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.
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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 $frac{3}{5})= 45$
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 $21 = 3times7$
Prime factorization of $30 = 3times2times5$
Common factors $= 3$
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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: $frac{3}{5},;frac{4}{6},;frac{9}{20}$
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: $frac{27}{60}ltfrac{36}{60}ltfrac{40}{60} Rightarrow frac{9}{20}ltfrac{3}{5}ltfrac{4}{6}$
Descending order: $frac{40}{60}gtfrac{36}{60}gtfrac{27}{60} Rightarrow frac{4}{6}gtfrac{3}{5}gtfrac{9}{20}$
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 $(frac{5}{6},;frac{9}{20}) = 60$
$frac{5times10}{6times10} = frac{50}{60}$
$frac{9times3}{20times3} = frac{27}{60}$
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 $(4,; 5) = 4times 5 = 20$
LCD$(frac{3}{4},;frac{1}{5}) = 20$
The fractions can be rewritten as $frac{15}{20}$ and $frac{4}{20}$.
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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 $(frac{21}{4},;frac{7}{3}) = 12$
$frac{21times3}{4times3} = frac{63}{12}$ and $frac{7times4}{3times4} = frac{28}{12}$
$frac{21}{4};-;frac{7}{3} = frac{63}{12};-;frac{28}{12} = frac{35}{12}$
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) $= 9times4 = 36$
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
Frequently Asked Questions on Least Common Denominator
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