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When working with fractions, finding the least common denominator (LCD) is a crucial step that allows us to add, subtract, or compare fractions with different denominators. The LCD is the smallest multiple that two or more denominators have in common. By determining the LCD, we can convert fractions with different denominators into equivalent fractions with the same denominator. This simplifies calculations and makes it easier to work with fractions. In this guide, we will explore methods and strategies to effectively find the least common denominator for fractions. Whether you are a student struggling to understand this concept or an individual looking to refresh your math skills, this guide will provide you with valuable insights to confidently find the LCD.
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] X Research 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] X Research 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] X Research 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] X Research 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] X Research 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] X Research 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 an essential skill in mathematics as it allows us to simplify fractions, add or subtract fractions with different denominators, and compare fractions more easily. By utilizing the prime factorization method or the listing method, we can determine the smallest common multiple of two or more numbers and obtain the least common denominator. It is important to remember that the least common denominator should be used for all fractions involved in an equation or problem to ensure accurate calculations and solutions. By practicing these techniques and understanding the concept of the least common denominator, we can become more proficient in working with fractions and enhance our overall mathematical skills.
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