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When working with fractions, it is often necessary to find a common denominator in order to perform various operations such as addition, subtraction, multiplication, and division. The least common denominator (LCD) plays a crucial role as it represents the smallest common multiple of the denominators involved. By finding the LCD, we can simplify fractions, compare them, and achieve a common base for mathematical operations. In this guide, we will explore various methods and strategies to easily identify the least common denominator, enabling us to work with fractions more efficiently and accurately. Whether you are a student struggling with homework or an adult dealing with real-life situations involving fractions, this topic will provide you with the tools to find the least common denominator with ease. Buckle up and get ready to master this fundamental skill in dealing with fractions!
This article was co-written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical modeling for genome evolution, and data science. Mario holds a bachelor’s degree in mathematics from California State University, Fresno, and a doctorate in applied mathematics from the University of California, Merced. Mario teaches at both the high school and college levels.
This article has been viewed 193,308 times.
To add or subtract fractions with different denominators, you must first find the least common denominator between them. It is the least common multiple of each of the initial denominators in the equation, or the smallest integer that can be divided by each denominator. [1] XResearch Source Determining the least common denominator allows you to convert the denominators to the same number so you can add and subtract them.
Steps
List Multiples [2] XResearch Sources
- Example: 1/2 + 1/3 + 1/5
- Multiples of 2: 2 * 1 = 2; 2 * 2 = 4; 2 * 3 = 6; 2 * 4 = 8; 2 * 5 = 10; 2 * 6 = 12; 2 * 7 = 14; etc
- Multiples of 3: 3 * 1 = 3; 3 * 2 = 6; 3 *3 = 9; 3 * 4 = 12; 3 * 5 = 15; 3 * 6 = 18; 3 * 7 = 21; etc
- Multiples of 5: 5 * 1 = 5; 5 * 2 = 10; 5 * 3 = 15; 5 * 4 = 20; 5 * 5 = 25; 5 * 6 = 30; 5 * 7 = 35; etc
- Note that if you still haven’t found a common denominator, you may have to keep writing multiples until you find a common multiple.
- This method is easier to use when the denominators are small numbers.
- In this example, the denominators that have only one in common are 30: 2 * 15 = 30 ; 3 * 10 = 30 ; 5 * 6 = 30
- So least common denominator = 30
- For example: (15/15) * (1/2); (10/10) * (1/3); (6/6) * (1/5)
- New equation: 15/30 + 10/30 + 6/30
- Example: 15/30 + 10/30 + 6/30 = 31/30 = 1 1/30
Using Greatest Common Factor [3] XResearch Source
- For example: 3/8 + 5/12.
- The factors of 8: 1, 2, 4, and 8
- The factors of 12: 1, 2, 3, 4, 6, 12
- In this example, 8 and 12 have the common factors of 1, 2, and 4.
- The greatest common factor is 4.
- In this example: 8 * 12 = 96
- Example: 96 / 4 = 24
- For example: 24 / 8 = 3; 24/12 = 2
- (3/3) * (3/8) = 9/24; (2/2) * (5/12) = 10/24
- 9/24 + 10/24
- Example: 9/24 + 10/24 = 19/24
Analyzing Each Denominator Product of Prime Factors [5] XResearch Source
- Example: 1/4 + 1/5 + 1/12
- Parsing 4 into primes: 2 * 2
- Parsing 5 into primes: 5
- Parse 12 into primes: 2 * 2 * 3
- Example: There are 2 numbers 2 out of 4; no 2 out of 5; 2 numbers 2 in 12
- There is no 3 in 4 and 5; a number 3 in 12
- There is no 5 in 4 and 12; a number 5 in 5
- Example: The maximum number of occurrences of 2 is two; of 3 is one; of 5 is one
- Example: 2, 2, 3, 5
- Example: 2 * 2 * 3 * 5 = 60
- Least common denominator = 60
- For example: 60/4 = 15; 60/5 = 12; 60/12 = 5
- 15 * (1/4) = 15/60; 12 * (1/5) = 12/60; 5 * (1/12) = 5/60
- 15/60 + 12/60 + 5/60
- Example: 15/60 + 12/60 + 5/60 = 32/60 = 8/15
Working with Integers and Mixed Numbers [7] XResearch Sources
- Example: 8 + 2 1/4 + 2/3
- 8 = 8/1
- 2 1/4; 2 * 4 + 1 = 8 + 1 = 9; 9/4
- Rewrite equation: 8/1 + 9/4 + 2/3
- Note that you don’t need to list multiples for 1 because any number multiplied by 1 equals itself; in other words, every number is a multiple of 1 .
- For example: 4 * 1 = 4; 4 * 2 = 8; 4 * 3 = 12 ; 4 * 4 = 16; etc
- 3 * 1 = 3; 3 * 2 = 6; 3 * 3 = 9; 3 * 4 = 12 ; etc
- Least common denominator = 12
- For example: (12/12) * (8/1) = 96/12; (3/3) * (9/4) = 12/27; (4/4) * (2/3) = 8/12
- 96/12 + 27/12 + 8/12
- Example: 96/12 + 27/12 + 8/12 = 131/12 = 10 11/12
Things you need
- Pencil
- Paper
- Computer (optional)
- Ruler
This article was co-written by Mario Banuelos, PhD. Mario Banuelos is an assistant professor of mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical modeling for genome evolution, and data science. Mario holds a bachelor’s degree in mathematics from California State University, Fresno, and a doctorate in applied mathematics from the University of California, Merced. Mario teaches at both the high school and college levels.
This article has been viewed 193,308 times.
To add or subtract fractions with different denominators, you must first find the least common denominator between them. It is the least common multiple of each of the initial denominators in the equation, or the smallest integer that can be divided by each denominator. [1] XResearch Source Determining the least common denominator allows you to convert the denominators to the same number so you can add and subtract them.
In conclusion, finding the least common denominator is a crucial skill in various mathematical operations, such as adding, subtracting, and comparing fractions. By following a systematic approach, including identifying the denominators of all given fractions, finding their common multiples, and selecting the smallest multiple, we can determine the least common denominator. This process helps ensure that fractions are compatible for mathematical operations, simplifies calculations, and allows for accurate comparisons between fractions. By understanding and applying the concept of the least common denominator, individuals can confidently navigate through fractions and excel in various mathematical endeavors.
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