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Least Common Denominator - LCD
When two or more fractions have the same denominators, they are termed as the common denominators. The least common denominator (LCD) refers to the smallest number that is a common denominator for a given set of fractions. For addition and subtraction of fractions and for comparing two or more fractions, the given fractions need to have common denominators. In this lesson, we will learn how to find the least common denominator in detail.
|1.||What is Least Common Denominator?|
|2.||How to Find the Least Common Denominator?|
|3.||Solved Examples on Least Common Denominator|
|4.||Practice Questions on Least Common Denominator|
|5.||FAQs on Least Common Denominator|
What is Least Common Denominator?
The least common denominator is defined as the smallest common multiple of all the common multiples of the denominators when 2 or more fractions are given.
Let’s add the fractions: (2/9)+(3/4)
For adding any two fractions, we first check if the denominators are the same or not as we can add or subtract only like fractions. Since the denominators are 9 and 4, we need to find a common number that is a multiple of both. This common multiple will help us simplify the problem. Thus, the least common multiple obtained for 9 and 4 is 36. Therefore, the expression can be written as:
(2/9)+(3/4) = (2/9 × 4/4) + (3/4 × 9/9) = (8/36) + (27/36) = 35/36
How to Find the Least Common Denominator?
In order to find the least common denominator, we can opt for either of the ways as given below:
- List the multiples of both denominators. For example, 2/15 and 1/25. The multiples of 15 are 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, ... and the multiples of 25 are 25, 50, 75, 100, 125, 150, 175, 200, 225, 250. Thus, the least common denominator will be 150 and the fractions will be 20/150 and 6/150 (by taking LCM)
- Multiply both the denominators. For example, 3/4 and 2/7. Here, the two denominators, 4 and 7 don't have any common multiple as such. Thus, we will multiply both the denominators. Thus, the least common denominator will be 28 and the fractions will be 21/28 and 8/28.
Apart from simplifying fractions, the least common denominator can be used to arrange fractions in ascending or descending order. For example, we can arrange the following fractions in ascending order by finding their LCD: (3/5, 9/20, 4/6). Thus, the least common multiple of the denominators 5, 20, and 6 is 60. Thus, the given fractions can be written as 36/60, 27/60, 40/60. Therefore, we can conclude that 27/60 < 36/60 < 40/60.
- A denominator can never be zero.
- The concept of the least common denominator for fractions is used to evaluate the result as a part of the whole.
Topics Related to Least Common Denominator
Solved Examples on Least Common Denominator
Example 1: Determine the least common denominator for the following fractions: 4/6, 8/9, and 3/12.
Solution: The denominators of the given fractions are 6, 9, and 12. Let's find out their LCD by listing multiples method:
- Multiples of 6 can be listed as 6, 12, 18, 24, 30, 36, ...
- Multiples of 9 can be listed as 9, 18, 24, 36, 45, 54, ...
- Multiples of 12 can be listed as 12, 24, 36, 48, 60, 72, ...
Thus, we can conclude that the least common multiple of 6, 9, and 12 is 36. Therefore, the least common denominator for the given fractions, 4/6, 8/9, and 3/12 is 36.
Example 2: Tom ate 2/3 of a cake and Ron ate 2/6 of that cake. Who ate more amount of cake?
Solution: The two fractions are 2/3 and 2/6. In order to find out who ate more amount of the cake, we need to find the LCD for the two given fractions. The LCD of 3 and 6 is 6. Thus, the fractions will be 4/6 and 2/6, and 4/6>2/6. Therefore, we can say that Tom ate more amount of cake.
Practice Questions on Least Common Denominator
FAQs on Least Common Denominator
What does Least Common Denominator Mean?
The least common denominator of the given non-zero denominators is the smallest whole number that is divisible by each of the denominators. It can also be used to add fractions, subtract fractions or compare them.
How Do You Find the Least Common Denominator?
To find the least common denominator, one can either list the multiples of each denominator and then look for the smallest number that appears in each list or multiply both the denominators, in case the denominators have no common multiple.
How to Find Least Common Denominator of Fractions?
In order to find the least common denominator for a given set of fractions, simply list the multiples of each denominator then look for the smallest multiple that is common in both the lists. For example, the LCD for the two fractions, 6/7 and 2/3 will be 21 as the only least common multiple to 7, and 3 (denominators of fractions) is 21.
What is the Least Common Denominator of the Exponents?
The least common denominator of the exponents is the lowest common denominator that divides the denominator of the given exponent terms. Let's consider the two denominators, 3x3y2z4 and 4xy5z2
- Step 1: Find the LCD of the coefficients. The LCD of 3 and 4 is 12.
- Step 2: Use all variables with the highest exponents on each variable, that is x3y5z4.
- Step 3: Write the result from step 1 and step 2 together, that is 12x3y5z4.
What is the Least Common Denominator of 2/3 and 5/8?
In the given fractions, 2/3 and 5/8, 3 and 8 are the denominators. The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, so on and the multiples of 8 are 8, 16, 24, 32, 40, so on. As we can see from the list of multiples, the smallest common multiple of 3 and 8 is 24, thus, the least common denominator of 2/3 and 5/8 is 24.
Can the Least Common Denominator be Negative?
No, the least common denominator cannot be negative as it represents the common multiples of the denominator. The least value of LCD can be 1 and not lesser than it which proves the point of LCD not being able to hold a negative value.
Are LCD and LCM the Same?
Although LCD and LCM require the same math processes, that is to find a common multiple of two or more given numbers. The key difference is that the LCD is the LCM in the denominators of the given fractions. In a way, the LCD or the least common denominators can be referred to as a special case of least common multiples.