}{9}$ and $\frac{3}{5})= 45$
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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 $\frac{5}{8}$ and $\frac{11}{12}$.
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 $(\frac{5}{8} ,\; \frac{11}{12}) =$ LCM (8,12) $= 24$
We can make the denominators of $\frac{5}{8}$ and $\frac{11}{12}$ same by finding the LCD. Multiply both numerator and denominator of $\frac{5}{8}$ with 3. Multiply both numerator and denominator of $\frac{11}{12}$ with 2.
$\frac{5}{8} \times \frac{3}{3}= \frac{15}{24}$
$\frac{11}{12}\times \frac{2}{2} = \frac{22}{24}$
Prime Factorization Method
Find the prime factorization of the denominators. Identify the common (matching) factors. Note down the remaining factors. Multiply them together.
Example: $\frac{5}{21},\; \frac{3}{30}$
Prime factorization of $21 = 3\times7$
Prime factorization of $30 = 3\times2\times5$
Common factors $= 3$
Uncommon factors $= 2,\; 7,\; 5$
LCD $= 2 \times7 \times5 \times3 = 210$
NOTE: If the two or more denominators have HCF $= 1$, simply multiply the denominators to find the LCD.
For example, $\frac{1}{9}$ and $\frac{4}{7}$.
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 $9 \times 7 = 63$.
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: $\frac{3}{5},\;\frac{4}{6},\;\frac{9}{20}$
| 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: $\frac{36}{60},\;\frac{40}{60},\;\frac{27}{60}$
Ascending order: $\frac{27}{60}\lt\frac{36}{60}\lt\frac{40}{60} \Rightarrow \frac{9}{20}\lt\frac{3}{5}\lt\frac{4}{6}$
Descending order: $\frac{40}{60}\gt\frac{36}{60}\gt\frac{27}{60} \Rightarrow \frac{4}{6}\gt\frac{3}{5}\gt\frac{9}{20}$
Adding and Subtracting Fractions Using LCD
Using the least common denominator, fractions can be added and subtracted.
Example 1: Find: $\frac{5}{6}\;-\;\frac{9}{20}$.
$6 = 2 \times 3$
$20 = 2 \times 2 \times 5$
LCM (6, 20) $= 2 \times2 \times3 \times5 = 60$
LCD $(\frac{5}{6},\;\frac{9}{20}) = 60$
$\frac{5\times10}{6\times10} = \frac{50}{60}$
$\frac{9\times3}{20\times3} = \frac{27}{60}$
We get
$\frac{5}{6}\;-\;\frac{9}{20} = \frac{50}{60}\;-\;\frac{27}{60} = \frac{13}{60}$
Example 2: Find $\frac{3}{4} + \frac{1}{5}$.
Since GCD$(4,\; 5) = 1$, LCM $(4,\; 5) = 4\times 5 = 20$
LCD$(\frac{3}{4},\;\frac{1}{5}) = 20$
The fractions can be rewritten as $\frac{15}{20}$ and $\frac{4}{20}$.
Sum $= \frac{15}{20} + \frac{4}{20} = \frac{19}{20}$
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 $\frac{2}{5},\;\frac{1}{7}$ and $\frac{4}{9}$.
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) $= 5 \times 7 \times 9 = 315$
LCD$(\frac{2}{5},\;\frac{1}{7},\;\frac{4}{9}) = 315$.
2. Simplify: $\frac{21}{4}\;-\;\frac{7}{3}$
Solution:
We will first find the LCD of the denominators.
LCM (3, 4) $= 12$
LCD $(\frac{21}{4},\;\frac{7}{3}) = 12$
$\frac{21\times3}{4\times3} = \frac{63}{12}$ and $\frac{7\times4}{3\times4} = \frac{28}{12}$
$\frac{21}{4}\;-\;\frac{7}{3} = \frac{63}{12}\;-\;\frac{28}{12} = \frac{35}{12}$
3. Find the LCD of $\frac{7}{8}$ and $\frac{1}{6}$ 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$(\frac{7}{8},\; \frac{1}{6}) = 24$
4. Compare the fractions $\frac{2}{9},\;\frac{3}{4}$.
Solution:
9 and 4 have no common factor other than 1.
Thus, LCM(4, 9) $= 9\times4 = 36$
Thus, LCD$(\frac{7}{8}$ and $\frac{1}{6}) = 36$
Let’s rewrite the fractions using the common denominator.
$\frac{2}{9} = \frac{8}{36}$ and $\frac{3}{4} = \frac{27}{36}$
Here, $\frac{8}{36} \lt \frac{27}{36}$
Thus, $\frac{2}{9} \lt \frac{3}{4}$
Practice Problems on Least Common Denominator
Least Common Denominator
Which of the following holds true?
LCD $=$ LCM $(6,\; 8) = 24$
$\frac{1\times4}{6\times4} = \frac{4}{24}$ and $\frac{5\times3}{8\times3} = \frac{15}{24}$
$\frac{4}{24}\lt\frac{15}{24}$
The Least Common Denominator of fractions is simply the ____ of all denominators.
The LCD of fractions is calculated by finding the LCM of the denominators.
The LCD of $\frac{1}{3}$ and $\frac{1}{4}$ is ____.
3 and 4 are coprimes. So, HCF$(3,\; 4) = 1$
LCM $(3,\; 4) = 12$
Thus, LCD of $\frac{1}{3}$ and $\frac{1}{4}$ is $3\times4 = 12$.
The LCD is the smallest number that is _____ all denominators.
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
What is the difference between LCM and LCD? Are LCM and LCD the same or different?
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.
How is LCD different from the common denominator?
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 $\frac{3}{5}$ and $\frac{2}{7}$, the least common denominator is 35. The common denominator can be 35, 70, 105, etc.
How are the LCD and GCF different?
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.
Can you find the LCD by simply multiplying the denominators?
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.
