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The least common denominator (LCD) is a crucial concept in mathematics, particularly in fraction operations. When working with fractions, it is often necessary to find a common denominator to simplify calculations and comparisons. The least common denominator refers to the smallest multiple that two or more denominators have in common. By finding the LCD, we can bring different fractions to a common denominator, facilitating addition, subtraction, and comparison. In this guide, we will explore various strategies and techniques to determine the least common denominator efficiently. Understanding how to find the LCD is a fundamental skill that will enhance your ability to handle fractions effectively and streamline mathematical computations.
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 a crucial step in simplifying and adding or subtracting fractions. By breaking down each denominator into its prime factors and identifying the highest power of each prime factor, we can determine the least common denominator. This process ensures that when fractions are added or subtracted, the resulting fraction has the same denominator. Various methods, such as using the prime factorization method or the comparison method, can be employed to find the least common denominator. The key is to identify the common factors and the highest powers, as well as to adjust the numerators accordingly. By finding the least common denominator, we can easily perform operations on fractions, making mathematical calculations simpler and more efficient.
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